Dynamical system

{{Short description|Mathematical model of the time dependence of a point in space}} {{about||the mechanical concept|Classical dynamics|the system theoretic perspective|Dynamical systems theory}} {{Redirect|Dynamical}}

[[File:Dynsys.png|thumb|right| A set of dynamical systems. Top left: a cellular automata. Top center: Exterior billiards. Top right a constrained 3-body problem. Bottom left a Poincare section of a Standard map (chaos arise in the dotted regions). Middle bottom: a chaotic Dynamical billiards (a symptom of chaos here are the trajectories filling the configuration space). Bottom right: A geodesic flow such as light on a surface, trajectories are geodesics i.e. minimum paths, in this case the phase space is a torus (stable orbits arise when the periods are rational, if irrational that is a path to chaos).]]

In mathematics, physics, engineering and systems theory, a '''dynamical system''' is the description of how a system evolves in time.

For example, an astronomer can experimentally record the positions of how the planets move in the sky, and this can be considered a complete enough description of a dynamical system. In the case of planets there is also enough knowledge to codify this information as a set of differential equations with initial conditions, or as a map from the present state to a future state in a predefined state space with a time parameter t, or as an orbit in phase space.<ref name="ChaosBook">{{cite book|title=Chaos: Classical and Quantum |url=https://chaosbook.org/version17/chapters/ChaosBook.pdf |last1=Cvitanović |first1=Predrag |last2=Artuso |first2=Roberto |last3=Mainieri |first3=Ronnie |last4=Tanner |first4=Gregor |last5=Vattay |first5=Gábor |edition=17 |access-date=7 April 2026 |url-status=live |archive-url=https://web.archive.org/web/20260407114501/https://chaosbook.org/version17/chapters/ChaosBook.pdf |archive-date=7 April 2026}}</ref>{{rp|at=remark ch. 2.1}}

The study of dynamical systems is the focus of ''dynamical systems theory'', which has applications to a wide variety of fields such as mathematics, physics,<ref>{{cite journal |last1=Melby |first1=Paul |last2=Weber |first2=Nicholas |last3=Hübler |first3=Alfred |title=Dynamics of self-adjusting systems with noise |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |date=September 2005 |volume=15 |issue=3 |article-number=033902 |doi=10.1063/1.1953147 |pmid=16252993 |bibcode=2005Chaos..15c3902M }}</ref><ref>{{cite journal |last1=Gintautas |first1=Vadas |last2=Foster |first2=Glenn |last3=Hübler |first3=Alfred W. |title=Resonant Forcing of Chaotic Dynamics |journal=Journal of Statistical Physics |date=February 2008 |volume=130 |issue=3 |pages=617–629 |doi=10.1007/s10955-007-9444-4 |arxiv=0705.0311 |bibcode=2008JSP...130..617G }}</ref> biology,<ref>{{cite book |title=Applications of Dynamical Systems in Biology and Medicine |series=The IMA Volumes in Mathematics and its Applications |date=2015 |volume=158 |doi=10.1007/978-1-4939-2782-1 |isbn=978-1-4939-2781-4 }}{{page needed|date=February 2026}}</ref> chemistry, engineering,<ref>{{cite book |first=Erwin |last=Kreyszig |title=Advanced Engineering Mathematics |location=Hoboken |publisher=Wiley |year=2011 |isbn=978-0-470-64613-7 }}</ref> economics,<ref>{{cite book |last=Gandolfo |first=Giancarlo |author-link=Giancarlo Gandolfo |title=Economic Dynamics: Methods and Models |location=Berlin |publisher=Springer |edition=4th |year=2009 |orig-year=1971 |isbn=978-3-642-13503-3 }}{{page needed|date=February 2026}}</ref> history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.

== Overview == [[File:Rudolphine_tables.jpg|thumb|Two pages of the Rudolphine tables showing eclipses of the Sun and Moon, data was collected from Tycho brahe and published by Kepler<ref name="ChaosBook"/>{{rp|at=Appendix 1.1}}]] The concept of a dynamical system has its origins in Newtonian mechanics and more precisely in celestial mechanics. There, as in other natural sciences and engineering disciplines, there is some need to predict the evolution of the system, but maybe also pose other questions such as stability, qualitative or long term behaviour, dependence on parameters, existence of periodic, stochastic or chaotic behaviour.<ref name="ChaosBook"/>{{rp|at=Part 1}} The relation from one state and another is either explicit such as a function in the parameter t predicting position and velocity of a particle or implicit such as a differential equation, difference equation or other time scale. Some times it may not be possible to define such a description, there may not even be a differential equation predicting stock price, or it maybe impossible to build one but still talk stock prices can be considered a dynamical system based on experimental data changing over time.<ref>Here is an example of a research group doing dynamical systems and their typical research areas such as networks and emergence https://vu.nl/en/about-vu/more-about/dynamical-systems</ref>

Important properties are existence and uniqueness of solutions, integrability (i.e. the existence of conserved quantities), the possibility to solve the system and be able to compute the state at any point in time. Other properties are whether the system is discrete, continuous, differentiable, smooth, deterministic, ergodic, stochastic or chaotic.<ref>Here is an example of Syllabus of an introductory course of dynamical systems https://pub.math.leidenuniv.nl/~rottschaeferv/IDS2013/IntroductionDynamicalSystems.html#:~:text=There%20are%20various%20kinds%20of,astronomy%20to%20ecology%20and%20neurophysiology.</ref><ref>Yet another syllabus https://chaosbook.org/course1/Syllabus.html</ref>

If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a ''trajectory'' or ''orbit''.

Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems.<ref>One of the first to get the intuition of numerical computations for weather forecasting is Richardson, he imagined a set of human people doing computations{{xref|(see also {{slink|Lewis Fry Richardson#Weather forecasting}})}}</ref> Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.<ref>{{cite journal |last1=Schultz |first1=David M. |last2=Lynch |first2=Peter |title=100 Years of L. F. Richardson's Weather Prediction by Numerical Process |journal=Monthly Weather Review |date=April 2022 |volume=150 |issue=4 |pages=693–695 |doi=10.1175/MWR-D-22-0068.1 }}</ref>

[[File:Stability_Diagram.png|thumb|500px|'''Stability diagram''' classifying Poincaré maps of linear autonomous system ]] For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: * The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.<ref name="ChaosBook"/>{{rp|at=Appendix 1.1}} * The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.<ref>{{cite book |title=Qualitative Theory of Dynamical Systems, Tools and Applications for Economic Modelling |series=Springer Proceedings in Complexity |date=2016 |doi=10.1007/978-3-319-33276-5 |isbn=978-3-319-33274-1 }}{{page needed|date=February 2026}}</ref> * The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.<ref>{{Cite web |title=Control Theory and Dynamical Systems |url=https://www.imperial.ac.uk/media/imperial-college/faculty-of-natural-sciences/department-of-mathematics/public/events/fritz/1_Introduction_2018_06_04.pdf}}</ref> * The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.<ref>https://researchoutreach.org/articles/exploring-transitions-chaos-complex-systems/#:~:text=More%20specifically%2C%20he%20explores%20the,and%20'quasi%2Dperiodicity'.</ref>

=== Examples === Simple examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake.

