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

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004

2. 

Department of Mathematics and Computer, Wuyi University, Wuyishan, 354300, China

*Corresponding author: Yonghui Xia

Received  September 2021 Revised  November 2021 Early access January 2022

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.

Citation: Hang Zheng, Yonghui Xia. Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021319
References:
[1]

S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., 257 (2015), 190-198.  doi: 10.1016/j.amc.2014.06.073.

[2]

J. Awrejcewicz and M. M. Holicke, Smooth and Non-smooth High Dimensional Chaos and the Melnikov-type Methods, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6542.

[3]

F. Battelli and M. Fečkan, On the chaotic behaviour of discontinuous systems, J. Dynam. Differential Equations, 23 (2011), 495-540.  doi: 10.1007/s10884-010-9197-7.

[4]

F. Battelli and M. Fečkan, Homoclinic trajectories in discontinuous systems, J. Dyn. Diff. Eqs., 20 (2008), 337-376.  doi: 10.1007/s10884-007-9087-9.

[5]

F. Battelli and M. Fečkan, Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems, Physica D, 241 (2012), 1962-1975.  doi: 10.1016/j.physd.2011.05.018.

[6]

F. Battelli and M. Fečkan, Bifurcation and chaos near sliding homoclinics, J. Diff. Eqs., 248 (2010), 2227-2262.  doi: 10.1016/j.jde.2009.11.003.

[7]

N. BlackbeardH. Erzgräber and S. Wieczorek, Shear-induced bifurcations and chaos in models of three coupled lasers, SIAM J. Appl. Dyn. Syst., 10 (2011), 469-509.  doi: 10.1137/100817383.

[8]

B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer-Verlag, Ltd., London, 1996. doi: 10.1007/978-1-4471-0557-2.

[9]

Q. J. CaoM. WiercigrochE. E. PavlovskaiaJ. M. T. Thompson and C. Grebogi, Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics, Philos. Trans. R. Soc. A, 366 (2008), 635-652.  doi: 10.1098/rsta.2007.2115.

[10]

V. ColaoL. Muglia and H. K. Xu, An existence result for a new class of impulsive functional differential equations with delay, J. Math. Anal. Appl., 441 (2016), 668-683.  doi: 10.1016/j.jmaa.2016.04.024.

[11]

Z. D. Du and W. N. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50 (2005), 445-458.  doi: 10.1016/j.camwa.2005.03.007.

[12]

M. Fečkan, Bifurcation of periodic solutions in differential inclusions, Appl. Math., 42 (1997), 369-393.  doi: 10.1023/A:1023010108956.

[13]

M. FečkanJ. R. Wang and Y. Zhou, Existence of periodic solutions for nonlinear evolution equations with non-instantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101. 

[14]

B. F. Feeny and F. C. Moon, Empirical dry-friction modeling in a forced oscillator using chaos, Nonlinear Dynam., 47 (2007), 129-141.  doi: 10.1007/s11071-006-9065-5.

[15]

U. Galvanetto and C. Knudsen, Event maps in a stick-slip system, Nonlinear Dynam., 13 (1997), 90-115.  doi: 10.1023/A:1008228120608.

[16]

G. R. Gautam and J. Dabas, Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses, Appl. Math. Comput., 259 (2015), 480-489.  doi: 10.1016/j.amc.2015.02.069.

[17]

A. GranadosS. J. Hogan and T. M. Seara, The Melnikov method and subharmonic orbits in a piecewise-smooth system, SIAM J. Appl. Dyn. Syst., 11 (2012), 801-830.  doi: 10.1137/110850359.

[18]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Sprigner-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1140-2.

[19]

Y. X. GuoW. H. Jiang and B. Niu, Bifurcation analysis in the control of chaos by extended delay feedback, J. Franklin Inst., 350 (2013), 155-170.  doi: 10.1016/j.jfranklin.2012.10.009.

[20]

E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2.

[21]

E. HernándezM. Pierri and D. O'Regan, On abstract differential equations with non instantaneous impulses, Topol. Methods Nonlinear Anal., 46 (2015), 1067-1088. 

[22]

P. Kukučka, Jumps of the fundamental solution matrix in discontinuous systems and applications, Nonlinear Anal., 66 (2007), 2529-2546.  doi: 10.1016/j.na.2006.03.037.

