A simple,yet accurate modi?ed multi-scale method(MMSM)for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed.This method depends on the clas...A simple,yet accurate modi?ed multi-scale method(MMSM)for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed.This method depends on the classical multi-scale method(MSM)and the method of variation of parameters.Assuming that the forced excitation is a constant,one could easily obtain the approximate analytical solution of the simpli?ed system based on the traditional MSM.Then,this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation.To certify the correctness and precision of the proposed analytical method,the van der Pol system with two scales subject to slowly periodic excitation is investigated;this system presents rich dynamical phenomena such as spiking(SP),spiking-quiescence(SP-QS),and quiescence(QS)responses.The approximate analytical expressions of the three types of responses are given by the MMSM,and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method(HBM).The results obtained by the present method are considerably better than those obtained by traditional methods,quantitatively and qualitatively,particularly when the excitation frequency is far less than the natural frequency of the system.展开更多
Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper. The synchronization mechanism of “slow-manifold adjustment” is firstly described o...Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper. The synchronization mechanism of “slow-manifold adjustment” is firstly described on the basis of geometric singular perturbation theory. Then, the impact of time delay on the structure of the slow manifold of synchronized system is obtained by using the method of stability switch, and thus, time-delay effects on synchronization features are stated. It is shown the time delay cannot qualitatively affect the synchronization mechanism, however, it can result in the drift of the optimal coupling strength.展开更多
When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion “symmetry” plays an important role. It is of one parameter on slow vector field. Then, ...When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion “symmetry” plays an important role. It is of one parameter on slow vector field. Then, it should be determined to introduce parameters to all slow/fast vectors. It is, however, there might be no way to explore for another potential in this system, because the geometrical structure is quite different from the system with one parameter. Even in this system, the “symmetry” is also useful to obtain the potentials classified by R. Thom. In this paper, via the coordinates changing, the possible way to explore for the potential will be shown. As it is analyzed on “hyper finite time line”, or done by using “non-standard analysis”, it is called “Hyper Catastrophe”. In the slow-fast system which includes a very small parameter , it is difficult to do precise analysis. Thus, it is useful to get the orbits as a singular limit. When trying to do simulations, it is also faced with difficulty due to singularity. Using very small time intervals corresponding small , we shall overcome the difficulty, because the difference equation on the small time interval adopts the standard differential equation. These small intervals are defined on hyper finite number N, which is nonstandard. As and the intervals are linked to use 1/N, the simulation should be done exactly.展开更多
Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a linkrotor structure connected by a flexible joint or ...Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a linkrotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slowfast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.展开更多
This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bi...This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.展开更多
Based on the traditional scheme for a nonlinear system with multiple time scales, the enveloping slow-fast analysis method is developed in the paper, which can be employed to investigate the dynamics of nonlinear fiel...Based on the traditional scheme for a nonlinear system with multiple time scales, the enveloping slow-fast analysis method is developed in the paper, which can be employed to investigate the dynamics of nonlinear fields with multiple time scales with periodic excitation. Upon using the method, the behaviors of the kinetic model of CO oxidation on the platinum group metals have been explored in detail. Two typicM bursting phenomena such as Fold/Fold/Hopf bursting and Fold/Fold bursting, are presented, the bifurcation mechanisms of which have been obtained. Furthermore, the dynamic difference between the two cases corresponding to relatively large and small perturbation frequencies, respectively, has been presented, which can be used to describe the influence of the frequencies involving in the evolution on the bursting behaviors in the system.展开更多
A modified slow-fast analysis method is presented for the periodically excited non-autonomous dynamical system with an order gap between the exciting frequency and the natural frequency.By regarding the exciting term ...A modified slow-fast analysis method is presented for the periodically excited non-autonomous dynamical system with an order gap between the exciting frequency and the natural frequency.By regarding the exciting term as a slow-varying parameter,a generalized autonomous fast subsystem can be defined,the equilibrium branches as well as the bifurcations of which can be employed to account for the mechanism of the bursting oscillations by combining the transformed phase portrait introduced.As an example,a typical periodically excited Hartley model is used to demonstrate the validness of the method,in which the exciting frequency is far less than the natural frequency.The equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying parameter are presented.Bursting oscillations for two typical cases are considered,which reveals that,fold bifurcation may cause the the trajectory to jump between different equilibrium branches,while Hopf bifurcation may cause the trajectory to oscillate around the stable limit cycle.展开更多
There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility t...There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.展开更多
基金the National Natural Science Foundation of China(Nos.11672191,11772206,and U1934201)the Hundred Excellent Innovative Talents Support Program in Hebei University(No.SLRC2017053)。
文摘A simple,yet accurate modi?ed multi-scale method(MMSM)for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed.This method depends on the classical multi-scale method(MSM)and the method of variation of parameters.Assuming that the forced excitation is a constant,one could easily obtain the approximate analytical solution of the simpli?ed system based on the traditional MSM.Then,this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation.To certify the correctness and precision of the proposed analytical method,the van der Pol system with two scales subject to slowly periodic excitation is investigated;this system presents rich dynamical phenomena such as spiking(SP),spiking-quiescence(SP-QS),and quiescence(QS)responses.The approximate analytical expressions of the three types of responses are given by the MMSM,and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method(HBM).The results obtained by the present method are considerably better than those obtained by traditional methods,quantitatively and qualitatively,particularly when the excitation frequency is far less than the natural frequency of the system.
