×

An equation for hyperchaos. (English) Zbl 0996.37502

The author investigates a four-variable oscillator that contains only one nonlinear quadratic term. By numerical simulation it is shown that the quadratic term produces chaos with two directions of hyperbolic instability on the attractor. He conjectures that in case this result is valid for systems of higher dimension than 4, turbulences may involve more than one direction of lateral instability, so turbulence may be hyperchaos of high order.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rossler, O. E., Z. Naturforsch., 31a, 1664 (1976)
[2] Rossler, O. E., Ann. NY Acad. Sci., 316, 376 (1979)
[3] Smale, S., Bull. Am. Math. Soc., 73, 747 (1967)
[4] Rossler, O. E., Chaotic oscillations - an example of hyperchaos, (Hoppensteadt, F. C., Nonlinear oscillations in biology. Nonlinear oscillations in biology, Lectures in Applied Mathematics, Vol. 17 (1979), AMS: AMS Providence, RI) · Zbl 0451.58024
[5] Rossler, O. E., Phys. Lett., 57A, 397 (1976)
[6] Rossler, O. E., Bull. Math. Biol., 39, 275 (1977)
[7] Rossler, O. E., Z. Naturforsch., 31a, 259 (1976)
[8] O.E. Rossler, I. Gumowski and C. Mira, in preparation.; O.E. Rossler, I. Gumowski and C. Mira, in preparation.
[9] Lorenz, E. N., J. Atmos. Sci., 20, 130 (1963)
[10] Haken, H., Phys. Lett., 53A, 77 (1975)
[11] Kuramoto, Y.; Yamada, T., Prog. Theor. Phys., 55, 679 (1976)
[12] Rossler, O. E., Z. Naturforsch., 31a, 1168 (1976)
[13] Rossler, O. E., Chaos, (Güttinger, W., Structural stability in physics. Structural stability in physics, Synergetics Series (1979), Springer: Springer Berlin) · Zbl 0372.58012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.