[23]

P. Kukučka, Melnikov method for discontinuous planar systems, Nonlinear Anal., 66 (2007), 2698-2719.  doi: 10.1016/j.na.2006.04.001.

[24]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.

[25]

M. Kunze and T. Küpper, Non-smooth dynamical systems: An overview, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, (eds. B. Fiedler), Springer, Berlin, (2001), 431–452. doi: 10.1007/978-3-642-56589-2_19.

[26]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bif. Chaos, 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.

[27]

S. B. LiS. ChaoW. Zhang and Y. X. Hao, The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewisesmooth systems and application, Nonlinear Dynam., 85 (2016), 1091-1104.  doi: 10.1007/s11071-016-2746-9.

[28]

S. B. LiX. J. GongW. Zhang and Y. X. Hao, The Melnikov Method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold, Nonlinear Dynam., 89 (2017), 939-953.  doi: 10.1007/s11071-017-3493-2.

[29]

S. B. Li, W. S. Ma, W. Zhang and Y. X. Hao, Melnikov method for a three-zonal planar hybrid piecewise-smooth system and application, Internat. J. Bifur. Chaos, 26 (2016), 1650014, 13 pp. doi: 10.1142/S0218127416500140.

[30]

S. B. LiT. T. Wang and X. L. Bian, Global dynamics for a class of new bistable nonlinear oscillators with bilateral elastic collisions, Int. J. Dyn. Control, 9 (2021), 885-900.  doi: 10.1007/s40435-020-00733-9.

[31]

S. B. LiW. Zhang and L. J. Gao, Perturbation analysis in parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities, Nonlinear Dynam., 71 (2013), 175-185.  doi: 10.1007/s11071-012-0649-y.

[32]

S. B. Li, W. Zhang and Y. X. Hao, Melnikov-type method for a class of discontinuous planar systems and applications, Internat. J. Bifur. Chaos, 24 (2014), 1450022, 10 pp. doi: 10.1142/S0218127414500229.

[33]

Y. Li and Z. C. Feng, Bifurcation and chaos in friction-induced vibration, ?International Mechanical Engineering Congress and Exposition, (2008), 471–479. doi: 10.1115/IMECE2002-32811.

[34]

J. LlibreE. Ponce and A. E. Teruel, Horseshoes near homoclinic orbits for piecewise linear differential systems in $\mathbb{R}^3$, Int. J. Bifur. Chaos, 17 (2007), 1171-1184.  doi: 10.1142/S0218127407017756.

[35]

V. Melnikov, On the stability of the center for time-periodic perturbations, TTrudy Moskov. Mat. Obš č, 12 (1963), 3-52. 

[36]

B. Niu and W. H. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, J. Math. Anal. Appl., 398 (2013), 362-371.  doi: 10.1016/j.jmaa.2012.08.051.

[37]

M. PierriD. O'Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219 (2013), 6743-6749.  doi: 10.1016/j.amc.2012.12.084.

[38]

R. L. Tian, Y. F. Zhou, Q. B. Wang and L. L. Zhang, Bifurcation and chaotic threshold of Duffing system with jump discontinuities, Eur. Phys. J. Plus, 131 (2016). doi: 10.1140/epjp/i2016-16015-9.

[39]

J. R. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., 73 (2017), 157-162.  doi: 10.1016/j.aml.2017.04.010.

[40]

Q. D. Wang, Periodically forced double homoclinic loops to a dissipative saddle, J. Differential Equations, 260 (2016), 4366-4392.  doi: 10.1016/j.jde.2015.11.011.

[41]

J. R. Wang and M. Fečkan, A general class of impulsive evolution equations, Topol. Methods Nonlinear Anal., 46 (2015), 915-933. 

[42]

J. R. Wang, M. M. Li, D. O'Regan and M. Fečkan, Robustness for linear evolution equations with non-instantaneous impulsive effects, Bull. Sci. Math., 159 (2020), 102827, 47 pp. doi: 10.1016/j.bulsci.2019.102827.

[43]

Z. C. Wei, I. Moroz, J. C. Sprott, A. Akgul and W. Zhang, Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolardisc dynamo, Chaos, 27 (2017), 033101, 10 pp. doi: 10.1063/1.4977417.

