In this paper, the complicated nonlinear dynamics at the equilibria of SD oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter α, are investigated. It is found that SD oscillator admits codimension-two bifurcation at the trivial equilibrium when α = 1. The universal unfolding for the codimensiontwo bifurcation is also found to be equivalent to the damped SD oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped SD oscillator as the perturbation parameters vary.
Critical transitions from one dynamical state to another contrasting state are observed in many complex systems. To understand the effects of stochastic events on critical transitions and to predict their occurrence as a control parameter varies are of utmost importance in various applications. In this paper, we carry out a prediction of noise-induced critical transitions using a bistable model as a prototype class of real systems. We find that the largest Lyapunov exponent and the Shannon entropy can act as general early warning indicators to predict noise-induced critical transitions, even for an earlier transition due to strong fluctuations. Furthermore, the concept of the parameter dependent basin of the unsafe regime is introduced via incorporating a suitable probabilistic notion. We find that this is an efficient tool to approximately quantify the range of the control parameter where noise-induced critical transitions may occur. Our method may serve as a paradigm to understand and predict noise-induced critical transitions in multistable systems or complex networks and even may be extended to a broad range of disciplines to address the issues of resilience.
A kind of impulsive differential system is constructed by the use of the non-smooth pendulum which is composed of a rigid wall and a pendulum. The pendulum is subjected to different types of impulsive excitations, which lead to the non-smooth homoclinic orbits. Specifically, the existence of non-smooth homoclinic orbits depends on both the classical heteroclinic orbits and type II periodic orbits. When the pendulum moves to the highest point, an impact impulsive excitation is considered. While the orbits arrived at the lowest point, other types of impulsive excitations are introduced. Hence, these non-smooth homoclinic orbits hold two classes of jump discontinuities. One is related to the direction of velocity and the other administered by the magnitude of velocity. In order to illustrate the criteria for chaotic motion of this kind of system, the well-known Melnikov theory for the smooth system is extended applying the Hamiltonian function, which reveals the effects of these impulsive excitations on the behaviors of nonlinear dynamical systems. The efficiency of the criteria for bifurcation and chaos mentioned above is verified by the phase portrait, Poincaré surface of section, and bifurcation diagrams.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.