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.

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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.

In this article, we present a new accurate iterative and asymptotic method to construct analytical periodic solutions for a strongly nonlinear system, even if it is not Z2-symmetric. This method is applicable not only to a conservative system but also to a non-conservative system with a limit cycle response. Distinct from the general harmonic balance method, it depends on balancing a few trigonometric terms (at most five terms) in the energy equation of the nonlinear system. According to this iterative approach, the dynamic frequency is a trigonometric function that varies with time t, which represents the influence of derivatives of the higher harmonic terms in a compact form and leads to a significant reduction of calculation workload. Two examples were solved and numerical solutions are presented to illustrate the effectiveness and convenience of the method. Based on the present method, we also outline a modified energy balance method to further simplify the procedure of higher order computation. Finally, a nonlinear strength index is introduced to automatically identify the strength of nonlinearity and classify the suitable strategies.

In real systems, the unpredictable jump changes of the random environment can induce the critical transitions (CTs) between two non-adjacent states, which are more catastrophic. Taking an asymmetric Lévy-noise-induced tri-stable model with desirable, sub-desirable, and undesirable states as a prototype class of real systems, a prediction of the noise-induced CTs from the desirable state directly to the undesirable one is carried out. We first calculate the region that the current state of the given model is absorbed into the undesirable state based on the escape probability, which is named as the absorbed region. Then, a new concept of the parameter dependent basin of the unsafe regime (PDBUR) under the asymmetric Lévy noise is introduced. It is an efficient tool for approximately quantifying the ranges of the parameters, where the noise-induced CTs from the desirable state directly to the undesirable one may occur. More importantly, it may provide theoretical guidance for us to adopt some measures to avert a noise-induced catastrophic CT.

In this paper, the midspan deflection of a beam bridge with vehicles passing through the bridge successively is investigated. The midspan deflection can be modeled as the vibration trace of smooth-and-discontinuous (SD) oscillator by considering the mode of the first order and up-and-down vibration. The nonlinear behaviors of the established model are studied and presented. KAM (Kolmogorov–Arnold–Moser) structures on the Poincaré section are constructed for the driven system without dissipation with generic KAM curve and a series of resonant points and the surrounding island chains connected by chaotic orbits. Introducing a series of complete elliptic integrals of the first and the second kind, the response curves of the system are detected, to which the effects of parameters are revealed. The relevant dynamics is depicted under external excitation exhibiting period leading to chaos. The efficiency of the bifurcation diagrams obtained in this paper is demonstrated via numerical simulations.