thumb|Folded-towel map attractor. A '''hyperchaotic system''' is a dynamical system with a bounded attractor set, on which there are at least two positive Lyapunov exponents.<ref>{{Cite journal |last1=Letellier |first1=Christophe |last2=Rossler |first2=Otto E. |date=2007-08-05 |title=Hyperchaos |journal=Scholarpedia |language=en |volume=2 |issue=8 |pages=1936 |doi=10.4249/scholarpedia.1936 |issn=1941-6016 |doi-access=free |bibcode=2007SchpJ...2.1936L }}</ref>
Since on an attractor, the sum of Lyapunov exponents is non-positive, there must be at least one negative Lyapunov exponent. If the system has continuous time, then along the trajectory, the Lyapunov exponent is zero, and so the minimal number of dimensions in which continuous-time hyperchaos can occur is 4.
Similarly, a discrete-time hyperchaos requires at least 3 dimensions.
== Mathematical examples ==
The first two hyperchaotic systems were proposed in 1979.<ref>{{Cite journal |last=Rossler |first=O.E. |date=April 1979 |title=An equation for hyperchaos |url=http://dx.doi.org/10.1016/0375-9601(79)90150-6 |journal=Physics Letters A |volume=71 |issue=2–3 |pages=155–157 |doi=10.1016/0375-9601(79)90150-6 |bibcode=1979PhLA...71..155R |issn=0375-9601|url-access=subscription }}</ref> One is a discrete-time system ("folded-towel map"): thumb|Folded-towel map attractor, animated. <math display="block">\begin{aligned} & x_{t+1}=3.8 x_t\left(1-x_t\right)-0.05\left(y_t+0.35\right)\left(1-2 z_t\right), \\ & y_{t+1}=0.1\left[\left(y_t+0.35\right)\left(1-2 z_t\right)-1\right]\left(1-1.9 x_t\right), \\ & z_{t+1}=3.78 z_t\left(1-z_t\right)+0.2 y_t . \end{aligned}</math>Another is a continuous-time system:<math display="block">\begin{array}{ll} \dot{x}=-y-z, & \dot{y}=x+0.25 y+w, \\ \dot{z}=3+x z, & \dot{w}=-0.5 z+0.05 w . \end{array}</math>More examples are found in.<ref>{{Cite book |last=Sprott |first=Julien C. |title=Elegant chaos : algebraically simple chaotic flows |date=2010 |publisher=World Scientific |isbn=978-981-283-882-7 |location=New Jersey |chapter=6.7: Hyperchaotic Systems |oclc=670430585}}</ref>
== Experimental examples == Only a few experimental hyperchaotic behaviors have been identified.
Examples include in an electronic circuit,<ref>{{Cite journal |last1=Matsumoto |first1=T. |last2=Chua |first2=L. |last3=Kobayashi |first3=K. |date=November 1986 |title=Hyper chaos: Laboratory experiment and numerical confirmation |journal=IEEE Transactions on Circuits and Systems |volume=33 |issue=11 |pages=1143–1147 |doi=10.1109/TCS.1986.1085862 |bibcode=1986ITCS...33.1143M |issn=1558-1276}}</ref> in a NMR laser,<ref>{{Cite journal |last1=Stoop |first1=R. |last2=Meier |first2=P. F. |date=1988-05-01 |title=Evaluation of Lyapunov exponents and scaling functions from time series |url=http://dx.doi.org/10.1364/josab.5.001037 |journal=Journal of the Optical Society of America B |volume=5 |issue=5 |pages=1037 |doi=10.1364/josab.5.001037 |bibcode=1988JOSAB...5.1037S |issn=0740-3224|url-access=subscription }}</ref> in a semiconductor system,<ref>{{Cite journal |last1=Stoop |first1=R. |last2=Peinke |first2=J. |last3=Parisi |first3=J. |last4=Röhricht |first4=B. |last5=Huebener |first5=R. P. |date=1989-05-01 |title=A p-Ge semiconductor experiment showing chaos and hyperchaos |url=https://dx.doi.org/10.1016/0167-2789%2889%2990078-X |journal=Physica D: Nonlinear Phenomena |language=en |volume=35 |issue=3 |pages=425–435 |doi=10.1016/0167-2789(89)90078-X |bibcode=1989PhyD...35..425S |issn=0167-2789|url-access=subscription }}</ref> and in a chemical system.<ref>{{Cite journal |last1=Eiswirth |first1=M. |last2=Kruel |first2=Th. -M. |last3=Ertl |first3=G. |last4=Schneider |first4=F. W. |date=1992-05-29 |title=Hyperchaos in a chemical reaction |url=https://dx.doi.org/10.1016/0009-2614%2892%2985672-W |journal=Chemical Physics Letters |language=en |volume=193 |issue=4 |pages=305–310 |doi=10.1016/0009-2614(92)85672-W |bibcode=1992CPL...193..305E |issn=0009-2614|url-access=subscription }}</ref>
