We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order...We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states（QSs） and the repetitive spiking states（SPs） may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.展开更多
We investigate the evolution of interfacial gravity-capillary waves propagating along the interface be-tween two dielectric fluids under the action of a horizontal electric field.There is a uniform backgroundflow in e...We investigate the evolution of interfacial gravity-capillary waves propagating along the interface be-tween two dielectric fluids under the action of a horizontal electric field.There is a uniform backgroundflow in each layer,and the relative motion tends to induce Kelvin-Helmholtz(KH)instability.The com-bined effects of gravity,surface tension and electrically induced forces are all taken into account.Underthe short-wave assumption,the expansion and truncation method of Dirichlet-Neumann(DN)operatorsis applied to derive a reduced dynamical model.When KH instability is suppressed linearly by a consider-ably large electric field,our numerical results reveal that in certain regions of parameter space,nonlinearsymmetric traveling wave solutions can be found near the minimum phase speed.Additionally,the de-tailed bifurcation structures are presented together with typical wave profiles.展开更多
The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influ- ences of the system parameter and the impulsive forces on the dynamics of the system are studied res...The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influ- ences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincar6 map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.展开更多
The mathematical model of CO oxidation with three time scales on platinum group metals is investigated, in which order gaps between the time scales related to external perturbation and the rates associated with differ...The mathematical model of CO oxidation with three time scales on platinum group metals is investigated, in which order gaps between the time scales related to external perturbation and the rates associated with different chemical reaction steps exist. Forced bursters, such as point–point type forced bursting and point–cycle type forced bursting, are presented. The bifurcation mechanism of forced bursting is novel, and the phenomenon where two different kinds of spiking states coexist in point–cycle type forced bursting has not been reported in previous work. A double-parameter bifurcation set of the fast subsystem is explored to reveal the transition mechanisms of different forced bursters with parameter variation.展开更多
Part II of this study detects the dominant decadal-centennial timescales in four SST indices up to the 2010/2011 winter and tries to relate them to the observed 11-yr and 88-yr solar activity with the sunspot number u...Part II of this study detects the dominant decadal-centennial timescales in four SST indices up to the 2010/2011 winter and tries to relate them to the observed 11-yr and 88-yr solar activity with the sunspot number up to Solar Cycle 24. To explore plausible solar origins of the observed decadal-centennial timescales in the SSTs and climate variability in general, we design a simple one-dimensional dynamical system forced by an annual cycle modulated by a small-amplitude single- or multi-scale ＂solar activity.＂ Results suggest that nonlinear harmonic and subharmonic resonance of the system to the forcing and period-doubling bifurcations are responsible for the dominant timescales in the system, including the 60-yr timescale that dominates the Atlantic Multidecadal Oscillation. The dominant timescales in the forced system depend on the system＇s parameter setting. Scale enhancement among the dominant response timescales may result in dramatic amplifications over a few decades and extreme values of the time series on various timescales. Three possible energy sources for such amplifications and extremes are proposed. Dynamical model results suggest that solar activity may play an important yet not well recognized role in the observed decadal-centennial climate variability. The atmospheric dynamical amplifying mechanism shown in Part I and the nonlinear resonant and bifurcation mechanisms shown in Part II help us to understand the solar source of the multi-scale climate change in the 20th century and the fact that different solar influenced dominant timescales for recurrent climate extremes for a given region or a parameter setting. Part II also indicates that solar influences on climate cannot be linearly compared with non-cyclic or sporadic thermal forcings because they cannot exert their influences on climate in the same way as the sun does.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
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 dyn...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.展开更多
The behaviors of a system that alternates between the R¨ossler oscillator and Chua＇s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state vari...The behaviors of a system that alternates between the R¨ossler oscillator and Chua＇s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.展开更多
