High-Dimensional Dynamics in the Delayed Henon Map

J. C. Sprott
Departments of Physics, University of Wisconsin, Madison, WI 53706, USA

Received 3 April 2006, Accepted 16 August 2006, Published 20 September 2006

ABSTRACT

A variant of the Henon map is described in which the linear term is replaced by one that involves a much earlier iterate of the map. By varying the time delay, this map can be used to explore the transition from low-dimensional to high-dimensional dynamics in a chaotic system with minimal algebraic complexity, including a detailed comparison of the Kaplan-Yorke and correlation dimensions. The high-dimensional limit exhibits universal features that may characterize a wide range of complex systems including the spawning of multiple coexisting attractors near the onset of chaos.

Ref: J. C. Sprott, Electronic Journal of Theoretical Physics 3, 19-35 (2006)

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Fig. 1. Regions of dynamical behaviors for Eq. (1) for various values of the time delay.
Fig. 1.

Fig. 2. Attractors for the system in Eq. (1) with a = 1.6 and b = 0.1 for various values of the time delay.
Fig. 2.


Fig. 3. Kaplan-Yorke dimension and Lyapunov exponents for the system in Eq. (1) with a = 1.6 and b = 0.1 versus time delay.
Fig. 3.


Fig. 4. Kaplan-Yorke dimension and Lyapunov exponents for the system in Eq. (1) with b = 0.1 showing the route to chaos at low dimension (d = 2) and high dimension (d = 100).
Fig. 4.

Fig. 5. Kaplan-Yorke dimension and a few of the largest Lyapunov exponents for the system in Eq. (1) with b = 0.1 and d = 100 showing in more detail the onset of chaos.
Fig. 5.

Fig. 6. Attractors for the system in Eq. (1) with b = 0.1 and d = 100 showing period doubling of a drift ring approaching the onset of chaos.
Fig. 6.

Fig. 7. Global bifurcations and multiple attractors for two values of b with d = 100.
Fig. 7.

Fig. 8. Relative probability of different values of <r2> for a = 0.7, b = 0.3, and d = 100, indicating the existence of at least seven distinct attractors.
Fig. 8.

Fig. 9. Four coexisting attractors for a = 0.7, b = 0.3, and d = 100 near the onset of chaos.
Fig. 9.