The phenomenon of stochastic bifurcation driven by the correlated non-Gaussian colored noise and the Gaussian white noise is investigated by the qualitative changes of steady states with the most probable phase portraits. To arrive at the Markovian approximation of the original non-Markovian stochastic process and derive the general approximate Fokker-Planck equation (FPE), we deal with the non-Gaussian colored noise and then adopt the uni¯ed colored noise approximation (UCNA). Subsequently, the theoretical equation concerning the most probable steady states is obtained by the maximum of the stationary probability density function (SPDF). The parameter of the uncorrelated additive noise intensity does enter the governing equation as a non-Markovian effect, which is in contrast to that of the uncorrelated Gaussian white noise case, where the parameter is absent from the governing equation, i.e., the most probable steady states are mainly controlled by the uncorrelated multiplicative noise. Additionally, in comparison with the deterministic counterpart, some peculiar bifurcation behaviors with regard to the most probable steady states induced by the correlation time of non-Gaussian colored noise, the noise intensity, and the non-Gaussian noise deviation parameter are discussed. Moreover, the symmetry of the stochastic bifurcation diagrams is destroyed when the correlation between noises is concerned. Furthermore, the feasibility and accuracy of the analytical predictions are verified compared with those of the Monte Carlo (MC) simulations of the original system.
相似文献Analytical approximation of heteroclinic bifurcation in a 3:1 subharmonic resonance is given in this paper. The system we consider that produces this bifurcation is a harmonically forced and self-excited nonlinear oscillator. This bifurcation mechanism, resulting from the disappearance of a stable slow flow limit cycle at the bifurcation point, gives rise to a synchronization phenomenon near the 3:1 resonance. The analytical approach used in this study is based on the collision criterion between the slow flow limit cycle and the three saddles involved in the bifurcation. The amplitudes of the 3:1 subharmonic response and of the slow flow limit cycle are approximated and the collision criterion is applied leading to an explicit analytical condition of heteroclinic connection. Numerical simulations are performed and compared to the analytical finding for validation.
相似文献Quarter car models of vehicles rolling on wavy roads lead to limit cycles of travel speed and acceleration with period doublings and bifurcation effects for appropriate driving force parameters. In case of narrow-banded road excitations, speed jumps occur, additionally. This has the consequence that the driving speed becomes turbulent. Bifurcation and jump effects vanish with growing vehicle damping. The same happens for increasing bandwidth of road excitations when, e.g., on flat highways there are no big road waves but only small noisy slope processes generated by rough road surfaces. The paper derives a new stability condition in mean. Numerical time integrations are stabilized by means of polar coordinates. Equivalently, Fourier series expansions are introduced in the angle domain. Phase portraits of travel speed and acceleration show new period-doublings of limit cycles when speed gets stuck before resonance. The paper extends these investigations to the stochastic case that road surfaces are random generated by filtered white noise. By means of Gaussian closure, a nonlinear mean speed equation is derived which includes the extreme cases of wavy roads and road noise.
相似文献This paper presents the bifurcation behaviors of a modified railway wheelset model to explore its instability mechanisms of hunting motion. Equivalent conicity data measured from China high-speed railway vehicle are used to modify the wheelset model. Firstly, the relationships between longitudinal stiffness, lateral stiffness, equivalent conicity and critical speed are taken into account by calculating the real parts of the eigenvalues of the Jacobian matrix and Hurwitz criterion for the corresponding linear model. Secondly, measured equivalent conicity data are fitted by a nonlinear function of the lateral displacement rather than are considered as a constant as usual. Nonlinear wheel–rail force function is used to describe the wheel–rail contact force. Based on these modifications, a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force is set up, and then, some instability mechanisms of China high-speed train vehicle are investigated based on Hopf bifurcation, fold (limit point) bifurcation of cycles, cusp bifurcation of cycles, Neimark–Sacker bifurcation of cycles and 1:1 resonance. In particular, fold bifurcation of cycles can produce a vast effect on the hunting motion of the modified wheelset model. One of the main reasons leading to hunting motion is due to the fold bifurcation structure of cycles, in which stable limit cycles and unstable limit cycles may coincide, and multiple nested limit cycles appear on a side of fold bifurcation curve of cycles. Unstable hunting motion mainly depends on the coexistence of equilibria and limit cycles and their positions; if the most outward limit cycle is stable, then the motion of high-speed vehicle should be safe in a reasonable range. Otherwise, if the initial values are chosen near the most outward unstable limit cycle or the system is perturbed by noises, the high-speed vehicle will take place unstable hunting motion and even lead to serious train derailment events. Therefore, in order to control hunting motions, it may be the easiest way in theory to guarantee the coexistence of the inner stable equilibrium and the most outward stable limit cycle in a wheelset system.
相似文献The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.
相似文献The three-dimensional Muthuswamy–Chua–Ginoux (MCG, for short) circuit system based on a thermistor is a generalization of the classical Muthuswamy–Chua circuit differential system. At present, there are only partial numerical simulations for the qualitative analysis of the MCG circuit system. In this work, we study local stability and Hopf bifurcations of the MCG circuit system depending on 8 parameters. The emerging of limit cycles under zero-Hopf bifurcation and Hopf bifurcation is investigated in detail by using the averaging method and the center manifolds theory, respectively. We provide sufficient conditions for a class of the circuit systems to have a prescribed number of limit cycles bifurcating from the zero-Hopf equilibria by making use of the third-order averaging method, as well as the methods of Gröbner basis and real solution classification from symbolic computation. Such algebraic analysis allows one to study the zero-Hopf bifurcation for any other differential system in dimension 3 or higher. After, the classical Hopf bifurcation of the circuit system is analyzed by computing the first three focus quantities near the Hopf equilibria. Some examples and numerical simulations are presented to verify the established theoretical results.
相似文献In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.
相似文献This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package.
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