Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic ...Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.展开更多
The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out tha...The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.展开更多
The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotati...The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotating rectangular coordinate system. Transforming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable, the equation of motion in complex variable form is obtained, in which the stiffness coefficient varies periodically with time. It presents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales (MMS) the averaged equation, the bifurcating response equations and local bifurcating set are obtained. Via the theory of singularity, the stability of constant solutions is analyzed and bifurcating response curves are obtained. This study shows that the rotating shaft has rich bifurcation phenomena.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
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
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
One of the basic problems in bifurcation theory is to understand the way in whichhorseshoes are created. In this paper, we study the bifurcation behavior exhibited by the toral Vander Pol equation subject to periodic ...One of the basic problems in bifurcation theory is to understand the way in whichhorseshoes are created. In this paper, we study the bifurcation behavior exhibited by the toral Vander Pol equation subject to periodic forcing. Our attention is focased on routes relevant to horseshoestype chaos.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
Large deflection theory of thin anisotropic circular plates was used to analyze the bifurcation behavior and chaotic phenomena of a corrugated thin circular plate with combined transverse periodic excitation and an in...Large deflection theory of thin anisotropic circular plates was used to analyze the bifurcation behavior and chaotic phenomena of a corrugated thin circular plate with combined transverse periodic excitation and an in-plane static boundary load. The nonlinear dynamic equation for the corrugated plate was derived by employing Galerkin＇s technique. The critical conditions for occurrence of the homoclinic and subharmonic bifurcations as well as chaos were studied theoretically using the Melnikov function method. The chaotic motion was also simulated numerically using Maple, with the Poincar~ map and phase curve used to evaluate when chaotic motion appears. The results indicate some chaotic motion in the corrugated plate. The method is directly applicable to chaotic analysis of an isotropic circular plate.展开更多
文摘Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.
基金Supported by the National Natural Science Foundation of China
文摘The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.
基金This project is supported by National Key Project of China(No.PD9521901).
文摘The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotating rectangular coordinate system. Transforming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable, the equation of motion in complex variable form is obtained, in which the stiffness coefficient varies periodically with time. It presents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales (MMS) the averaged equation, the bifurcating response equations and local bifurcating set are obtained. Via the theory of singularity, the stability of constant solutions is analyzed and bifurcating response curves are obtained. This study shows that the rotating shaft has rich bifurcation phenomena.
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.
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.
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.
基金The project supported by National Natural Science Foundation of China
文摘One of the basic problems in bifurcation theory is to understand the way in whichhorseshoes are created. In this paper, we study the bifurcation behavior exhibited by the toral Vander Pol equation subject to periodic forcing. Our attention is focased on routes relevant to horseshoestype chaos.
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Large deflection theory of thin anisotropic circular plates was used to analyze the bifurcation behavior and chaotic phenomena of a corrugated thin circular plate with combined transverse periodic excitation and an in-plane static boundary load. The nonlinear dynamic equation for the corrugated plate was derived by employing Galerkin＇s technique. The critical conditions for occurrence of the homoclinic and subharmonic bifurcations as well as chaos were studied theoretically using the Melnikov function method. The chaotic motion was also simulated numerically using Maple, with the Poincar~ map and phase curve used to evaluate when chaotic motion appears. The results indicate some chaotic motion in the corrugated plate. The method is directly applicable to chaotic analysis of an isotropic circular plate.