In this paper, the dynamic behavior analysis of the electromechanical coupling characteristics of a flywheel energy storage system （FESS） with a permanent magnet （PM） brushless direct-current （DC） motor （BLDCM...In this paper, the dynamic behavior analysis of the electromechanical coupling characteristics of a flywheel energy storage system （FESS） with a permanent magnet （PM） brushless direct-current （DC） motor （BLDCM） is studied. The Hopf bifurcation theory and nonlinear methods are used to investigate the generation process and mechanism of the coupled dynamic behavior for the average current controlled FESS in the charging mode. First, the universal nonlinear dynamic model of the FESS based on the BLDCM is derived. Then, for a 0.01 kWh/1.6 kW FESS platform in the Key Laboratory of the Smart Grid at Tianjin University, the phase trajectory of the FESS from a stable state towards chaos is presented using numerical and stroboscopic methods, and all dynamic behaviors of the system in this process are captured. The characteristics of the low-frequency oscillation and the mechanism of the Hopf bifurcation are investigated based on the Routh stability criterion and nonlinear dynamic theory. It is shown that the Hopf bifurcation is directly due to the loss of control over the inductor current, which is caused by the system control parameters exceeding certain ranges. This coupling nonlinear process of the FESS affects the stability of the motor running and the efficiency of energy transfer. In this paper, we investigate into the effects of control parameter change on the stability and the stability regions of these parameters based on the averaged-model approach. Furthermore, the effect of the quantization error in the digital control system is considered to modify the stability regions of the control parameters. Finally, these theoretical results are verified through platform experiments.展开更多
By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibri...By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibrium points.Two types of periodic oscillation,such as 2T-focus/cycle periodic switching and 2T-focus/focus periodic switching,have been observed,the mechanism of which is presented through the switching relationship.The distribution of eigenvalues related to the equilibrium points determined by two subsystems is discussed to interpret oscillation-increasing and oscillation-decreasing cascades of the periodic oscillations.Furthermore,the invariant subspaces of the equilibrium point are investigated to reveal the mechanism of dynamical phenomena in the periodic switching.展开更多
By introducing the periodic parameter-switching signal to the Lorenz oscillator, a switched dynamic model is established. In order to investigate the mechanism of the behaviors of the whole system, bifurcation sets of...By introducing the periodic parameter-switching signal to the Lorenz oscillator, a switched dynamic model is established. In order to investigate the mechanism of the behaviors of the whole system, bifurcation sets of the subsystems are derived and the Poincar6 map of the switched system is defined by suitable local sections and local maps. Under certain parameter conditions, symmetric periodic oscillations may be observed. With the variation of parameter, the symmetry-breaking bifurcation mecha- nisms of the symmetric periodic oscillations can be understood by calculating the associated maximal Lyapunov exponent and Floquet multiplies. Meanwhile, the parameter values of the related local bifurcations, such as saddle-node, pitchfork and peri- od-doubling bifurcations are calculated based on the Floquet multiplies.展开更多
The main purpose of the paper is to display the relaxation oscillations, known as the bursting phenomena, for the coupled oscillators with periodic excitation with an order gap between the exciting frequency and the n...The main purpose of the paper is to display the relaxation oscillations, known as the bursting phenomena, for the coupled oscillators with periodic excitation with an order gap between the exciting frequency and the natural frequency. For the case when the exciting frequency is much smaller than the natural frequency, different types of bursting oscillations such as fold/fold, Hopf/Hopf bursting oscillations can be observed. By regarding the whole exciting term as a slow-varying parameter on the fact that the exciting term changes on a much smaller time scale, bifurcations sets of the generalized autonomous system is derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two cases with typical bifurcation patterns are focused on as examples to explore the dynamical evolution with the variation of the amplitude of the external excitation. Bursting oscillations which alternate between quiescent states (QSs) and repetitive spiking states (SPs) can be obtained, the mechanism of which is presented by introducing the transformed phase portraits overlapping with the bifurcation diagrams of the generalized autonomous system. It is found that not only the forms of QSs and SPs, but also the bifurcations at the transition points between QSs and SPs, may influence the structures of bursting attractors. Furthermore, the amplitudes and the frequencies related to SPs may depend on the bifurcation patterns from the quiescent sates.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.21276115)
文摘We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states（QSs） and the repetitive spiking states（SPs） may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.
基金supported by the National Natural Science Foundation of China(Grant 12071429).