文摘Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper. The synchronization mechanism of “slow-manifold adjustment” is firstly described on the basis of geometric singular perturbation theory. Then, the impact of time delay on the structure of the slow manifold of synchronized system is obtained by using the method of stability switch, and thus, time-delay effects on synchronization features are stated. It is shown the time delay cannot qualitatively affect the synchronization mechanism, however, it can result in the drift of the optimal coupling strength.
文摘When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion “symmetry” plays an important role. It is of one parameter on slow vector field. Then, it should be determined to introduce parameters to all slow/fast vectors. It is, however, there might be no way to explore for another potential in this system, because the geometrical structure is quite different from the system with one parameter. Even in this system, the “symmetry” is also useful to obtain the potentials classified by R. Thom. In this paper, via the coordinates changing, the possible way to explore for the potential will be shown. As it is analyzed on “hyper finite time line”, or done by using “non-standard analysis”, it is called “Hyper Catastrophe”. In the slow-fast system which includes a very small parameter , it is difficult to do precise analysis. Thus, it is useful to get the orbits as a singular limit. When trying to do simulations, it is also faced with difficulty due to singularity. Using very small time intervals corresponding small , we shall overcome the difficulty, because the difference equation on the small time interval adopts the standard differential equation. These small intervals are defined on hyper finite number N, which is nonstandard. As and the intervals are linked to use 1/N, the simulation should be done exactly.
基金supported by the National Natural Science Foundation of China(11032009 and 11272236)
文摘Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a linkrotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slowfast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.
基金supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No.200430)in part by the National Natural Science Foundation of China (Grant Nos.10825207,10532050)
文摘This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 20976075,20976041,and 10972091)
文摘Based on the traditional scheme for a nonlinear system with multiple time scales, the enveloping slow-fast analysis method is developed in the paper, which can be employed to investigate the dynamics of nonlinear fields with multiple time scales with periodic excitation. Upon using the method, the behaviors of the kinetic model of CO oxidation on the platinum group metals have been explored in detail. Two typicM bursting phenomena such as Fold/Fold/Hopf bursting and Fold/Fold bursting, are presented, the bifurcation mechanisms of which have been obtained. Furthermore, the dynamic difference between the two cases corresponding to relatively large and small perturbation frequencies, respectively, has been presented, which can be used to describe the influence of the frequencies involving in the evolution on the bursting behaviors in the system.
基金supported by the National Natural Science Foundation of China(Grants11632008 and 11872189)
文摘A modified slow-fast analysis method is presented for the periodically excited non-autonomous dynamical system with an order gap between the exciting frequency and the natural frequency.By regarding the exciting term as a slow-varying parameter,a generalized autonomous fast subsystem can be defined,the equilibrium branches as well as the bifurcations of which can be employed to account for the mechanism of the bursting oscillations by combining the transformed phase portrait introduced.As an example,a typical periodically excited Hartley model is used to demonstrate the validness of the method,in which the exciting frequency is far less than the natural frequency.The equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying parameter are presented.Bursting oscillations for two typical cases are considered,which reveals that,fold bifurcation may cause the the trajectory to jump between different equilibrium branches,while Hopf bifurcation may cause the trajectory to oscillate around the stable limit cycle.
文摘There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.