<gallery mode="packed" heights="140"> File:Three-body Problem Animation with COM.gif|Three body problem: Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. File:Arnoldcatmap.svg|Arnold cat map: picture showing how the linear map stretches the unit square and how its pieces are rearranged when the modulo operation is performed. The lines with the arrows show the direction of the contracting and expanding eigenspaces File:Ising-tartan.png|Baker's map: Example of a measure that is invariant under the action of the (unrotated) baker's map: an invariant measure. Applying the baker's map to this image always results in exactly the same image. File:Stadium billiard.gif|alt=|Billiards: A particle moving inside the Bunimovich stadium, a well-known chaotic billiard. File:OuterBilliardsDefinition.png|Outer billiards: defined relative to a pentagon File:Bouncing_ball_strobe_edit.jpg|Bouncing ball dynamics: The motion is not quite parabolic due to air resistance. File:Circle map bifurcation.jpeg|Bifurcation diagram for a Circle map. Black regions correspond to Arnold tongues. File:Miimcr.png|The recursive application of a Complex quadratic polynomial as a complex plane map gives a Dynamical system. Here there is a Dynamical plane with a Julia set and critical orbit. File:Double pendulum simulation.gif|right|Motion of the double compound pendulum (from numerical integration of the equations of motion) File:Dyadic trans.gif|right|Dyadic transformation ''xy'' plot where ''x'' = ''x''0 ∈ [0, 1] is rational and ''y'' = ''x''''n'' for all ''n'' File:Lorenz attractor yb.svg|The Lorenz attractor arises in the study of the Lorenz oscillator, a dynamical system. File:Heard Island Karman vortex street.jpg|An example of a Kármán vortex street, an emergent phenomenon from Fluid dynamics. File:Kicked Rotor Phase Portrait.png| The Kicked Rotor, a famous chaotic system File:PIA17173 Titan resonances in Saturn's C ring.jpg|Orbital resonance in Saturn's rings. File:Hyperion true.jpg|The chaotic rotation of Hyperion. The Solar System as a whole is full of examples of dynamical systems from Celestial mechanics</gallery>

Other classical examples include: {{Div col|colwidth=25em}} * Hénon map * Irrational rotation * Kaplan–Yorke map * Lorenz system * Quadratic map simulation system * Rössler map * Swinging Atwood's machine * Tent map {{Div col end}}

Any mathematical map can be treated as the definition of a dynamical system for example: {{Div col|colwidth=25em}} * List of chaotic maps {{Div col end}}

=== History === [[File:Forced Duffing equation Poincaré section.png|300px|thumb|A two-dimensional Poincaré section of the forced Duffing equation]] Many people regard French mathematician Henri Poincaré as the founder of dynamical systems.<ref>{{cite journal |last1=Holmes |first1=Philip |title=Poincaré, celestial mechanics, dynamical-systems theory and 'chaos' |journal=Physics Reports |date=September 1990 |volume=193 |issue=3 |pages=137–163 |doi=10.1016/0370-1573(90)90012-Q }}</ref> Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.<ref name="ChaosBook"/>{{rp|at=Appendix 1.1.1}}

Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.<ref name="ChaosBook"/>{{rp|at=Appendix 1.1}} In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his ''Dynamical Systems''.<ref>{{cite web |url=https://archive.org/details/dynamicalsystems00birk/ |title=Dynamical Systems |last=Birkhoff |first=G.D. |date=1927 |location=New York |publisher=American Mathematical Society Collection |language=en}}</ref>

Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.<ref name="ChaosBook"/>{{rp|at=Appendix 1.2}}

[[File:Smale Horseshoe Map.svg|left|thumb|The Smale horseshoe map {{math| ''f'' }} is the composition of three geometrical transformations.]] Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.

Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.<ref name="ChaosBook"/>{{rp|at=Appendix 1.4}} [[File:RS-68 rocket engine test.jpg|thumb|right|RS-68 being tested at NASA's Stennis Space Center]] In the late 20th century, the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems.<ref name="Rega">{{cite book |last1=Rega |first1=Giuseppe |title=IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems |chapter=Tribute to Ali H. Nayfeh (1933–2017) |series=IUTAM Bookseries |date=2020 |volume=37 |pages=1–13 |doi=10.1007/978-3-030-23692-2_1 |isbn=978-3-030-23691-5 }}</ref> His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft.<ref name="fi">{{cite web |title=Ali Hasan Nayfeh |url=https://www.fi.edu/laureates/ali-hasan-nayfeh |website=Franklin Institute Awards |publisher=The Franklin Institute |access-date=25 August 2019 |date=4 February 2014}}</ref>

=== Generalizations === The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory, physics such as Phase space, quantum state and thermodynamic state, engineering such as system theory, control theory and even information theory.

==== Mathematical intuition ==== From a mathematics perspective in the most general case the state space X is treated as a generic set of abstract algebra. This space X has a semi-group structure on it (i.e. where only associativity is required) and there is most often a natural choice for an Identity element, which is typically attached to the origin of the chosen reference frame. This semi-group can be intuitively interpreted as the time coordinate t.<ref>Dynamical systems, Oliver Knill, Harvard University, Spring semester, 2005 https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf Classes of Dynamical systems pp 6</ref> Time in fact has an addition operation and an origin, the identity, like a group. The action of the semi-group on X is a set of maps from X to itself parametric in the time t, and this is intuitively the time evolution.<ref>Dynamical systems, Oliver Knill, Harvard University, Spring semester, 2005 https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf A FANCY DEFINITION pp 6</ref>

==== Generalizing the state space ==== It is possible to allow different choices of the state space such as a function space (e.g. the pressure, temperature and velocity of a gas in a rocket are a function in the space of solutions of some fluid dynamics PDEs and they may vary over time),<ref>{{xref|(see also Koopman–von Neumann classical mechanics)}}</ref><ref name="Quantum Dynamical Systems">{{cite book |editor1-first=Stéphane |editor1-last=Attal |editor2-first=Alain |editor2-last=Joye |editor3-first=Claude-Alain |editor3-last=Pillet |title=Open Quantum Systems I |series=Lecture Notes in Mathematics |date=2006 |volume=1880 |doi=10.1007/b128449 |isbn=978-3-540-30991-8 |url=https://hal.science/hal-00128867v1 }}{{page needed|date=February 2026}}</ref> a Quantum state space (e.g. the state of an atom can be described by a set of functions in an hilbert space and a set of probabilities for these),<ref name="Quantum Dynamical Systems"/><ref>{{cite book |last1=Alicki |first1=Robert |last2=Fannes |first2=Mark |title=Quantum Dynamical Systems |date=2001 |doi=10.1093/acprof:oso/9780198504009.001.0001 |isbn=978-0-19-850400-9 }}{{page needed|date=February 2026}}</ref> or a manifold (e.g. the state of a black hole can be described by a metric tensor on a Riemann manifold and its position will be a vector in the same manifold).<ref>{{cite book |last1=Giulini |first1=Domenico |title=Springer Handbook of Spacetime |chapter=Dynamical and Hamiltonian Formulation of General Relativity |series=Springer Handbooks |date=2014 |pages=323–362 |doi=10.1007/978-3-642-41992-8_17 |isbn=978-3-642-41991-1 }}</ref> Other choices can be a Phase space, a Configuration space or even a discrete space (e.g. the set of prime numbers or a finite field).<ref>Dynamical Systems in Number Theory https://www.math.uzh.ch/gorodnik/dyn_num/index.html</ref>

==== Time as a multidimensional manifold ==== [[File:STS-3 Canadarm captures PDP.jpg|left|thumb|Considering two control variables of a robotic arm which are typically angles and assuming complete rotations of 360 degrees the space of configurations will then be a torus]] Time can be generalized too as a generic set of continuous parameters, for example the control parameters of a robot can be a manifold. There is no need that time has a direction, that is smooth or even that it has whatsoever meaning similar to the intuition of time, in fact it can be generalized to even more general algebraic objects.<ref>Here a few examples where time is generalized into real, complex or discrete n-dimensional vector spaces https://mathoverflow.net/questions/105071/dynamical-systems-with-multidimensional-complex-and-other-exotic-kinds-of-time</ref>

A general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, and for example in image processing.{{citation needed|date=February 2026}}

Time typically is considered often an external parameter as in classical and quantum mechanics, and this is typically called time domain representation, and it goes hand in hand with the Hamiltonian mechanics formulation. This is not necessarily always the case: general relativity for example is frame independent,<ref>{{cite book |title=Springer Handbook of Spacetime |series=Springer Handbooks |date=2014 |doi=10.1007/978-3-642-41992-8 |isbn=978-3-642-41991-1 }}</ref> and gravity has an influence over time too, and in quantum electrodynamics the use of the Lagrangian mechanics formulation is more common<ref>{{cite journal |last1=Doikou |first1=Anastasia |last2=Evangelisti |first2=Stefano |last3=Feverati |first3=Giovanni |last4=Karaiskos |first4=Nikos |title=Introduction to Quantum Integrability |journal=International Journal of Modern Physics A |date=10 July 2010 |volume=25 |issue=17 |pages=3307–3351 |doi=10.1142/S0217751X10049803 |arxiv=0912.3350 }}</ref> where time and space are on same footing. In both cases the literature still talks about dynamical systems.

