Generalization of the Simplest Autonomous Chaotic System

Buncha Munmuangsaena, Banlue Srisuchinwonga, J.C. Sprottb
aSirindhorn International Institute of Technology (SIIT), Thammasat University, Pathum-Thani 12000, Thailand
bDepartment of Physics, University of Wisconsin, Madison, WI 53706, USA

Received 19 October 2010, Received in revised form 7 February 2011, Accepted 12 February 2011, Available online 16 February 2011

ABSTRACT

An extensive numerical search of jerk systems of the form x''' + x'' + x = f (x') revealed many cases with chaotic solutions in addition to the one with f (x') = x'2 that has long been known. Particularly simple is the piecewise-linear case with f (x') = α(1− x') for x' > 1 and zero otherwise, which produces chaos even in the limit of α→∞. The dynamics in this limit can be calculated exactly, leading to a two-dimensional map. Such a nonlinearity suggests an elegant electronic circuit implementation using a single diode.

Ref: B. Munmuangsaen, B. Srisuchinwong, and J.C. Sprott, Phys. Lett. A 375, 1445-1450 (2011)

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Fig. 1. Attractors of Eq. (3) for each of the nonlinear functions in Table 1.
Figure 1

Fig. 2. The largest Lyapunov exponent and bifurcation diagram of Eq. (3) for f (x') = −A exp(x') with 0 < A < 0.5.
Figure 1

Fig. 3. Homoclinic orbit in Eq. (3) for f (x') = −A exp(x') with A = 0.1306.
Figure 3

Fig. 4. Attractor for the piecewise-linear system.
Figure 4

Fig. 5. Poincaré section at x' = 0 for the piecewise-linear system.
Figure 5

Fig. 6. Return map for the maximum value of x for the piecewise-linear system.
Figure 6