The chaotic motions of axial compressed nonlinear elastic beam subjected to transverse load were studied. The damping force in the system is nonlinear. Considering material and geometric nonlinearity, nonlinear govern...The chaotic motions of axial compressed nonlinear elastic beam subjected to transverse load were studied. The damping force in the system is nonlinear. Considering material and geometric nonlinearity, nonlinear governing equation of the system was derived. By use of nonlinear Galerkin method, differential dynamic system was set up. Melnikov method was used to analyze the characters of the system.The results showed that chaos may occur in the system when the load parameters P 0 and f satisfy some conditions. The zone of chaotic motion was belted. The route from subharmonic bifurcation to chaos was analyzed. The critical conditions that chaos occurs were determined.展开更多
A famous model, the chemical reaction-Brussel model with periodic force, is investigated.We study the regilar Hopf bifurcation and singular Hopf bifurcation from a basic equilibrium, and show the existence of the subh...A famous model, the chemical reaction-Brussel model with periodic force, is investigated.We study the regilar Hopf bifurcation and singular Hopf bifurcation from a basic equilibrium, and show the existence of the subharmonic solutions by using the averaging method and perturbed methods and bifurcation equations. By our analysis it can be shown that the homoclinic orbits do not occur, so we can conjecture that the harmonic oscillation can make successive subharmonic bifurcations, until a chaotic state ultimately develops. The results and methods in this paper are our first step in theoretically treating the transition to a chaotic state in the Brussel model and are appropriate to investigating the general nonlinear oscillation with periodic force.展开更多
Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for th...Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for the existence of subharmonic solutions of order m is obtained. An example is given in the end of the paper.展开更多
Ferroresonance is a complex and little known electrotechnical phenomenon. This lack of knowledge means that it is voluntarily considered responsible for a number of unexplained destructions or malfunctioning of equipm...Ferroresonance is a complex and little known electrotechnical phenomenon. This lack of knowledge means that it is voluntarily considered responsible for a number of unexplained destructions or malfunctioning of equipment. The mathematical framework most suited to the general study of this phenomenon is the bifurcation theory, the main tool of which is the continuation method. Nevertheless, the use of a continuation process is not devoid of difficulties. In fact, to continue the solutions isolats which are closed curves, it is necessary to know a solution belonging to this isolated curve (isolat) to initialise the continuation method. The principal contribution of this article is to develop an analytical method allowing systematic calculation of this initial solution for various periodic ferroresonant modes (fundamental, harmonic and subharmonic) appearing on nonlinear electric system. The approach proposed uses a problem formulation in the frequency domain. This method enables to directly determine the solution in steady state without computing of the transient state. When we apply this method to the single-phase ferroresonant circuits (series and parallels configurations), we could easily calculate an initial solution for each ferroresonant mode that can be established. Knowing this first solution, we show how to use this analytical approach in a continuation technique to find the other solutions. The totality of the obtained solutions is represented in a plane where the abscissa is the amplitude of the supply voltage and the ordinate the amplitude of the system’s state variable (flux or voltage). The curve thus obtained is called “bifurcation diagram”. We will be able to then obtain a synthetic knowledge of the possible behaviors of the two circuits and particularly the limits of the dangerous zones of the various periodic ferroresonant modes that may appear. General results related to the series ferroresonance and parallel ferroresonance, obtained numerically starting from the theoretical and real cases,展开更多
The dynamics behavior of tension bar with periodic tension velocity was presented. Melnikov method war used to study the dynamic system. The results show that material nonlinear may result in anomalous dynamics respon...The dynamics behavior of tension bar with periodic tension velocity was presented. Melnikov method war used to study the dynamic system. The results show that material nonlinear may result in anomalous dynamics response. The subharmonic bifurcation and chaos may occur in the determined system when the tension velocity exceeds the critical value.展开更多
Considering Peierls-Nabarro （P-N） force and viscous effect of material, the dynamic behavior of one-dimensional infinite metallic thin bar subjected to axially periodic load is investigated. Governing equation, whic...Considering Peierls-Nabarro （P-N） force and viscous effect of material, the dynamic behavior of one-dimensional infinite metallic thin bar subjected to axially periodic load is investigated. Governing equation, which is sine-Gordon type equation, is derived. By means of collective-coordinates, the partial equation can be reduced to ordinary differential dynamical system to describe motion of breather. Nonlinear dynamic analysis shows that the amplitude and frequency of P-N force would influence positions of hyperbolic saddle points and change subharmonic bifurcation point, while the path to chaos through odd subharmonic bifurcations remains. Several examples are taken to indicate the effects of amplitude and period of P-N force on the dynamical response of the bar. The simulation states that the area of chaos is half-infinite. This area increases along with enhancement of the amplitude of P-N force. And the frequency of P-N force has similar influence on the system.展开更多
文摘The chaotic motions of axial compressed nonlinear elastic beam subjected to transverse load were studied. The damping force in the system is nonlinear. Considering material and geometric nonlinearity, nonlinear governing equation of the system was derived. By use of nonlinear Galerkin method, differential dynamic system was set up. Melnikov method was used to analyze the characters of the system.The results showed that chaos may occur in the system when the load parameters P 0 and f satisfy some conditions. The zone of chaotic motion was belted. The route from subharmonic bifurcation to chaos was analyzed. The critical conditions that chaos occurs were determined.