==== Discrete dynamical system ==== {{main |Dynamical system#Discrete dynamical systems}} Time can also be a discrete parameter. When time is generalized to the multi-dimensional case, i.e. as a general set of control or external parameters, this space can be interpreted as a Lattice, i.e. as the discrete points of a manifold or the tics of a stock price.<ref>{{cite journal |last1=Fabretti |first1=Annalisa |title=A Dynamical Model for Financial Market: Among Common Market Strategies Who and How Moves the Price to Fluctuate, Inflate, and Burst? |journal=Mathematics |date=22 February 2022 |volume=10 |issue=5 |pages=679 |doi=10.3390/math10050679 |doi-access=free |hdl=2108/311561 |hdl-access=free }}</ref> Discrete Time events therefore can be counted by integers, for example like the measurements of the position of the planets in the sky, but this can be very different than the intuition of time as a clock that has equispaced time events. One of the tasks is typically to extract some mathematical model from the data.<ref>{{cite journal |last1=Haller |first1=George |last2=Kaszás |first2=Bálint |title=Data-driven linearization of dynamical systems |journal=Nonlinear Dynamics |date=November 2024 |volume=112 |issue=21 |pages=18639–18663 |doi=10.1007/s11071-024-10026-x |pmid=39219721 |pmc=11362512 }}</ref>

==== Not deterministic ==== [[File:Difference between deterministic and Nondeterministic.svg|left|thumb|A Nondeterministic Turing machine is an example of discrete dynamical system where computation cannot be represented sequentially but only on a tree, given each input state may have multiple output states]] The ''evolution rule'' of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state.<ref>{{cite book |last=Strogatz |first=S. H. |year=2001 |title=Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry |publisher=Perseus }}</ref><ref>{{cite book |first1=A. |last1=Katok |first2=B. |last2=Hasselblatt |title=Introduction to the Modern Theory of Dynamical Systems |location=Cambridge |publisher=Cambridge University Press |year=1995 |isbn=978-0-521-34187-5 |url-access=registration |url=https://archive.org/details/introductiontomo0000kato }}</ref> However, some systems are not deterministic they may allow multiple future states (i.e. the maps are generalized into multivalued functions and not uniquely defined everywhere) and the system can be subject to a bifurcation.

==== Stochastic ==== {{Main|random dynamical systems}} {{see also|stochastic system}}

Some systems are also stochastic, either in the input parameters such as an oscillator with a random force, or in the initial conditions, or in the predicted variables as in a Stochastic differential equation. In that random events also affect the evolution of the state variables, and this includes stochastic jump processes which are not continuous, a prototype example of a stochastic dynamical system are stock prices.<ref>{{cite arXiv |last1=Sip |first1=Viktor |last2=Breyton |first2=Martin |last3=Petkoski |first3=Spase |last4=Jirsa |first4=Viktor |date=2025 |title=Dynamical system reconstruction from partial observations using stochastic dynamics |class=cs.LG |eprint=2510.01089 }}</ref>

==== Chaotic and Quantum systems ==== right|thumb|False color image of the far field of a submerged turbulent jet Last but not least there are chaotic systems (i.e. typically deterministic but not predictable) such as: * complex dynamics * Hyperbolic dynamics * multiplicative chaos<ref>Gaussian multiplicative chaos revisited https://hal.science/hal-00293830v1/document</ref><ref>Complex Gaussian multiplicative chaos https://ui.adsabs.harvard.edu/abs/2013arXiv1307.6117L/abstract</ref><ref>Random bounded analytic functions by Random measure (Gaussian multiplicative chaos) https://ymsc.tsinghua.edu.cn/__local/4/8D/F5/4AC71574298685B0A526A63A96F_DE8276B0_2DB582.pdf</ref><ref>Gaussian multiplicative chaos a review https://arxiv.org/abs/1305.6221</ref> * non-deterministic chaos<ref>{{xref|(see also jump process and Duffing oscillator)}}</ref><ref>"Non-deterministic chaos is a new dynamical paradigm where a non-deterministic system is influenced by random perturbations" https://ui.adsabs.harvard.edu/abs/1994chao.dyn..8001D/abstract</ref>

And quantum systems (i.e. deterministic until they are measured), or quantum chaotic systems.<ref name="ChaosBook"/>

== Formal definition == Assume that X is a non empty set with elements called states. Assume a general transformation: <math display="block">T : X \to X </math>

It is possible to interpret X as a state space and T as the evolution between states.<ref>{{Cite web |date=2008-01-09 |title=254A, Lecture 1: Overview |url=https://terrytao.wordpress.com/2008/01/08/254a-lecture-1-overview/ |access-date=2026-04-07 |website=What's new |language=en}}</ref> Adding different structures on T and on X allows to model different properties of the dynamical system.

It is possible to model time evolution: <math>\hat{T}</math> can be a semigroup with one parameter <math>t</math> called time that will also belong to a semi-group such as <math>N (t>0)</math> in the discrete time case, <math>R^{+} (t>0)</math> in the continuous time case.

A semigroup structure introduces associativity <math display="block">\hat{T_1}(\hat{T_2}\hat{T_3})=(\hat{T_1}\hat{T_2})\hat{T_3}</math> which implies a composition law between different time evolutions:<ref>A fancy definition https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf</ref> <math display="block">\hat{T}(t_1+t_2)=\hat{T}(t_1)\hat{T}(t_2)</math> this is also ultimately a homomorphism.

It is possible to define an origin of time <math>t=0</math> adding an identity to the semi-group <math display="block">\hat{T}(0)=\mathbf{1}</math> and it is finally possible also to model reversible time evolution: T can be a group such as <math>\mathbf{Z}</math> or <math>R</math>, and being a group this in fact has a definition of inverse transformations:<ref>Classes of dynamical systems https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf</ref> <math display="block">\exists! \hat{T}^{-1}: \hat{T}^{-1} = \hat{T}(-t), \hat{T}(-t)\hat{T}(t) = \mathbf{1}</math>

More commonly there are multiple classes of definitions for a dynamical system: a first one is motivated by ordinary differential equations and is geometrical in flavor, there is an additional differentiability structure; a second one is motivated by ergodic theory and is measure theoretical in flavor, there is an additional topological structure and a last one which is motivated by Category theory and is more abstract in flavour.

=== Geometrical definition === {{More footnotes needed|section|date=February 2026|partial=y|reason=need at least one per section to tell which reference to look at, mandatory changes are applicable from here downwards, the rest was cleaned, still it can be eventually further improved}}

In the geometrical definition, a dynamical system is the tuple <math> \langle \mathcal{T}, \mathcal{M}, f\rangle </math>. <math>\mathcal{T}</math> is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. <math>\mathcal{M}</math> is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. ''f'' is an evolution rule ''t'' → ''f'' ''t'' (with <math>t\in\mathcal{T}</math>) such that ''f t'' is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain <math> \mathcal{T}</math> into the space of diffeomorphisms of the manifold to itself. In other terms, ''f''(''t'') is a diffeomorphism, for every time ''t'' in the domain <math> \mathcal{T}</math>.{{citation needed|reason=it would be better to have another source that uses something close to this definition|date=March 2026}}

==== Algebraic dynamical systems ==== {{main|Dynamical system#Algebraic dynamical system}} An important class of systems from a mathematical perspective is when the map <math>f</math> is algebraic or in general when the map is implicitly defined by a set of algebraic equations and the manifold <math>\mathcal{M}</math> is ideally defined on a generic field.<ref>Dynamical systems of algebraic origin: https://link.springer.com/book/10.1007/978-3-0348-9236-0 </ref>

==== Real dynamical system ==== A ''real dynamical system'', ''real-time dynamical system'', ''continuous time dynamical system'', or ''flow'' is a tuple (''T'', ''M'', Φ) with ''T'' an open interval in the real numbers '''R''', ''M'' a manifold typically but not necessarily locally homeomorphic to a Banach space, and Φ a continuous function.<ref>Smooth Dynamical Systems M. C. IRWIN, 1.1</ref>

===== Differentiability ===== If Φ is continuously differentiable the system is called a ''differentiable dynamical system''. The manifold ''M'' is then typically but not necessarily locally diffeomorphic to a Banach space.<ref>dynamics of flows defined by differential equations: https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf</ref>

