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The damping term makes the Smalehorseshoe heteroclinic chaotic motion easier
1.  School of Basic Sciences for Aviation, Naval Aviation University, Yantai 264001, Shandong, China 
2.  Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China 
The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smalehorseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smalehorseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasiperiodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasiperiodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.
References:
[1] 
G. R. Abdollahzade1, M. Bayat, M. Shahidi, G. Domairry and M. Rostamian, Analysis of dynamic model of a structure with nonlinear damped behavior, International J. Engineering and Technology, 2 (2012), 160168. 
[2] 
S. R. Bishop and M. J. Clifford, Zones of chaotic behavior in the parametrically excited pendulum, J. Sound Vibration, 189 (1996), 142147. doi: 10.1006/jsvi.1996.0011. 
[3] 
C. Dou, J. Fan, C. Li, J. Cao and M. Gao, On discontinuous dynamics of a class of frictioninfluenced oscillators with nonlinear damping under bilateral rigid constraints, Mechanism and Machine Theory, 147 (2020). doi: 10.1016/j.mechmachtheory.2019.103750. 
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A. ElíasZúniga, Analytical solution of the damped HelmholtzDuffing equation, Appl. Math. Lett., 25 (2012), 23492353. doi: 10.1016/j.aml.2012.06.030. 
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T. S. Jang, H. Baek, H. S. Choi and Su nGuLee, A new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems, Mechanical Systems and Signal Processing, 25 (2011), 22192228. doi: 10.1016/j.ymssp.2011.01.012. 
[6] 
P. Kumar and S. Narayanan, Chaos and bifurcation analysis of stochastically excited discontinuous nonlinear oscillators, Nonlinear Dyn, 102 (2020), 927950. doi: 10.1007/s11071020059605. 
[7] 
D. Li and S. W. Shaw, The effects of nonlinear damping on degenerate parametric amplification, Nonlinear Dyn, 102 (2020), 24332452. 
[8] 
S. Li, S. Yang and W. Guo, Investigation on chaotic motion in hysteretic nonlinear suspension system with multifrequency excitationss, Mechanics Research Communications, 31 (2004), 229236. 
[9] 
G. Litak, G. SpuzSzpos, K. Szabelski and J. Warmiński, Vibration analysis of a selfexcited system with parametric forcing and nonlinear stiffness, Int. J. Bifurcation and Chaos, 9 (1999), 493504. doi: 10.1142/S021812749900033X. 
[10] 
H. E. Nusse and J. A. Yorke, Dynamics: Numerical Explorations, SpringerVerlag, New York, 1991. 
[11] 
M. Siewe Siewe, H. J. Cao and A. F. Sanjuán Miguel, Effect of nonlinear dissipation on the basin boundaries of a driven twowell RayleighDuffing oscillator, Chaos, Solitons and Fractal, 39 (2009), 10921099. doi: 10.1016/j.chaos.2007.05.007. 
[12] 
J. L. Trueba, J. Rams and A. F. Sanjuán Miguel, Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 22572267. doi: 10.1142/S0218127400001419. 
[13] 
B. Tang and M. J. Brennan, A comparison of the effects of nonlinear damping on the free vibration of a singledegree of dreedom system, J. Vibration and Acoustics, 134 (2012), 15. 
[14] 
Y. Ueda, Nonlinear Theory and Its Applications, IEICE, 5 (2014), 252258. 
[15] 
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2 of Texts in Applied Mathematics, SpringerVerlag, NewYork, NY, 1990. doi: 10.1007/9781475740677. 
[16] 
H. Youzera, A. Meftah Sid, N. Challamel and A. Tounsi, Nonlinear damping and forced vibration analysis of laminated composite beams, Composites: Part B, 43 (2012), 11471154. doi: 10.1016/j.compositesb.2012.01.008. 