[44]

Z. C. WeiP. YuW. Zhang and M. H. Yao, Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system, Nonlinear Dynam., 82 (2015), 131-141.  doi: 10.1007/s11071-015-2144-8.

[45]

Z. C. Wei and W. Zhang, Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium, Internat. J. Bifur. Chaos, 24 (2014), 1450127.  doi: 10.1142/S0218127414501272.

[46]

Z. C. WeiW. ZhangZ. Wang and M. H. Yao, Hidden attractors and dynamical behaviors in an extended Rikitake system, Internat. J. Bifur. Chaos, 25 (2015), 1550028.  doi: 10.1142/S0218127415500285.

[47]

Z. C. WeiW. Zhang and M. H. Yao, On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system, Nonlinear Dynam., 82 (2015), 1251-1258.  doi: 10.1007/s11071-015-2230-y.

[48]

S. Wieczorek and W. W. Chow, Chaos in practically isolated microcavity lasers, Phys. Rev. Lett., 92 (2004), 213901.  doi: 10.1103/PhysRevLett.92.213901.

[49]

S. WieczorekB. Krauskopf and D. Lenstra, Sudden chaotic transitions in an optically injected semiconductor laser, Opt. Lett., 26 (2001), 816-818.  doi: 10.1364/OL.26.000816.

[50]

S. Wiggins, Introduction To Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/b97481.

[51]

P. YangJ. R. Wang and M. Fečkan, Boundedness, periodicity, and conditional stability of noninstantaneous impulsive evolution equations, Math. Meth. Appl. Sci., 43 (2020), 5905-5926.  doi: 10.1002/mma.6332.

[52]

W. XuJ. Q. Feng and H. W. Rong, Melnikov's method for a general nonlinear vibro-impact oscillator, Nonlinear Anal., 71 (2009), 418-426.  doi: 10.1016/j.na.2008.10.120.

show all references

References:
[1]

S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., 257 (2015), 190-198.  doi: 10.1016/j.amc.2014.06.073.

[2]

J. Awrejcewicz and M. M. Holicke, Smooth and Non-smooth High Dimensional Chaos and the Melnikov-type Methods, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6542.

[3]

F. Battelli and M. Fečkan, On the chaotic behaviour of discontinuous systems, J. Dynam. Differential Equations, 23 (2011), 495-540.  doi: 10.1007/s10884-010-9197-7.

[4]

F. Battelli and M. Fečkan, Homoclinic trajectories in discontinuous systems, J. Dyn. Diff. Eqs., 20 (2008), 337-376.  doi: 10.1007/s10884-007-9087-9.

[5]

F. Battelli and M. Fečkan, Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems, Physica D, 241 (2012), 1962-1975.  doi: 10.1016/j.physd.2011.05.018.

[6]

F. Battelli and M. Fečkan, Bifurcation and chaos near sliding homoclinics, J. Diff. Eqs., 248 (2010), 2227-2262.  doi: 10.1016/j.jde.2009.11.003.

[7]

N. BlackbeardH. Erzgräber and S. Wieczorek, Shear-induced bifurcations and chaos in models of three coupled lasers, SIAM J. Appl. Dyn. Syst., 10 (2011), 469-509.  doi: 10.1137/100817383.

[8]

B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer-Verlag, Ltd., London, 1996. doi: 10.1007/978-1-4471-0557-2.

[9]

Q. J. CaoM. WiercigrochE. E. PavlovskaiaJ. M. T. Thompson and C. Grebogi, Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics, Philos. Trans. R. Soc. A, 366 (2008), 635-652.  doi: 10.1098/rsta.2007.2115.

[10]

V. ColaoL. Muglia and H. K. Xu, An existence result for a new class of impulsive functional differential equations with delay, J. Math. Anal. Appl., 441 (2016), 668-683.  doi: 10.1016/j.jmaa.2016.04.024.

[11]

Z. D. Du and W. N. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50 (2005), 445-458.  doi: 10.1016/j.camwa.2005.03.007.

[12]

M. Fečkan, Bifurcation of periodic solutions in differential inclusions, Appl. Math., 42 (1997), 369-393.  doi: 10.1023/A:1023010108956.

[13]

M. FečkanJ. R. Wang and Y. Zhou, Existence of periodic solutions for nonlinear evolution equations with non-instantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101. 

