Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of themean value.Afterwards,the stochastic vibro-impact systemcan be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincarésections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the randomintensitymay result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.

Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model

Based on the chaotic geomagnetic field model, a non-smooth factor is introduced to explore complex dynamical behaviors of a system with multiple time scales. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamic behaviors. Due to the existence of non-smooth factors, different types of bifurcations are presented in spiking states, such as grazing-sliding bifurcation and across-sliding bifurcation. In addition, the non-smooth fold bifurcation may lead to the appearance of a special quiescent state in the interface as well as a non-smooth homoclinic bifurcation phenomenon. Due to these bifurcation behaviors, a special transition between spiking and quiescent state can also occur.

Generalized Hopf Bifurcation for Non-smooth Planar Dynamical Systems:the Corner Case

Piece-wise smooth systems are an important class of ordinary differential equations whosedynamics are known to exhibit complex bifurcation scenarios and chaos. Broadly speaking,piece-wise smooth systems can undergo all the bifurcation that smooth ones can. Moreinterestingly, there is a whole class of bifurcation that are unique to piece-wise smoothsystems, such as the bifurcation caused by the geometric shape of the region in which the

Non-smooth bifurcations in a double-scroll circuit with periodic excitation

The fast-slow effect can be observed in a typical non-smooth electric circuit with order gap between the natural frequency and the excitation frequency. Numerical simulations are employed to show complicated behaviours, especially different types of busting phenomena. The bifurcation mechanism for the bursting solutions is analysed by assuming the forms of the solutions and introducing the generalized Jacobian matrix at the non-smooth boundaries, which can also be used to account for the evolution of the complicated structures of the phase portraits with the variation of the parameter. Period-adding bifurcation has been explored through the computation of the eigenvalues related to the solutions. At the non-smooth boundaries the so-called ＇single crossing bifurcation＇ can occur, corresponding to the case where the eigenvalues jump only once across the imaginary axis, which leads the periodic burster to have a quasi-periodic oscillation.

Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles

This article describes the implementation of a novel method for detection and continuation of bifurcations in non-smooth complex dynamic systems. The method is an alternative to existing ones for the follow-up of associated phenomena, precisely in the circumstances in which the traditional ones have limitations (simultaneous impact, Filippov and first derivative discontinuities and multiple discontinuous boundaries). The topology of cycles in non-smooth systems is determined by a group of ordered segments and points of different regions and their boundaries. In this article, we compare the limit cycles of non-smooth systems against the sequences of elements, in order to find patterns. To achieve this goal, a method was used, which characterizes and records the elements comprising the cycles in the order that they appear during the integration process. The characterization discriminates: a) types of points and segments;b) direction of sliding segments;and c) regions or discontinuity boundaries to which each element belongs. When a change takes place in the value of a parameter of a system, our comparison method is an alternative to determine topological changes and hence bifurcations and associated phenomena. This comparison has been tested in systems with discontinuities of three types: 1) impact;2) Filippov and 3) first derivative discontinuities. By coding well-known cycles as sequences of elements, an initial comparison database was built. Our comparison method offers a convenient approach for large systems with more than two regions and more than two sliding segments.

Symmetric bursting behaviour in non-smooth Chua's circuit

The dynamics of a non-smooth electric circuit with an order gap between its parameters is investigated in this paper. Different types of symmetric bursting phenomena can be observed in numerical simulations. Their dynamical behaviours are discussed by means of slow-fast analysis. Furthermore, the generalized Jacobian matrix at the non-smooth boundaries is introduced to explore the bifurcation mechanism for the bursting solutions, which can also be used to account for the evolution of the complicated structures of the phase portraits. With the variation of the parameter, the periodic symmetric bursting can evolve into chaotic symmetric bursting via period-doubling bifurcation.

Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation

A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function （PDF） of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.

Modeling the dynamics of information propagation in the temporal and spatial environment

In this paper, we try to establish a non-smooth susceptible–infected–recovered(SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Secondly, we divide the system into two parts and discuss the existence of equilibrium points for each of them. For the left part, we define R_(0) to study the relationship between R_(0) and the existence of equilibrium points. For the right part, we classify many different cases by discussing the coefficients of the equilibrium point equation. Then, on this basis, we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation. Further, we consider the effect of diffusion. We specifically analyze the stability of equilibrium points. In addition, we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points. According to the Lyapunov number, we also determine the direction of the bifurcation. When I = I_(c), we discuss conditions for the existence of discontinuous Hopf bifurcation. Finally, through numerical simulations and combined with the practical meaning of the parameters, we prove the correctness of the previous theoretical theorem.