[17] 
P. P. Zhou and H. J. Cao, The effect of symmetrybreaking on the parameterically excited pendulum, Chaos, Solitons and Fractals, 38 (2008), 590597. doi: 10.1016/j.chaos.2007.06.073. 
show all references
References:
[1] 
G. R. Abdollahzade1, M. Bayat, M. Shahidi, G. Domairry and M. Rostamian, Analysis of dynamic model of a structure with nonlinear damped behavior, International J. Engineering and Technology, 2 (2012), 160168. 
[2] 
S. R. Bishop and M. J. Clifford, Zones of chaotic behavior in the parametrically excited pendulum, J. Sound Vibration, 189 (1996), 142147. doi: 10.1006/jsvi.1996.0011. 
[3] 
C. Dou, J. Fan, C. Li, J. Cao and M. Gao, On discontinuous dynamics of a class of frictioninfluenced oscillators with nonlinear damping under bilateral rigid constraints, Mechanism and Machine Theory, 147 (2020). doi: 10.1016/j.mechmachtheory.2019.103750. 
[4] 
A. ElíasZúniga, Analytical solution of the damped HelmholtzDuffing equation, Appl. Math. Lett., 25 (2012), 23492353. doi: 10.1016/j.aml.2012.06.030. 
[5] 
T. S. Jang, H. Baek, H. S. Choi and Su nGuLee, A new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems, Mechanical Systems and Signal Processing, 25 (2011), 22192228. doi: 10.1016/j.ymssp.2011.01.012. 
[6] 
P. Kumar and S. Narayanan, Chaos and bifurcation analysis of stochastically excited discontinuous nonlinear oscillators, Nonlinear Dyn, 102 (2020), 927950. doi: 10.1007/s11071020059605. 
[7] 
D. Li and S. W. Shaw, The effects of nonlinear damping on degenerate parametric amplification, Nonlinear Dyn, 102 (2020), 24332452. 
[8] 
S. Li, S. Yang and W. Guo, Investigation on chaotic motion in hysteretic nonlinear suspension system with multifrequency excitationss, Mechanics Research Communications, 31 (2004), 229236. 
[9] 
G. Litak, G. SpuzSzpos, K. Szabelski and J. Warmiński, Vibration analysis of a selfexcited system with parametric forcing and nonlinear stiffness, Int. J. Bifurcation and Chaos, 9 (1999), 493504. doi: 10.1142/S021812749900033X. 
[10] 
H. E. Nusse and J. A. Yorke, Dynamics: Numerical Explorations, SpringerVerlag, New York, 1991. 
[11] 
M. Siewe Siewe, H. J. Cao and A. F. Sanjuán Miguel, Effect of nonlinear dissipation on the basin boundaries of a driven twowell RayleighDuffing oscillator, Chaos, Solitons and Fractal, 39 (2009), 10921099. doi: 10.1016/j.chaos.2007.05.007. 
[12] 
J. L. Trueba, J. Rams and A. F. Sanjuán Miguel, Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 22572267. doi: 10.1142/S0218127400001419. 
[13] 
B. Tang and M. J. Brennan, A comparison of the effects of nonlinear damping on the free vibration of a singledegree of dreedom system, J. Vibration and Acoustics, 134 (2012), 15. 
[14] 
Y. Ueda, Nonlinear Theory and Its Applications, IEICE, 5 (2014), 252258. 
[15] 
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2 of Texts in Applied Mathematics, SpringerVerlag, NewYork, NY, 1990. doi: 10.1007/9781475740677. 
[16] 
H. Youzera, A. Meftah Sid, N. Challamel and A. Tounsi, Nonlinear damping and forced vibration analysis of laminated composite beams, Composites: Part B, 43 (2012), 11471154. doi: 10.1016/j.compositesb.2012.01.008. 
[17] 
P. P. Zhou and H. J. Cao, The effect of symmetrybreaking on the parameterically excited pendulum, Chaos, Solitons and Fractals, 38 (2008), 590597. doi: 10.1016/j.chaos.2007.06.073. 
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