== Applications ==
Hyperchaotic systems have been investigated for applications that exploit their high dimensionality and extreme sensitivity to initial conditions, resulting from the presence of multiple positive Lyapunov exponents.<ref name="LetellierRossler">{{cite journal |last1=Letellier |first1=Christophe |last2=Rössler |first2=Otto E. |title=Hyperchaos |journal=Scholarpedia |volume=2 |issue=9 |pages=1931 |year=2007 |doi=10.4249/scholarpedia.1931 |doi-broken-date=29 December 2025 |doi-access=free |url=https://www.scholarpedia.org/article/Hyperchaos}}</ref>
=== Secure communications and cryptography === One of the most extensively studied applications of hyperchaos is in secure communication and cryptography. Hyperchaotic signals have been used to mask information within broadband, noise-like waveforms, while message recovery is achieved through synchronization of identical hyperchaotic systems at the receiver. Compared with lower-dimensional chaotic systems, hyperchaotic systems generally offer increased resistance to phase-space reconstruction and parameter estimation attacks.<ref>{{cite journal |last=Smaoui |first=Nidhal |title=Secure communications based on the synchronization of two hyperchaotic Chen systems and two unified chaotic systems |journal=Communications in Nonlinear Science and Numerical Simulation |volume=16 |issue=9 |pages=3613–3623 |year=2011 |doi=10.1016/j.cnsns.2010.12.021}}</ref>
=== Electronic circuits and oscillators === Hyperchaos has been experimentally realized in electronic circuit implementations of nonlinear oscillators, enabling the study of high-dimensional chaotic dynamics and their robustness under physical constraints. Such circuit realizations are frequently used in demonstrations of hyperchaotic synchronization and communication schemes.<ref>{{cite journal |last1=Yu |first1=J. |last2=Li |first2=C. |last3=Zhang |first3=S. |title=A new hyperchaotic system and its circuit implementation |journal=Communications in Nonlinear Science and Numerical Simulation |volume=15 |issue=2 |pages=336–345 |year=2010 |doi=10.1016/j.cnsns.2009.04.029}}</ref>
=== Image encryption and random number generation === Hyperchaotic maps and flows have been applied in image encryption and pseudorandom number generator design. Their sensitivity to initial conditions and parameter variations supports effective permutation and diffusion mechanisms in digital images and signals, particularly in proposed chaos-based cryptographic schemes.<ref>{{cite journal |last1=Zhou |first1=M. |last2=Wang |first2=C. |last3=Sun |first3=J. |title=A novel image encryption scheme based on conservative hyper-chaotic system and closed-loop diffusion between blocks |journal=Signal Processing |volume=171 |article-number=107484 |year=2020 |doi=10.1016/j.sigpro.2020.107484 |bibcode=2020SigPr.17107484Z }}</ref>
=== Nonlinear systems and control === In physics and engineering, hyperchaotic systems are used as models of high-dimensional nonlinear system behavior and as test cases for synchronization, control, and stability analysis in complex dynamical networks.<ref name="LetellierRossler" />
== See also == * Chaos theory
== References == {{Reflist}}
Category:Chaotic maps Category:Nonlinear systems Category:Articles containing video clips