===== Dimensionality ===== If the manifold ''M'' is locally diffeomorphic to '''R'''''n'', the dynamical system is ''finite-dimensional''; if not, the dynamical system is ''infinite-dimensional''.<ref>The Korteweg–De Vries equation is an example with infinite degrees of freedom and infinite integrals of motion</ref>{{please clarify|reason=The Korteg De vries is a PDE that uses a wave function phi(x,t) here it is a bit unclear how you move from PDEs to ODEs|date=March 2026}}

===== Flows ===== When ''T'' is taken to be the reals, the dynamical system is called ''global'' or a ''flow''; and if ''T'' is restricted to the non-negative reals, then the dynamical system is a ''semi-flow''.{{please clarify|reason=are other combinations relevant?|date=April 2026}}

===== Classical definition ===== The modern geometrical definition assumes a map that provides an explicit description of the dynamical system, this is motivated by ergodic theory, by partial differential equations and by mathematical techniques that go beyond differential equations. An explicit description is often not available, the classical geometrical definition is implicit, rooted in classical mechanics, and based on a standard set of ordinary differential equations and a finite set of degrees of freedom: <div style="background: #F0F0F0; width: 50%;"> <div style="text-indent: 2em; border-top: thin solid #999999; border-bottom: thin solid #999999; border-left: thin solid #999999; border-right: thin solid #999999; padding-left: 1em; padding-right: 1em; margin-top: 0.5em; margin-bottom: 1em;"> The totality of states of motion may be set into one-to-one correspondence with the points, P, of a closed n-dimensional manifold, M, in such wise that for suitable coordinates <math>x_1,...,x_n</math> the differential equations of motion may be written: <math display="block">\frac{dx_i}{dt} = u_i(x_1,...,x_n,t);(i=1,...,n)</math> There can be different regularity conditions to the functions <math>u_i</math> such as being differentiable or analytic.<ref>Birkhoff, Dynamical systems (1927), page 193: https://archive.org/details/dynamicalsystems00birk/page/188/mode/2up</ref> </div></div> This definition implies the existence and uniqueness of solutions of such equations.

===== Lagrangian Dynamical system ===== {{main |Lagrangian system}} It is also possible to cast the geometrical definition in terms of a variational principle: <div style="background: #F0F0F0; width: 50%;"> <div style="text-indent: 2em; border-top: thin solid #999999; border-bottom: thin solid #999999;border-left: thin solid #999999; border-right: thin solid #999999; padding-left: 1em; padding-right: 1em; margin-top: 0.5em; margin-bottom: 1em;"> Let M be a differentiable manifold, TM its tangent bundle, and <math>L: TM\to \mathbb{R}</math> a differentiable function. A map <math>\gamma: \mathbb{R}\to M</math> is called a motion in the Lagrangian system, with configuration manifold M and Lagrangian L, if <math>\gamma</math> is an extremal of the functional: <math display="block">\Phi(\gamma)=\int_{t_0}^{t_1} L(\gamma,\dot{\gamma}) dt </math> where <math>\dot{\gamma}\in TM_{\gamma(t)}</math> is called velocity vector.<ref>Arnold Mathematical methods of Classical Mechanics(1989), sec 19, pp 83</ref> </div></div>

===== Hamiltonian Dynamical system ===== {{main |Hamiltonian system}} Dually to the Lagrangian it is possible to use a Hamiltonian formulation which includes a Symplectic or Poisson manifold structure on the phase space.<ref>Arnold Mathematical methods of Classical Mechanics(1989), sec 19, pp 83, example end of page</ref>

===== Non integrable systems ===== To be complete there are also systems that are typically not integrable systems such as dissipative systems, nonholonomic systems and systems that have a contact manifold structure for example systems that have a no slip boundary condition (i.e. some constraint on the velocity on the boundary).

==== Discrete dynamical system ==== {{main|Dynamical system#Discrete dynamical systems}} A ''discrete-time dynamical system'' is a tuple (''T'', ''M'', Φ), where ''M'' is a manifold locally diffeomorphic to a Banach space, and Φ is a function. ''T'' can be taken to be the integers or the non negative integers. The manifold itself can be a graph or made discrete for example with a discrete topology.<ref>A graph and a general discrete space can be a Hausdorff space, can have a measure, and at least if it is finite is compact. It is not strictly a good example of a Banach space because Cauchy sequences may not make sense. This boundary between finite and infinite is interesting in the field of arithmetic geometry.</ref>

=== Measure theoretical definition === {{main|Measure-preserving dynamical system}}

A dynamical system may be defined formally as a measure-preserving transformation of a measure space, the triplet (''T'', (''X'', Σ, ''μ''), Φ). Here, ''T'' is a monoid (usually the non-negative integers), ''X'' is a set, and (''X'', Σ, ''μ'') is a probability space, meaning that Σ is a sigma-algebra on ''X'' and μ is a finite measure on (''X'', Σ). A map Φ: ''X'' → ''X'' is said to be Σ-measurable if and only if, for every σ in Σ, one has <math>\Phi^{-1}\sigma \in \Sigma</math>. A map Φ is said to '''preserve the measure''' if and only if, for every ''σ'' in Σ, one has <math>\mu(\Phi^{-1}\sigma ) = \mu(\sigma)</math>. Combining the above, a map Φ is said to be a '''measure-preserving transformation of ''X'' ''', if it is a map from ''X'' to itself, it is Σ-measurable, and is measure-preserving. The triplet (''T'', (''X'', Σ, ''μ''), Φ), for such a Φ, is then defined to be a '''dynamical system'''.<ref>Ergodic theory, introduction, Glossary: https://link.springer.com/referencework/10.1007/978-1-0716-2388-6</ref>

The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates <math>\Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi</math> for every integer ''n'' are studied.{{citation needed|reason=iteration and recursive relations are just special cases, each map from <math>t_n</math> to <math>t_{n+1}</math>, can be fully different, iteration is just a special case actually of <math>\Phi = \Phi{t_n} \circ \Phi{t_{n-1}} \circ \dots \circ \Phi_{t_0}</math> |date=February 2026}} For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.{{citation needed|reason=more complicated??? |date=February 2026}}

==== Relation to geometric definition ==== The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the Krylov–Bogolyubov theorem) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.

Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution.

For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.{{please clarify|reason=probably a strong example with microcanonical ensemble statistics or liouville measure would help here|date=March 2026}}

==== Topological dynamical system ==== A topological dynamical system is a global dynamical system ('''T''', ''X'', Φ) on a locally compact and Hausdorff topological space ''X''.{{clarification needed|reason=unclear what is the requirement to be global and if it is needed, also unclear if being locally compact vs being Hausdorff is needed, isn't being Hausdorff general enough ?|date=February 2026}} T is a topological isomorphism and therefore a homeomorphism.<ref>Terry tao lecture 1</ref>

===== Compactification ===== It is often useful to study the continuous extension Φ* of Φ to the one-point compactification ''X*'' of ''X''. Even after losing the differential structure of the original system, there are compactness arguments to analyze the new system ('''R''', ''X*'', Φ*). This is similar in spirit to Projective geometry where all limit points to infinity are the same point.

Another more general technique is to use Stone–Čech compactification<ref>Terry tao lecture 3</ref> which is similar in spirit to affine geometry where all limit points at infinity are considered different.

===== Relevance ===== In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected.

As an example in a topological dynamical system the limit orbit of an attractor is contained within the manifold itself. This is a non trivial statement for multiple reasons: limit orbits may never be reached; limit orbits may have Lebesgue measure zero; attaching a probability to a limit orbit would be non trivial; an attractor may also have multiple limit orbits and the distinction between different compactifications may be relevant.

