Abstract
<p style='text-indent:20px;'>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 Smale-horseshoe 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 Smale-horseshoe 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, quasi-periodic 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 quasi-periodic 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.</p>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics
Reference17 articles.
1. G. R. Abdollahzade1, M. Bayat, M. Shahidi, G. Domairry, M. Rostamian.Analysis of dynamic model of a structure with nonlinear damped behavior, International J. Engineering and Technology, 2 (2012), 160-168.
2. S. R. Bishop, M. J. Clifford.Zones of chaotic behavior in the parametrically excited pendulum, J. Sound Vibration, 189 (1996), 142-147.
3. C. Dou, J. Fan, C. Li, J. Cao and M. Gao, On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints, Mechanism and Machine Theory, 147 (2020).
4. A. Elías-Zúniga.Analytical solution of the damped Helmholtz-Duffing equation, Appl. Math. Lett., 25 (2012), 2349-2353.
5. T. S. Jang, H. Baek, H. S. Choi, Su n-GuLee.A new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems, Mechanical Systems and Signal Processing, 25 (2011), 2219-2228.