Hyperchaos

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. ^[1]

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. ^[2] One is a discrete-time system ("folded-towel map"):

Folded-towel map attractor, animated.

$${\displaystyle {\begin{aligned}&x_{t+1}=3.8x_{t}\left(1-x_{t}\right)-0.05\left(y_{t}+0.35\right)\left(1-2z_{t}\right),\\&y_{t+1}=0.1\left[\left(y_{t}+0.35\right)\left(1-2z_{t}\right)-1\right]\left(1-1.9x_{t}\right),\\&z_{t+1}=3.78z_{t}\left(1-z_{t}\right)+0.2y_{t}.\end{aligned}}}$$Another is a continuous-time system:$${\displaystyle {\begin{array}{ll}{\dot {x}}=-y-z,&{\dot {y}}=x+0.25y+w,\\{\dot {z}}=3+xz,&{\dot {w}}=-0.5z+0.05w.\end{array}}}$$More examples are found in. ^[3]

Experimental examples

Only a few experimental hyperchaotic behaviors have been identified.

Examples include in an electronic circuit, ^[4] in a NMR laser, ^[5] in a semiconductor system, ^[6] and in a chemical system. ^[7]

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. ^[8]

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. ^[9]

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. ^[10]

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. ^[11]

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. ^[8]

References

1. Letellier, Christophe; Rossler, Otto E. (2007-08-05). "Hyperchaos". Scholarpedia. 2 (8): 1936. Bibcode:2007SchpJ...2.1936L. doi: 10.4249/scholarpedia.1936 . ISSN   1941-6016.
2. Rossler, O.E. (April 1979). "An equation for hyperchaos" . Physics Letters A. 71 (2–3): 155–157. Bibcode:1979PhLA...71..155R. doi:10.1016/0375-9601(79)90150-6. ISSN   0375-9601.
3. Sprott, Julien C. (2010). "6.7: Hyperchaotic Systems". Elegant chaos : algebraically simple chaotic flows. New Jersey: World Scientific. ISBN   978-981-283-882-7. OCLC   670430585.
4. Matsumoto, T.; Chua, L.; Kobayashi, K. (November 1986). "Hyper chaos: Laboratory experiment and numerical confirmation". IEEE Transactions on Circuits and Systems. 33 (11): 1143–1147. Bibcode:1986ITCS...33.1143M. doi:10.1109/TCS.1986.1085862. ISSN   1558-1276.
5. Stoop, R.; Meier, P. F. (1988-05-01). "Evaluation of Lyapunov exponents and scaling functions from time series" . Journal of the Optical Society of America B. 5 (5): 1037. Bibcode:1988JOSAB...5.1037S. doi:10.1364/josab.5.001037. ISSN   0740-3224.
6. Stoop, R.; Peinke, J.; Parisi, J.; Röhricht, B.; Huebener, R. P. (1989-05-01). "A p-Ge semiconductor experiment showing chaos and hyperchaos" . Physica D: Nonlinear Phenomena. 35 (3): 425–435. Bibcode:1989PhyD...35..425S. doi:10.1016/0167-2789(89)90078-X. ISSN   0167-2789.
7. Eiswirth, M.; Kruel, Th. -M.; Ertl, G.; Schneider, F. W. (1992-05-29). "Hyperchaos in a chemical reaction" . Chemical Physics Letters. 193 (4): 305–310. Bibcode:1992CPL...193..305E. doi:10.1016/0009-2614(92)85672-W. ISSN   0009-2614.
8. Letellier, Christophe; Rössler, Otto E. (2007). "Hyperchaos". Scholarpedia . 2 (9): 1931. doi: 10.4249/scholarpedia.1931 (inactive 29 December 2025).
9. Smaoui, Nidhal (2011). "Secure communications based on the synchronization of two hyperchaotic Chen systems and two unified chaotic systems". Communications in Nonlinear Science and Numerical Simulation. 16 (9): 3613–3623. doi:10.1016/j.cnsns.2010.12.021.
10. Yu, J.; Li, C.; Zhang, S. (2010). "A new hyperchaotic system and its circuit implementation". Communications in Nonlinear Science and Numerical Simulation. 15 (2): 336–345. doi:10.1016/j.cnsns.2009.04.029.
11. Zhou, M.; Wang, C.; Sun, J. (2020). "A novel image encryption scheme based on conservative hyper-chaotic system and closed-loop diffusion between blocks". Signal Processing. 171 107484. Bibcode:2020SigPr.17107484Z. doi:10.1016/j.sigpro.2020.107484.