文摘We investigate the evolution of interfacial gravity-capillary waves propagating along the interface be-tween two dielectric fluids under the action of a horizontal electric field.There is a uniform backgroundflow in each layer,and the relative motion tends to induce Kelvin-Helmholtz(KH)instability.The com-bined effects of gravity,surface tension and electrically induced forces are all taken into account.Underthe short-wave assumption,the expansion and truncation method of Dirichlet-Neumann(DN)operatorsis applied to derive a reduced dynamical model.When KH instability is suppressed linearly by a consider-ably large electric field,our numerical results reveal that in certain regions of parameter space,nonlinearsymmetric traveling wave solutions can be found near the minimum phase speed.Additionally,the de-tailed bifurcation structures are presented together with typical wave profiles.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11202180,61273106,and 11171290)the Natural Science Foundation of Jiangsu Province,China(Grant Nos.BK2010292 and BK2010293)+2 种基金the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.10KJB510026)the National Training Programs of Innovation and Entrepreneurship for Undergraduates,China(Grant No.201210324009)the Training Programs of Practice and Innovation for Jiangsu College Students,China(Grant No.2012JSSPITP2378)
文摘The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influ- ences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincar6 map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 20976075, 10972091, and 11002093)the College Graduate Student Scientific Research Innovation Foundation of Jiangsu Province, China (Grant No. CXLX12-0619)
文摘The mathematical model of CO oxidation with three time scales on platinum group metals is investigated, in which order gaps between the time scales related to external perturbation and the rates associated with different chemical reaction steps exist. Forced bursters, such as point–point type forced bursting and point–cycle type forced bursting, are presented. The bifurcation mechanism of forced bursting is novel, and the phenomenon where two different kinds of spiking states coexist in point–cycle type forced bursting has not been reported in previous work. A double-parameter bifurcation set of the fast subsystem is explored to reveal the transition mechanisms of different forced bursters with parameter variation.
基金provided by the LASG State Key Laboratory Special Fund for this research project
文摘Part II of this study detects the dominant decadal-centennial timescales in four SST indices up to the 2010/2011 winter and tries to relate them to the observed 11-yr and 88-yr solar activity with the sunspot number up to Solar Cycle 24. To explore plausible solar origins of the observed decadal-centennial timescales in the SSTs and climate variability in general, we design a simple one-dimensional dynamical system forced by an annual cycle modulated by a small-amplitude single- or multi-scale ＂solar activity.＂ Results suggest that nonlinear harmonic and subharmonic resonance of the system to the forcing and period-doubling bifurcations are responsible for the dominant timescales in the system, including the 60-yr timescale that dominates the Atlantic Multidecadal Oscillation. The dominant timescales in the forced system depend on the system＇s parameter setting. Scale enhancement among the dominant response timescales may result in dramatic amplifications over a few decades and extreme values of the time series on various timescales. Three possible energy sources for such amplifications and extremes are proposed. Dynamical model results suggest that solar activity may play an important yet not well recognized role in the observed decadal-centennial climate variability. The atmospheric dynamical amplifying mechanism shown in Part I and the nonlinear resonant and bifurcation mechanisms shown in Part II help us to understand the solar source of the multi-scale climate change in the 20th century and the fact that different solar influenced dominant timescales for recurrent climate extremes for a given region or a parameter setting. Part II also indicates that solar influences on climate cannot be linearly compared with non-cyclic or sporadic thermal forcings because they cannot exert their influences on climate in the same way as the sun does.
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Project supported by the National Natural Science Foundation of China(Grant Nos.10972091,20976075 and 10872080)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant No. 20976075)
文摘The behaviors of a system that alternates between the R¨ossler oscillator and Chua＇s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.