文摘A famous model, the chemical reaction-Brussel model with periodic force, is investigated.We study the regilar Hopf bifurcation and singular Hopf bifurcation from a basic equilibrium, and show the existence of the subharmonic solutions by using the averaging method and perturbed methods and bifurcation equations. By our analysis it can be shown that the homoclinic orbits do not occur, so we can conjecture that the harmonic oscillation can make successive subharmonic bifurcations, until a chaotic state ultimately develops. The results and methods in this paper are our first step in theoretically treating the transition to a chaotic state in the Brussel model and are appropriate to investigating the general nonlinear oscillation with periodic force.
文摘Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for the existence of subharmonic solutions of order m is obtained. An example is given in the end of the paper.
文摘Ferroresonance is a complex and little known electrotechnical phenomenon. This lack of knowledge means that it is voluntarily considered responsible for a number of unexplained destructions or malfunctioning of equipment. The mathematical framework most suited to the general study of this phenomenon is the bifurcation theory, the main tool of which is the continuation method. Nevertheless, the use of a continuation process is not devoid of difficulties. In fact, to continue the solutions isolats which are closed curves, it is necessary to know a solution belonging to this isolated curve (isolat) to initialise the continuation method. The principal contribution of this article is to develop an analytical method allowing systematic calculation of this initial solution for various periodic ferroresonant modes (fundamental, harmonic and subharmonic) appearing on nonlinear electric system. The approach proposed uses a problem formulation in the frequency domain. This method enables to directly determine the solution in steady state without computing of the transient state. When we apply this method to the single-phase ferroresonant circuits (series and parallels configurations), we could easily calculate an initial solution for each ferroresonant mode that can be established. Knowing this first solution, we show how to use this analytical approach in a continuation technique to find the other solutions. The totality of the obtained solutions is represented in a plane where the abscissa is the amplitude of the supply voltage and the ordinate the amplitude of the system’s state variable (flux or voltage). The curve thus obtained is called “bifurcation diagram”. We will be able to then obtain a synthetic knowledge of the possible behaviors of the two circuits and particularly the limits of the dangerous zones of the various periodic ferroresonant modes that may appear. General results related to the series ferroresonance and parallel ferroresonance, obtained numerically starting from the theoretical and real cases,
文摘The dynamics behavior of tension bar with periodic tension velocity was presented. Melnikov method war used to study the dynamic system. The results show that material nonlinear may result in anomalous dynamics response. The subharmonic bifurcation and chaos may occur in the determined system when the tension velocity exceeds the critical value.
基金Project supported by the National Natural Science Foundation of China (Nos.10172063, 10672112) the Youth Science Foundation of Shanxi Province (No.20051004) the Youth Academic Leader Foundation of Shanxi Province
文摘Considering Peierls-Nabarro （P-N） force and viscous effect of material, the dynamic behavior of one-dimensional infinite metallic thin bar subjected to axially periodic load is investigated. Governing equation, which is sine-Gordon type equation, is derived. By means of collective-coordinates, the partial equation can be reduced to ordinary differential dynamical system to describe motion of breather. Nonlinear dynamic analysis shows that the amplitude and frequency of P-N force would influence positions of hyperbolic saddle points and change subharmonic bifurcation point, while the path to chaos through odd subharmonic bifurcations remains. Several examples are taken to indicate the effects of amplitude and period of P-N force on the dynamical response of the bar. The simulation states that the area of chaos is half-infinite. This area increases along with enhancement of the amplitude of P-N force. And the frequency of P-N force has similar influence on the system.