=== Definition with Category theory === ==== Categories vs semi-groups ==== {{bq|text=A category X of mathematical objects has a semigroup G of homomorphisms acting on it (topological spaces have continuous maps, sets have arbitrary maps, groups, rings fields or algebras have homomorphisms, measure spaces have measurable maps). We can view each of these categories as a dynamical system. One can even include the category of dynamical systems with suitable homomorphisms. But this viewpoint is not a very useful in itself.<ref>A final link: https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf</ref>}}

==== Definition with monoids ==== In the context of category theory, categories are always defined together with an Identity map, therefore these definitions are based on monoids instead of semi-groups.<ref>Note that it is always possible to add an Identity on the semi-group, define a monoid and pull back the monoid structure onto the original semi-group</ref>

A '''dynamical system'''<ref>Giunti M. and Mazzola C. (2012), "[https://www.researchgate.net/publication/272943599_Dynamical_Systems_on_Monoids_Toward_a_General_Theory_of_Deterministic_Systems_and_Motion Dynamical systems on monoids: Toward a general theory of deterministic systems and motion]". In Minati G., Abram M., Pessa E. (eds.), ''Methods, models, simulations and approaches towards a general theory of change'', pp. 173–185, Singapore: World Scientific. {{ISBN|978-981-4383-32-5}} </ref><ref>Mazzola C. and Giunti M. (2012), "[https://www.researchgate.net/publication/281244041_Reversible_dynamics_and_the_directionality_of_time Reversible dynamics and the directionality of time]". In Minati G., Abram M., Pessa E. (eds.), ''Methods, models, simulations and approaches towards a general theory of change'', pp. 161–171, Singapore: World Scientific. {{ISBN|978-981-4383-32-5}}.</ref>{{Citation needed|reason=seems to be some shape of primary sources: "However, according to Giunti and Mazzola [2] (this volume), a dynamical system on a monoid is", would be best to have a more intelligible definition from a non primary source. This is even more funny: https://math.stackexchange.com/questions/475462/dynamical-systems "If I would have explained this way I would have emptied my class" |date=February 2026}} is a tuple (''T'', ''X'', Φ) where ''T'' is a monoid,{{citation needed|reason=questionable if semi group is a more genric definition, other literatures seems to refer to semi-groups always. Only this guy from cagliari seems to be fixated with monoids. There is typically a natural choice of identity or time=0 too, there is an ambiguity in the identity too because ultimately the origin of the reference frame is arbitrary|date=February 2026}} written additively, ''X'' is a non-empty set and Φ is a function: <math display="block">\Phi: U \subseteq (T \times X) \to X</math> with: <math display="block">\mathrm{proj}_{2}(U) = X</math> (where <math>\mathrm{proj}_{2}</math> is the 2nd projection map{{Citation needed|reason=what is the 2nd projection map ? it seems a trivial statement to project back on X, i.e. X is the second space that was put in the cartersian product. Actually It recalls fiber bundles, but in that case the base space can be time, control parameters or even space, I would see it instead natuaral to define the base space as the state space and the fiber as the semi-group structure on it, but I am not sure if this is a standard definition either|date=February 2026}}) and for any ''x'' in ''X'': <math display="block">\Phi(0,x) = x</math> <math display="block">\Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x),</math> for <math>\, t_1,\, t_2 + t_1 \in I(x)</math> and <math>\ t_2 \in I(\Phi(t_1, x)) </math>, where we have defined the set <math> I(x) := \{ t \in T : (t,x) \in U \}</math> for any ''x'' in ''X''.

In particular, in the case that <math> U = T \times X </math> we have for every ''x'' in ''X'' that <math> I(x) = T </math> and thus that Φ defines a monoid action of ''T'' on ''X''.

The function Φ(''t'',''x'') is called the '''evolution function''' of the dynamical system: it associates to every point ''x'' in the set ''X'' a unique image, depending on the variable ''t'', called the '''evolution parameter'''. ''X'' is called '''phase space''' or '''state space''', while the variable ''x'' represents an '''initial state''' of the system.

We often write: <math display="block">\Phi_x(t) \equiv \Phi(t,x)</math> <math display="block">\Phi^t(x) \equiv \Phi(t,x)</math> if we take one of the variables as constant. The function <math display="block">\Phi_x:I(x) \to X</math> is called the '''flow''' through ''x'' and its graph is called the '''trajectory''' through ''x''. The set <math display="block">\gamma_x \equiv\{\Phi(t,x) : t \in I(x)\}</math> is called the '''orbit''' through ''x''. The orbit through ''x'' is the image of the flow through ''x''.

A subset ''S'' of the state space ''X'' is called Φ-'''invariant''' if for all ''x'' in ''S'' and all ''t'' in ''T'' <math display="block">\Phi(t,x) \in S.</math> Thus, in particular, if ''S'' is Φ-'''invariant''', <math>I(x) = T</math> for all ''x'' in ''S''. That is, the flow through ''x'' must be defined for all time for every element of ''S''.

== Construction of dynamical systems == The concept of ''evolution in time'' is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems. But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as the following: <math display="block">\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})</math> <math display="block">\boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0</math> where * <math>\dot{\boldsymbol{x}}</math> represents the velocity of the material point '''x''' * ''M'' is a finite dimensional manifold * '''v''': ''T'' × ''M'' → ''TM'' is a vector field in '''R'''''n'' or '''C'''''n'' and represents the change of velocity induced by the known forces acting on the given material point in the phase space ''M''. The change is not a vector in the phase space ''M'', but is instead in the tangent space ''TM''.

There is no need for higher order derivatives in the equation, nor for the parameter ''t'' in ''v''(''t'',''x''), because these can be eliminated by considering systems of higher dimensions.

Depending on the properties of this vector field, the mechanical system is called * '''autonomous''', when '''v'''(''t'', '''x''') = '''v'''('''x''') * '''homogeneous''' when '''v'''(''t'', '''0''') = 0 for all ''t''

The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above <math display="block">\boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0)</math>

The dynamical system is then (''T'', ''M'', Φ).

Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy <math display="block">\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0</math> where <math>\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}</math> is a functional from the set of evolution functions to the field of the complex numbers.

This equation is useful when modeling mechanical systems with complicated constraints.

Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations.

== Discrete dynamical systems == [[File:YF-17_aircraft_Plot.jpg|thumb|right| A computational fluid dynamics mesh is an example of a discretization of a dynamical system, typically both in space and time, for computational purposes]] A discrete dynamical system is when either time or space or both are discrete. Typically for both space and time, there is a finite or countable sets of points and bounded maps and operators, that can be manipulated on a computer given some general assumptions on the boundaries.

=== Mathematical definition === In the general context of mathematics, it is possible to define the dynamical system as a general discrete map <ref>John Gemmer, Chapter 14, Discrete dynamical systems https://www.dam.brown.edu/people/jgemmer/GreenwellCh14.pdf</ref> as in the Formal definition. A generic sequence is already per se a discrete dynamical system.<ref name="JohnGemmer">John Gemmer, Chapter 14</ref>{{rp|at=Example 1}} Recursion and interation of maps is another such case.<ref name="JohnGemmer"/>{{rp|at=Examples 2 and 3}} A prototype of this is the Logistic map.<ref name="JohnGemmer"/>{{rp|at=Example 4}}

=== Empirical definition === From an empirical perspective, all dynamical systems derived from temporal data are discrete, Gauss for example proved that with the measurement of 3 positions and times of Ceres in the sky is possible to fully determine the orbit, therefore be able to compute any possible position and velocity of the asteroid in the past or the future and therefore fully characterize the dynamical system.<ref>https://sites.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html</ref> Typical tasks with experimental data are to derive a mathematical model.<ref>{{cite journal |last1=Moore |first1=Samuel A. |last2=Mann |first2=Brian P. |last3=Chen |first3=Boyuan |title=Automated global analysis of experimental dynamics through low-dimensional linear embeddings |journal=npj Complexity |date=17 December 2025 |volume=2 |issue=1 |article-number=36 |doi=10.1038/s44260-025-00062-y |doi-access=free }}</ref>

=== Examples === <gallery mode="packed" heights="140"> File:Fibonacci Rabbits.svg|Fibonacci Rabbits: Population model with infinite resources that generates the Fibonacci numbers.<ref>{{Cite web |title=Fibonacci's Rabbits |url=https://mathcenter.oxford.emory.edu/site/math125/fibonacciRabbits/ |access-date=2026-04-09 |website=mathcenter.oxford.emory.edu}}</ref> File:LogisticCobwebChaos.gif|Logistic map A generic map applied to itself recursively makes a dynamical system File:Predator-prey toy model.gif|Discrete example of predator prey model<ref>{{cite journal |last1=Liu |first1=Xiaoli |last2=Xiao |first2=Dongmei |title=Complex dynamic behaviors of a discrete-time predator–prey system |journal=Chaos, Solitons & Fractals |date=April 2007 |volume=32 |issue=1 |pages=80–94 |doi=10.1016/j.chaos.2005.10.081 }}</ref> File:Page_rank_animation.gif|PageRank is an example of discrete dynamical system that predicts the future probability of a user o visit a page <ref>example 9.1.2: https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter9/DynSystDiscrete.html</ref> </gallery>

