The effects of constant excitation on the recently proposed smooth-and-discontinuous(SD)oscillator are investigated,which may lead to the variation of equilibrium and the property of phase portrait.By solving a quarti...The effects of constant excitation on the recently proposed smooth-and-discontinuous(SD)oscillator are investigated,which may lead to the variation of equilibrium and the property of phase portrait.By solving a quartic algebraic equation,the transition set and bifurcation for SD oscillator under constant excitation(CSD)are presented,while the number of equilibria depends on the values of the smoothness parameter and the constant excitation.Complicated structures of Kolmogorov–Arnold–Moser(KAM)structures on the Poincarésection are depicted for the driven system without dissipation.Chaotic behaviour is also demonstrated numerically for the system perturbed by both viscous-damping and external excitation.The results show that CSD is an unsymmetrical system that displays different dynamical behaviours from an SD oscillator and will enrich the range of the SD oscillator in research and application.展开更多
Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the g...Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor＇s expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov＇s method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. （Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soe. A 366 635）.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos 11002093 and 11172183the Science and Technology Plan Project of Hebei Science and Technology Department(No 11215643).
文摘The effects of constant excitation on the recently proposed smooth-and-discontinuous(SD)oscillator are investigated,which may lead to the variation of equilibrium and the property of phase portrait.By solving a quartic algebraic equation,the transition set and bifurcation for SD oscillator under constant excitation(CSD)are presented,while the number of equilibria depends on the values of the smoothness parameter and the constant excitation.Complicated structures of Kolmogorov–Arnold–Moser(KAM)structures on the Poincarésection are depicted for the driven system without dissipation.Chaotic behaviour is also demonstrated numerically for the system perturbed by both viscous-damping and external excitation.The results show that CSD is an unsymmetrical system that displays different dynamical behaviours from an SD oscillator and will enrich the range of the SD oscillator in research and application.
基金supported by the National Natural Science Foundation of China (Grant Nos.11002093,11072065,and 10872136)the Science Foundation of the Science and Technology Department of Hebei Province of China (Grant No.11215643)
文摘Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor＇s expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov＇s method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. （Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soe. A 366 635）.
