Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function

Hang Zheng; Yonghui Xia

doi:10.3934/dcdsb.2021319

Article Contents

2022, Volume 27, Issue 11: 6353-6371. Doi: 10.3934/dcdsb.2021319

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Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function

Hang Zheng^1,2, and
Yonghui Xia^1, ,

1.
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004
2.
Department of Mathematics and Computer, Wuyi University, Wuyishan, 354300, China

^*Corresponding author: Yonghui Xia
^*Corresponding author: Yonghui Xia

Received: September 2021

Revised: November 2021

Early access: January 2022

Published: November 2022

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Abstract

In this paper, we study the chaotic behavior of a class of hybrid piecewise-smooth system incorporated into an impulsive effect (HPSS-IE) under a periodic perturbation. More precisely, we assume that the unperturbed system with a homoclinic orbit, it transversally jumps across the first switching manifold by an impulsive stimulation and continuously crosses the second switching manifold. Then the corresponding Melnikov-type function is derived. Based on the new Melnikov-type function, the bifurcation and chaotic threshold of the perturbed HPSS-IE are analyzed. Furthermore, numerical simulations are precisely demonstrated through a concrete example. The results indicate that it is an extension work of previous references.

Keywords:

Mathematics Subject Classification: Primary: 34A36, 34A38, 37D45; Secondary: 65P20.

Citation:

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Figure 1. Piecewise smooth homoclinic orbit of the unperturbed system (5)

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Figure 2. The stable and unstable manifolds of perturbed homoclinic orbit for system (4)

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Figure 3. The stable and unstable manifolds of perturbed homoclinic orbit for system (4)

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Figure 4. Threshold curves of chaos and parameters bifurcation diagram for system (22)

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Figure 5. The phase portraits and time history curves of system (22) where (a) the phase portraits of $(x,y)$, (b) time history curves of $(t,x)$, (c) time history curves of $(t,y)$. Taking $f_0 = 20$, $\omega = 1.2$, $\mu = 2$ and $\varepsilon = 0.01$

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Figure 6. The phase portraits and time history curves of system (22) where (a) the phase portraits of $(x,y)$, (b) time history curves of $(t,x)$, (c) time history curves of $(t,y)$. Taking $f_0 = 30$, $\omega = 1.2$, $\mu = 2$ and $\varepsilon = 0.01$

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Figure 7. The phase portraits, Poincaré section and time history curves of system (22), taking $f_0 = 40$, $\omega = 1.2$, $\mu = 2$ and $\varepsilon = 0.01$

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Figure 8. The phase portraits, Poincaré section and time history curves of system (22), taking $f_0 = 50$, $\omega = 1.2$, $\mu = 2$ and $\varepsilon = 0.01$

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