=== Applied mathematics and physics ===  [[File:10 PM March 12 surface analysis of Great Blizzard of 1888.png|thumb|left|Weather forecasts are an application of dynamical systems in the context of Computational Fluid Dynamics]] In the context of applied mathematics such as physics, biology or engineering, the starting point is typically a finite difference equation<ref>{{cite book |last1=Galor |first1=Oded |title=Discrete Dynamical Systems |chapter=One-Dimensional, First-Order Systems |date=2007 |pages=1–26 |doi=10.1007/3-540-36776-4_1 |isbn=978-3-540-36775-8 }}</ref> such as the most simple possible one here: <math display="block">y_{t+1} = a x_t + b</math>

More generally this can be generalized into a generic discrete map from a n-dimensional manifold to itself: <math display="block">y^i_{t+1} = f^i_t(x_1,..,x_n),i=1,...,n</math>

In the context of Hamiltonian flows,<ref>{{xref|(see also Fluid flow)}}</ref> motion itself can be considered a canonical transformation (i.e. ultimately a map) and therefore a discrete set of these in a discrete time interval <math>\Delta t</math> is again a shape of characterization of the full discrete dynamical system.

There are also cases of dense orbits,<ref>{{xref|(see also chaos theory)}}</ref> where in essence the state phase space is not compact, and unbounded operators, like in quantum mechanics, where the evolution maps are not compact.

An example of this is a weather forecast of Earth where the data points are separated in space from each other. The system can be put on a lattice, and formulas can be used to compute and predict certain variables, like in the case of the discretization of Navier–Stokes equations.

=== Cascades === [[File: backtang2.png|thumb|right|Self-organized criticality is a set of emergent phenomena discovered in cellular automata the attractor is a critical point in a phase transition]] Discrete dynamical systems are often also called cascades, when the concept of passing over information from one step to the next is predominant. Typical examples are avalanches<ref>Complex networks and cascade dynamics, Introduction https://pdodds.w3.uvm.edu/teaching/courses/2009-08UVM-300/docs/others/2011/hackett2011b.pdf</ref><ref>{{xref|(see also Abelian sandpile model)}}</ref> and also period doubling cascades.<ref>{{xref|(see also logistic map)}}</ref> When ''T'' is taken to be the integers, it is a ''cascade'' or a ''map''. If ''T'' is restricted to the non-negative integers the system is called a ''semi-cascade''.<ref>{{Cite book|title=Discrete Dynamical Systems|last=Galor|first=Oded|publisher=Springer|year=2010}}</ref>{{clarify|reason=unclear if this semi-cascade is useful at all, to the extent of the discourse, maybe just something that can be omitted|date=February 2026}}

=== Cellular Automaton === {{Main|cellular automata}} A ''cellular automaton'' is a tuple (''T'', ''M'', Φ), with ''T'' a lattice such as the integers or a higher-dimensional integer grid, ''M'' is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in ''M'' represents the "space" lattice, while the one in ''T'' represents the "time" lattice.{{clarify|reason=double check this definition, why M is a manifold of functions?|date=February 2026}}

=== Other notable examples === * Symbolic dynamics * Finite state automata * Turing machines * Arithmetic dynamics * Graph dynamical system * By its discrete nature graph theory and in general any form of discrete mathematics can be seen in the lense of discrete dynamical systems * Same apply to any form of finite mathematics, finite geometry, finite groups or countable mathematical structures such as coxeter groups

Some of these are separate subfields in their own right such as number theory or network theory.

== Linear dynamical systems == {{Main|Linear dynamical system}} Linear dynamical systems are at the heart of any system engineering and system theory curriculum. Historically linear systems up to the 1970s reflected most of system theory as a whole (i.e. before the widespread availability of computers). They include the basic features of any dynamical system, such as attenuation saturation and oscillation, and at least locally they can approximate also any non linear systems.

Linear dynamical systems can be solved in terms of simple functions such as exponentials and simple trigonometric functions (i.e. complex exponentials), and the behavior of all orbits can be classified.

In a linear system the phase space is the ''N''-dimensional Euclidean space, so any point in phase space can be represented by a vector with ''N'' numbers. N dimensional Linear dynamical systems are also not chaotic

The analysis of linear systems is also simplified and possible because they satisfy a superposition principle: if ''u''(''t'') and ''w''(''t'') satisfy a linear differential equation that describe a system, then so will a linear combination <math>\alpha u(t) + \beta w(t)</math>. With superposition is possible to generate new solutions from known ones, therefore is just necessary to classify the fundamental solutions to know all of them.

=== Flows === For a flow, the vector field v(''x'') is an affine function of the position in the phase space, that is, <math display="block"> \dot{x} = v(x) = A x + b,</math> with ''A'' a matrix, ''b'' a vector of numbers and ''x'' the position vector. The solution to this system can be found by using the superposition principle (linearity). The case ''b'' ≠ 0 with ''A'' = 0 is just a straight line in the direction of ''b'': <math display="block">\Phi^t(x_1) = x_1 + b t. </math>

When ''b'' is zero and ''A'' ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if ''x''0 = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point ''x''0, <math display="block">\Phi^t(x_0) = e^{t A} x_0. </math>

When ''b'' = 0, the eigenvalues of ''A'' determine the structure of the phase space. From the eigenvalues and the eigenvectors of ''A'' it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.

The distance between two different initial conditions in the case ''A'' ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.

thumb|500px|center|Linear vector fields and a few trajectories. {{Clear}}

=== Maps === A discrete-time, affine dynamical system has the form of a matrix difference equation: <math display="block"> x_{n+1} = A x_n + b, </math> with ''A'' a matrix and ''b'' a vector. As in the continuous case, the change of coordinates ''x'' → ''x'' + (1 − ''A'') –1''b'' removes the term ''b'' from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system ''A'' ''n''''x''0. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of ''A'' determine the structure of phase space. For example, if ''u''1 is an eigenvector of ''A'', with a real eigenvalue smaller than one, then the straight lines given by the points along ''α'' ''u''1, with ''α'' ∈ '''R''', is an invariant curve of the map. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems such as Chaotic maps.

=== Local dynamics === The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a ''singular point'' of the vector field (a point where ''v''(''x'') = 0) will remain a singular point under smooth transformations; a ''periodic orbit'' is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

=== Rectification === A flow in most small patches of the phase space can be made very simple. If ''y'' is a point where the vector field ''v''(''y'') ≠ 0, then there is a change of coordinates for a region around ''y'' where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The ''rectification theorem'' says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space ''M'' the dynamical system is ''integrable''. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where ''v''(''x'') = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

=== Near periodic orbits === In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point ''x''0 in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to ''v''(''x''0). These points are a Poincaré section ''S''(''γ'', ''x''0), of the orbit. The flow now defines a map, the Poincaré map ''F'' : ''S'' → ''S'', for points starting in ''S'' and returning to ''S''. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes ''x''0.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map ''F''. By a translation, the point can be assumed to be at ''x'' = 0. The Taylor series of the map is ''F''(''x'') = ''J'' · ''x'' + O(''x''2), so a change of coordinates ''h'' can only be expected to simplify ''F'' to its linear part <math display="block"> h^{-1} \circ F \circ h(x) = J \cdot x.</math>

This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If ''λ''1, ..., ''λ''''ν'' are the eigenvalues of ''J'' they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form ''λ''''i'' – Σ (multiples of other eigenvalues) occurs in the denominator of the terms for the function ''h'', the non-resonant condition is also known as the small divisor problem.

=== Conjugation results === The results on the existence of a solution to the conjugation equation depend on the eigenvalues of ''J'' and the degree of smoothness required from ''h''. As ''J'' does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of ''J'' are not in the unit circle, the dynamics near the fixed point ''x''0 of ''F'' is called ''hyperbolic'' and when the eigenvalues are on the unit circle and complex, the dynamics is called ''elliptic''.

