BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING

Zhiyan Yang; Tao Jiang; Zhujun Jing

doi:10.11948/2013030

2013 Volume 3 Issue 4

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Article navigation > Journal of Applied Analysis & Computation > 2013 > 3(4): 405-431

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Zhiyan Yang, Tao Jiang, Zhujun Jing. BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING[J]. Journal of Applied Analysis & Computation, 2013, 3(4): 405-431. doi: 10.11948/2013030

Citation:

Zhiyan Yang, Tao Jiang, Zhujun Jing. BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING[J]. Journal of Applied Analysis & Computation, 2013, 3(4): 405-431. doi: 10.11948/2013030

BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING

1 The Information School, Beijing Wuzi University, 101149 Beijing, China;
2 The center for Dynamical systems, Academy of Mathematical and system sciences, Chinese Academy of Sciences, Beijing, 100080, China

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Abstract

Abstract

The Duffing-Van der Pol equation with fifth nonlinear-restoring force and one external forcing term is investigated in detail:the existence and bifurcations of harmonic and second-order subharmonic, and third-order subharmonic, third-order superharmonic and m-order subharmonic under small perturbations are obtained by using second-order averaging method and subharmonic Melnikov function; the threshold values of existence of chaotic motion are obtained by using Melnikov method. The numerical simulation results including the influences of periodic and quasi-periodic and all parameters exhibit more new complex dynamical behaviors. We show that the reverse period-doubling bifurcation to chaos, period-doubling bifurcation to chaos, quasi-periodic orbits route to chaos, onset of chaos, and chaos suddenly disappearing, and chaos suddenly converting to period orbits, different chaotic regions with a great abundance of periodic windows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, and so on), and more wide period-one window, and varied chaotic attractors including small size and maximum Lyapunov exponent approximate to zero but positive, and the symmetry-breaking of periodic orbits. In particular, the system can leave chaotic region to periodic motion by adjusting the parameters p, β, γ, f and ω, which can be considered as a control strategy.
- Duffing-Van der Pol equation /
- bifurcations /
- periodic solutions /
- chaotic attractors /
- complex period windows
MSC: 37G15;37D45

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References

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