基金supported by the National Basic Research Program of China (Grant No. 2009CB2197)the National Natural Science Foundation of China (Grant No. 51177108)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110032110066)
文摘In this paper, the dynamic behavior analysis of the electromechanical coupling characteristics of a flywheel energy storage system （FESS） with a permanent magnet （PM） brushless direct-current （DC） motor （BLDCM） is studied. The Hopf bifurcation theory and nonlinear methods are used to investigate the generation process and mechanism of the coupled dynamic behavior for the average current controlled FESS in the charging mode. First, the universal nonlinear dynamic model of the FESS based on the BLDCM is derived. Then, for a 0.01 kWh/1.6 kW FESS platform in the Key Laboratory of the Smart Grid at Tianjin University, the phase trajectory of the FESS from a stable state towards chaos is presented using numerical and stroboscopic methods, and all dynamic behaviors of the system in this process are captured. The characteristics of the low-frequency oscillation and the mechanism of the Hopf bifurcation are investigated based on the Routh stability criterion and nonlinear dynamic theory. It is shown that the Hopf bifurcation is directly due to the loss of control over the inductor current, which is caused by the system control parameters exceeding certain ranges. This coupling nonlinear process of the FESS affects the stability of the motor running and the efficiency of energy transfer. In this paper, we investigate into the effects of control parameter change on the stability and the stability regions of these parameters based on the averaged-model approach. Furthermore, the effect of the quantization error in the digital control system is considered to modify the stability regions of the control parameters. Finally, these theoretical results are verified through platform experiments.
基金supported by the National Natural Science Foundation of China (Grant Nos. 20976075 and 10972091)College Graduate Student Scientific Research Innovation Foundation of Jiangsu,China (Grant No. CXLX12-0619)
文摘By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibrium points.Two types of periodic oscillation,such as 2T-focus/cycle periodic switching and 2T-focus/focus periodic switching,have been observed,the mechanism of which is presented through the switching relationship.The distribution of eigenvalues related to the equilibrium points determined by two subsystems is discussed to interpret oscillation-increasing and oscillation-decreasing cascades of the periodic oscillations.Furthermore,the invariant subspaces of the equilibrium point are investigated to reveal the mechanism of dynamical phenomena in the periodic switching.
基金supported by the National Natural Science Foundation of China (Grant Nos. 21276115, 11272135, 11202085)the Scientific Research Innovation Foundation of Jiangsu Province (Grant No. CXZZ13-0653)the Natural Science Foundation for Colleges and Universities of Jiangsu Province (Grant No. 11KJB130001)
文摘By introducing the periodic parameter-switching signal to the Lorenz oscillator, a switched dynamic model is established. In order to investigate the mechanism of the behaviors of the whole system, bifurcation sets of the subsystems are derived and the Poincar6 map of the switched system is defined by suitable local sections and local maps. Under certain parameter conditions, symmetric periodic oscillations may be observed. With the variation of parameter, the symmetry-breaking bifurcation mecha- nisms of the symmetric periodic oscillations can be understood by calculating the associated maximal Lyapunov exponent and Floquet multiplies. Meanwhile, the parameter values of the related local bifurcations, such as saddle-node, pitchfork and peri- od-doubling bifurcations are calculated based on the Floquet multiplies.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11272135, 21276115, 11472115 & 11472116)the Scientific Research Innovation Foundation of Jiangsu Province (Grant No.1291480004)
文摘The main purpose of the paper is to display the relaxation oscillations, known as the bursting phenomena, for the coupled oscillators with periodic excitation with an order gap between the exciting frequency and the natural frequency. For the case when the exciting frequency is much smaller than the natural frequency, different types of bursting oscillations such as fold/fold, Hopf/Hopf bursting oscillations can be observed. By regarding the whole exciting term as a slow-varying parameter on the fact that the exciting term changes on a much smaller time scale, bifurcations sets of the generalized autonomous system is derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two cases with typical bifurcation patterns are focused on as examples to explore the dynamical evolution with the variation of the amplitude of the external excitation. Bursting oscillations which alternate between quiescent states (QSs) and repetitive spiking states (SPs) can be obtained, the mechanism of which is presented by introducing the transformed phase portraits overlapping with the bifurcation diagrams of the generalized autonomous system. It is found that not only the forms of QSs and SPs, but also the bifurcations at the transition points between QSs and SPs, may influence the structures of bursting attractors. Furthermore, the amplitudes and the frequencies related to SPs may depend on the bifurcation patterns from the quiescent sates.