In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map ''J'' · ''x''. The hyperbolic case is also ''structurally stable''. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of ''J'' in the complex plane, implying that the map is still hyperbolic.

The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.

== Bifurcation theory == {{Main|Bifurcation theory}} right|thumb|bifurcation with saddle point and equilibrium point When the evolution map Φ''t'' (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value ''μ''0 is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter ''μ''. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

[[File:Chaosorderchaos.png|left|thumb|Period halving bifurcations, followed by equilibrium, followed by period-doubling bifurcations]] The bifurcations of a hyperbolic fixed point ''x''0 of a system family ''Fμ'' can be characterized by the eigenvalues of the first derivative of the system ''DF''''μ''(''x''0) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of ''DFμ'' on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.

Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.

== Ergodic systems == {{Main|Ergodic theory}} In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset ''A'' into the points Φ ''t''(''A'') and invariance of the phase space means that <math display="block"> \mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ). </math> In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let ''F'' be a phase space volume-preserving map and ''A'' a subset of the phase space. Then almost every point of ''A'' returns to ''A'' infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region ''A'' is vol(''A'')/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable ''a'' is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ t. This introduces an operator ''U'' ''t'', the transfer operator, <math display="block"> (U^t a)(x) = a(\Phi^{-t}(x)). </math>

By studying the spectral properties of the linear operator ''U'' it becomes possible to classify the ergodic properties of Φ ''t''. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ ''t'' gets mapped into an infinite-dimensional linear problem involving ''U''.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−β''H''). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

== Nonlinear dynamical systems and chaos == === Chaotic systems === {{Main|Chaos theory}} Simple nonlinear dynamical systems, including piecewise linear systems, can exhibit strongly unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This unpredictable behavior has been called ''chaos''. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent spaces perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the ''stable manifold'') and another of the points that diverge from the orbit (the ''unstable manifold'').

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

The chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The Pomeau–Manneville scenario of the logistic map and the Fermi–Pasta–Ulam–Tsingou problem arose with just second-degree polynomials; the horseshoe map is piecewise linear.

=== Solutions of finite duration === For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,<ref>{{cite book |author = Vardia T. Haimo |title = 1985 24th IEEE Conference on Decision and Control |chapter = Finite Time Differential Equations |year = 1985 |pages = 1729–1733 |doi = 10.1109/CDC.1985.268832 }}</ref> meaning here that in these solutions the system will reach the value zero at some time, called an ending time, and then stay there forever after. This can occur only when system trajectories are not uniquely determined forwards and backwards in time by the dynamics, thus solutions of finite duration imply a form of "backwards-in-time unpredictability" closely related to the forwards-in-time unpredictability of chaos. This behavior cannot happen for Lipschitz continuous differential equations according to the proof of the Picard-Lindelof theorem. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line.

As example, the equation: <math display="block">y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1</math> Admits the finite duration solution: <math display="block">y(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2</math> that is zero for <math>t \geq 2</math> and is not Lipschitz continuous at its ending time <math>t = 2.</math>

== Algebraic dynamical system == {{Expand section|date=April 2026}}

Algebraic dynamical systems can be considered as a special algebraic case of the classical geometric definition based on a system of ODEs. These systems are often defined as a set of algebraic equations, and as an algebraic variety, they are often studied with methods from algebraic geometry and Galois theory.

One such examples is the Poncelet map, where a point is moving in successive steps between two given conics, and the equations are algebraic (i.e. tangents and intersections).

Another example can be a billiard with a border that is an algebraic curve, such as an elliptic billiard, in this case the dynamics is defined by reflections.<ref>The Poncelet grid and the billiard in an ellipse: https://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2006/15.pdf</ref>

Note that algebraic curves are differentiable almost every where (a part from a finite number of points), therefore one can study them with analytic methods. In these two examples, the continuous dynamics is piecewise linear, the important events are typically discrete, albeit may be still an infinite or even measurable set (such as an ergodic trajectory in the billiard), the important events are when an algebraic equation applies (such as a reflection on a border). These systems can therefore be studied as discrete dynamical systems too.

== Category theory for dynamical systems == {{Expand section|date=February 2026}} In the period between 2000 and 2020, category theory has been applied to system theory (e.g. to open systems and subsystems)<ref>{{Cite web |last=Jaz Myers |first=David |date=3 September 2023 |title=Categorical Systems Theory |url=https://www.davidjaz.com/Papers/DynamicalBook.pdf |access-date=10 April 2026}}</ref> and to dynamical systems<ref>{{Cite web |last=Kelly |first=James |title=A Category Theoretic Approach to Dynamical Systems |url=https://srs.amsi.org.au/wp-content/uploads/sites/92/2023/11/kelly_james_srs-report.pdf |access-date=10 April 2026}}</ref><ref>{{Cite web |title=category theory for dynamical systems in nLab |url=https://ncatlab.org/nlab/show/category+theory+for+dynamical+systems |access-date=2026-04-09 |website=ncatlab.org |language=en}}</ref><ref>{{Cite web |last=jadeedenstarmaster |date=2019-03-31 |title=Dynamical Systems with Category Theory? Yes! |url=https://jadeedenstarmaster.wordpress.com/2019/03/31/dynamical-systems-with-category-theory-yes/ |access-date=2026-04-09 |website=Cat Blog |language=en}}</ref>. The motivation is to study common properties across dynamical systems, topological dynamical systems (i.e. with compact state space), and measure preserving dynamical systems (e.g. hamiltonian systems)<ref>{{Cite web |date=2008-01-11 |title=254A, Lecture 2: Three categories of dynamical systems |url=https://terrytao.wordpress.com/2008/01/10/254a-lecture-2-three-categories-of-dynamical-systems/ |access-date=2026-04-09 |website=What's new |language=en}}</ref>. It is also possible to draw an analogy between group representation theory (such as irreducible representations) and ergodic decomposition<ref>{{Cite web |title=ergodic decomposition theorem in nLab |url=https://ncatlab.org/nlab/show/ergodic+decomposition+theorem |access-date=2026-04-09 |website=ncatlab.org}}</ref> i.e. that every invariant (i.e. conservative) measure is a mixture of ergodic ones, in an analogous fashion to the central limit theorem. Ultimately this can be compared to the fundamental theorem of arithmetic and to prime number decomposition.<ref>{{Cite web |last=Moreira |first=Joel |date=2013-09-20 |title=Ergodic Decomposition |url=https://joelmoreira.wordpress.com/2013/09/20/ergodic-decomposition/ |access-date=2026-04-09 |website=I Can't Believe It's Not Random! |language=en}}</ref>

== References == {{reflist}}

== Further reading == {{further reading cleanup|date=February 2025}}

{{refbegin|2}} *{{cite book |first=Vladimir I. |last=Arnold |author-link=Vladimir Arnold |chapter=Fundamental concepts |title=Ordinary Differential Equations |location=Berlin |publisher=Springer Verlag |year=2006 |isbn=3-540-34563-9 }} *{{cite book |first=I. D. |last=Chueshov |title=Introduction to the Theory of Infinite-Dimensional Dissipative Systems }} online version of first edition on the EMIS site [http://www.emis.de/monographs/Chueshov/]. * {{cite book |title=Infinite-Dimensional Dynamical Systems in Mechanics and Physics |series=Applied Mathematical Sciences |date=1997 |volume=68 |doi=10.1007/978-1-4612-0645-3 |isbn=978-1-4612-6853-6 }}

Works providing a broad coverage: * {{cite book | author=Ralph Abraham and Jerrold E. Marsden | title= Foundations of mechanics | publisher= Benjamin–Cummings | year= 1978 | isbn=978-0-8053-0102-1}} (available as a reprint: {{ISBN|0-201-40840-6}}) * ''Encyclopaedia of Mathematical Sciences'' ({{ISSN|0938-0396}}) has a sub-series on dynamical systems with reviews of current research. * {{cite book |author1=Christian Bonatti |author2=Lorenzo J. Díaz |author3=Marcelo Viana | title= Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective| publisher= Springer | year= 2005 | isbn=978-3-540-22066-4}} * {{cite journal | doi=10.1090/S0002-9904-1967-11798-1 | author=Stephen Smale | title= Differentiable dynamical systems | journal= Bulletin of the American Mathematical Society | year= 1967 |volume= 73 |pages= 747–817 | issue=6| author-link=Stephen Smale | doi-access= free | bibcode=1967BAMaS..73..747S }}