[14]

B. F. Feeny and F. C. Moon, Empirical dry-friction modeling in a forced oscillator using chaos, Nonlinear Dynam., 47 (2007), 129-141.  doi: 10.1007/s11071-006-9065-5.

[15]

U. Galvanetto and C. Knudsen, Event maps in a stick-slip system, Nonlinear Dynam., 13 (1997), 90-115.  doi: 10.1023/A:1008228120608.

[16]

G. R. Gautam and J. Dabas, Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses, Appl. Math. Comput., 259 (2015), 480-489.  doi: 10.1016/j.amc.2015.02.069.

[17]

A. GranadosS. J. Hogan and T. M. Seara, The Melnikov method and subharmonic orbits in a piecewise-smooth system, SIAM J. Appl. Dyn. Syst., 11 (2012), 801-830.  doi: 10.1137/110850359.

[18]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Sprigner-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1140-2.

[19]

Y. X. GuoW. H. Jiang and B. Niu, Bifurcation analysis in the control of chaos by extended delay feedback, J. Franklin Inst., 350 (2013), 155-170.  doi: 10.1016/j.jfranklin.2012.10.009.

[20]

E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2.

[21]

E. HernándezM. Pierri and D. O'Regan, On abstract differential equations with non instantaneous impulses, Topol. Methods Nonlinear Anal., 46 (2015), 1067-1088. 

[22]

P. Kukučka, Jumps of the fundamental solution matrix in discontinuous systems and applications, Nonlinear Anal., 66 (2007), 2529-2546.  doi: 10.1016/j.na.2006.03.037.

[23]

P. Kukučka, Melnikov method for discontinuous planar systems, Nonlinear Anal., 66 (2007), 2698-2719.  doi: 10.1016/j.na.2006.04.001.

[24]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.

[25]

M. Kunze and T. Küpper, Non-smooth dynamical systems: An overview, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, (eds. B. Fiedler), Springer, Berlin, (2001), 431–452. doi: 10.1007/978-3-642-56589-2_19.

[26]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bif. Chaos, 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.

[27]

S. B. LiS. ChaoW. Zhang and Y. X. Hao, The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewisesmooth systems and application, Nonlinear Dynam., 85 (2016), 1091-1104.  doi: 10.1007/s11071-016-2746-9.

[28]

S. B. LiX. J. GongW. Zhang and Y. X. Hao, The Melnikov Method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold, Nonlinear Dynam., 89 (2017), 939-953.  doi: 10.1007/s11071-017-3493-2.

[29]

S. B. Li, W. S. Ma, W. Zhang and Y. X. Hao, Melnikov method for a three-zonal planar hybrid piecewise-smooth system and application, Internat. J. Bifur. Chaos, 26 (2016), 1650014, 13 pp. doi: 10.1142/S0218127416500140.

[30]

S. B. LiT. T. Wang and X. L. Bian, Global dynamics for a class of new bistable nonlinear oscillators with bilateral elastic collisions, Int. J. Dyn. Control, 9 (2021), 885-900.  doi: 10.1007/s40435-020-00733-9.

[31]

S. B. LiW. Zhang and L. J. Gao, Perturbation analysis in parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities, Nonlinear Dynam., 71 (2013), 175-185.  doi: 10.1007/s11071-012-0649-y.

[32]

S. B. Li, W. Zhang and Y. X. Hao, Melnikov-type method for a class of discontinuous planar systems and applications, Internat. J. Bifur. Chaos, 24 (2014), 1450022, 10 pp. doi: 10.1142/S0218127414500229.

[33]

Y. Li and Z. C. Feng, Bifurcation and chaos in friction-induced vibration, ?International Mechanical Engineering Congress and Exposition, (2008), 471–479. doi: 10.1115/IMECE2002-32811.

[34]

J. LlibreE. Ponce and A. E. Teruel, Horseshoes near homoclinic orbits for piecewise linear differential systems in $\mathbb{R}^3$, Int. J. Bifur. Chaos, 17 (2007), 1171-1184.  doi: 10.1142/S0218127407017756.

[35]

V. Melnikov, On the stability of the center for time-periodic perturbations, TTrudy Moskov. Mat. Obš č, 12 (1963), 3-52. 

[36]

B. Niu and W. H. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, J. Math. Anal. Appl., 398 (2013), 362-371.  doi: 10.1016/j.jmaa.2012.08.051.

