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]

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. 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.