Introductory texts with a unique perspective: * {{cite book | author=V. I. Arnold | title=Mathematical methods of classical mechanics | publisher=Springer-Verlag | year=1982 | isbn=978-0-387-96890-2 | author-link=Vladimir Arnold | url-access=registration | url=https://archive.org/details/mathematicalmeth0000arno }} * {{cite book | author=Jacob Palis and Welington de Melo | title=Geometric theory of dynamical systems: an introduction | publisher=Springer-Verlag | year=1982 | isbn=978-0-387-90668-3 | url-access=registration | url=https://archive.org/details/geometrictheoryo0000pali }} * {{cite book | author=David Ruelle | title=Elements of Differentiable Dynamics and Bifurcation Theory | publisher=Academic Press | year=1989 | isbn=978-0-12-601710-6| author-link=David Ruelle }} * {{cite book | author=Tim Bedford, Michael Keane and Caroline Series, ''eds.'' | title= Ergodic theory, symbolic dynamics and hyperbolic spaces | publisher= Oxford University Press | year= 1991 | isbn= 978-0-19-853390-0 }} * {{cite book | author= Ralph H. Abraham and Christopher D. Shaw | title= Dynamics – the geometry of behavior| edition=2nd | publisher= Addison-Wesley | year= 1992 | isbn= 978-0-201-56716-8 }}

Textbooks * {{cite book | author= Kathleen T. Alligood, Tim D. Sauer and James A. Yorke | title= Chaos. An introduction to dynamical systems | publisher= Springer Verlag | year= 2000 | isbn=978-0-387-94677-1}} * {{cite book | author= Oded Galor | title= ''Discrete Dynamical Systems'' | publisher= Springer | year= 2011 | isbn=978-3-642-07185-0}} * {{cite book | author= Morris W. Hirsch, Stephen Smale and Robert L. Devaney | title= Differential Equations, dynamical systems, and an introduction to chaos | publisher= Academic Press | year= 2003 | isbn=978-0-12-349703-1}} * {{cite book |author1=Anatole Katok |author2=Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | isbn=978-0-521-57557-7}} * {{cite book | author= Stephen Lynch | title= Dynamical Systems with Applications using Maple |edition=2nd | publisher= Springer | year= 2010|isbn = 978-0-8176-4389-8 }} * {{cite book | title= Dynamical Systems with Applications using MATLAB |edition=2nd | publisher= Springer International Publishing | year= 2014|isbn = 978-3319068190 | author= Stephen Lynch }} * {{cite book | author= Stephen Lynch | title= Dynamical Systems with Applications using Mathematica|edition= 2nd | publisher= Springer | year= 2017|isbn = 978-3-319-61485-4 }} * {{cite book | title= Dynamical Systems with Applications using Python | publisher= Springer International Publishing | year= 2018|isbn = 978-3-319-78145-7 | author= Stephen Lynch }} * {{cite book | author= James Meiss | title= Differential Dynamical Systems | publisher= SIAM | year= 2007|isbn = 978-0-89871-635-1 }} * {{cite book | author= David D. Nolte | title= Introduction to Modern Dynamics: Chaos, Networks, Space and Time | publisher= Oxford University Press | year= 2015 | isbn=978-0199657032 }} * {{cite book | author= Julien Clinton Sprott | title= ''Chaos and time-series analysis'' | publisher= Oxford University Press | year= 2003 | isbn=978-0-19-850839-7}} * {{cite book | author=Steven H. Strogatz | title= Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering | publisher= Addison Wesley | year= 1994|isbn = 978-0-201-54344-5 | author-link= Steven Strogatz }} * {{cite book| last = Teschl| given = Gerald|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=American Mathematical Society| place = Providence| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}} * {{cite book | author= Stephen Wiggins | title= Introduction to Applied Dynamical Systems and Chaos | publisher= Springer | year= 2003 | isbn= 978-0-387-00177-7 }}

Popularizations: * {{cite book | author=Florin Diacu and Philip Holmes | title= Celestial Encounters | publisher= Princeton | year= 1996 | isbn= 978-0-691-02743-2}} * {{cite book | author=James Gleick | title= Chaos: Making a New Science | publisher= Penguin | year= 1988 | isbn= 978-0-14-009250-9| author-link= James Gleick | title-link= Chaos: Making a New Science }} * {{cite book | author-link=Ivar Ekeland | author=Ivar Ekeland | title= Mathematics and the Unexpected (Paperback) | publisher= University Of Chicago Press | year= 1990 | isbn= 978-0-226-19990-0}} * {{cite book | author=Ian Stewart | year = 1997 | title = Does God Play Dice? The New Mathematics of Chaos | publisher = Penguin | isbn = 978-0-14-025602-4}} {{refend}}

== External links == {{external links|date=February 2025}} {{Commons category|Dynamical systems}} * Public domain Course on dynamical systems [https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf Harvard] *[http://www.arxiv.org/list/math.DS/recent Arxiv preprint server] has daily submissions of (non-refereed) manuscripts in dynamical systems. *[http://www.scholarpedia.org/article/Encyclopedia_of_Dynamical_Systems Encyclopedia of dynamical systems] A part of Scholarpedia – peer-reviewed and written by invited experts. *[http://www.egwald.ca/nonlineardynamics/index.php Nonlinear Dynamics]. Models of bifurcation and chaos by Elmer G. Wiens *[http://amath.colorado.edu/faculty/jdm/faq-Contents.html Sci.Nonlinear FAQ 2.0 (Sept 2003)] provides definitions, explanations and resources related to nonlinear science

'''Online books or lecture notes''' *[https://arxiv.org/abs/math.HO/0111177 Geometrical theory of dynamical systems]. Nils Berglund's lecture notes for a course at ETH at the advanced undergraduate level. *[https://archive.org/details/dynamicalsystems00birk Dynamical systems]. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems. *[http://chaosbook.org/ Chaos: classical and quantum]. An introduction to dynamical systems from the periodic orbit point of view. *[https://www.cs.brown.edu/research/ai/dynamics/tutorial/home.html Learning Dynamical Systems]. Tutorial on learning dynamical systems. *[https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ Ordinary Differential Equations and Dynamical Systems]. Lecture notes by Gerald Teschl

'''Research groups''' *[http://www.math.rug.nl/~broer/ Dynamical Systems Group Groningen], IWI, University of Groningen. *[http://www-chaos.umd.edu/ Chaos @ UMD]. Concentrates on the applications of dynamical systems. *[http://www.math.stonybrook.edu/dynamical-systems], SUNY Stony Brook. Lists of conferences, researchers, and some open problems. *[http://www.math.psu.edu/dynsys/ Center for Dynamics and Geometry], Penn State. *[http://www.cds.caltech.edu/ Control and Dynamical Systems], Caltech. *[https://web.archive.org/web/20061018031023/http://lanoswww.epfl.ch/ Laboratory of Nonlinear Systems], Ecole Polytechnique Fédérale de Lausanne (EPFL). *[https://web.archive.org/web/20070208153906/http://www.math.uni-bremen.de/ids.html Center for Dynamical Systems], University of Bremen *[https://web.archive.org/web/20070406053155/http://www.eng.ox.ac.uk/samp/ Systems Analysis, Modelling and Prediction Group], University of Oxford *[http://sd.ist.utl.pt/ Non-Linear Dynamics Group], Instituto Superior Técnico, Technical University of Lisbon *[http://www.impa.br/ Dynamical Systems] {{Webarchive|url=https://web.archive.org/web/20170602221933/http://www.impa.br/ |date=2017-06-02 }}, IMPA, Instituto Nacional de Matemática Pura e Applicada. *[http://ndw.cs.cas.cz/ Nonlinear Dynamics Workgroup] {{Webarchive|url=https://web.archive.org/web/20150121174532/http://ndw.cs.cas.cz/ |date=2015-01-21 }}, Institute of Computer Science, Czech Academy of Sciences. *[https://dynamicalsystems.upc.edu/ UPC Dynamical Systems Group Barcelona], Polytechnical University of Catalonia. *[https://www.ccdc.ucsb.edu/ Center for Control, Dynamical Systems, and Computation], University of California, Santa Barbara.

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