[37]

M. PierriD. O'Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219 (2013), 6743-6749.  doi: 10.1016/j.amc.2012.12.084.

[38]

R. L. Tian, Y. F. Zhou, Q. B. Wang and L. L. Zhang, Bifurcation and chaotic threshold of Duffing system with jump discontinuities, Eur. Phys. J. Plus, 131 (2016). doi: 10.1140/epjp/i2016-16015-9.

[39]

J. R. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., 73 (2017), 157-162.  doi: 10.1016/j.aml.2017.04.010.

[40]

Q. D. Wang, Periodically forced double homoclinic loops to a dissipative saddle, J. Differential Equations, 260 (2016), 4366-4392.  doi: 10.1016/j.jde.2015.11.011.

[41]

J. R. Wang and M. Fečkan, A general class of impulsive evolution equations, Topol. Methods Nonlinear Anal., 46 (2015), 915-933. 

[42]

J. R. Wang, M. M. Li, D. O'Regan and M. Fečkan, Robustness for linear evolution equations with non-instantaneous impulsive effects, Bull. Sci. Math., 159 (2020), 102827, 47 pp. doi: 10.1016/j.bulsci.2019.102827.

[43]

Z. C. Wei, I. Moroz, J. C. Sprott, A. Akgul and W. Zhang, Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolardisc dynamo, Chaos, 27 (2017), 033101, 10 pp. doi: 10.1063/1.4977417.

[44]

Z. C. WeiP. YuW. Zhang and M. H. Yao, Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system, Nonlinear Dynam., 82 (2015), 131-141.  doi: 10.1007/s11071-015-2144-8.

[45]

Z. C. Wei and W. Zhang, Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium, Internat. J. Bifur. Chaos, 24 (2014), 1450127.  doi: 10.1142/S0218127414501272.

[46]

Z. C. WeiW. ZhangZ. Wang and M. H. Yao, Hidden attractors and dynamical behaviors in an extended Rikitake system, Internat. J. Bifur. Chaos, 25 (2015), 1550028.  doi: 10.1142/S0218127415500285.

[47]

Z. C. WeiW. Zhang and M. H. Yao, On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system, Nonlinear Dynam., 82 (2015), 1251-1258.  doi: 10.1007/s11071-015-2230-y.

[48]

S. Wieczorek and W. W. Chow, Chaos in practically isolated microcavity lasers, Phys. Rev. Lett., 92 (2004), 213901.  doi: 10.1103/PhysRevLett.92.213901.

[49]

S. WieczorekB. Krauskopf and D. Lenstra, Sudden chaotic transitions in an optically injected semiconductor laser, Opt. Lett., 26 (2001), 816-818.  doi: 10.1364/OL.26.000816.

[50]

S. Wiggins, Introduction To Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/b97481.

[51]

P. YangJ. R. Wang and M. Fečkan, Boundedness, periodicity, and conditional stability of noninstantaneous impulsive evolution equations, Math. Meth. Appl. Sci., 43 (2020), 5905-5926.  doi: 10.1002/mma.6332.

[52]

W. XuJ. Q. Feng and H. W. Rong, Melnikov's method for a general nonlinear vibro-impact oscillator, Nonlinear Anal., 71 (2009), 418-426.  doi: 10.1016/j.na.2008.10.120.

Figure 1.  Piecewise smooth homoclinic orbit of the unperturbed system (5)
Figure 2.  The stable and unstable manifolds of perturbed homoclinic orbit for system (4)
Figure 3.  The stable and unstable manifolds of perturbed homoclinic orbit for system (4)
Figure 4.  Threshold curves of chaos and parameters bifurcation diagram for system (22)
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 $\varepsilon = 0.01$
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 $\varepsilon = 0.01$
Figure 7.  The phase portraits, Poincaré section and time history curves of system (22), taking $f_0 = 40$, $\omega = 1.2$, $\mu = 2$ and $\varepsilon = 0.01$
Figure 8.  The phase portraits, Poincaré section and time history curves of system (22), taking $f_0 = 50$, $\omega = 1.2$, $\mu = 2$ and $\varepsilon = 0.01$
[1]

Kazuyuki Yagasaki. Application of the subharmonic Melnikov method to piecewise-smooth systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2189-2209. doi: 10.3934/dcds.2013.33.2189

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