The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues...The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.展开更多
In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And...In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.展开更多
Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the p...Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the previous work.The present paper aims to explore composite fast-slow dynamics when the frequency ratio is variable.As a result,a novel route to composite fast-slow dynamics is obtained.We find that,when presented with variable frequency ratios in a 1:n fashion,the sliding fast-slow oscillations may turn into the ones characterized by the fact that the clusters of large-amplitude oscillations of relaxational type are exhibited in each period of the oscillations,and hence the mixedmode fast-slow oscillations.Depending on whether the transition of the trajectory is from the upper subsystem via the fold bifurcation or not,these interesting oscillations are divided into two classes,both of which are investigated numerically.Our study shows that,when the frequency ratio n is increased from n=3,newly created boundary equilibrium bifurcation points may appear on the original sliding boundary line,which is divided into smaller parts,showing sliding and downward crossing dynamical characteristics.This is the root cause of the clusters,showing large-amplitude oscillations of relaxational type,resulting in the formation of mixed-mode fast-slow oscillations.Thus,a novel route to composite fast-slow dynamics by frequency switching is explained.Besides,the effects of the forcing on the mixed-mode fast-slow oscillations are explored.The magnitude of the forcing frequency may have some effects on the number of large-amplitude oscillations in the clusters.The magnitude of the forcing amplitude determines whether the fast-slow characteristics can be produced.展开更多
The purpose of this paper is to investigate the macroeconomic effect of pollution emission in an overlapping generation system. This is done through constructing an endogenous economy which generates pollution and env...The purpose of this paper is to investigate the macroeconomic effect of pollution emission in an overlapping generation system. This is done through constructing an endogenous economy which generates pollution and environment deterioration. It shows how different level of pollution emission could cause complexity such as fold bifurcation in this system. In addition, numerical simulations are provided to show the emergency of complexity.展开更多
Dynamical behaviors of a class-B laser system with dissipative strength are analyzed for a model in which the polarization is adiabatically eliminated. The results show that the injected signal has an important effect...Dynamical behaviors of a class-B laser system with dissipative strength are analyzed for a model in which the polarization is adiabatically eliminated. The results show that the injected signal has an important effect on the dynamical behaviors of the system. When the injected signal is zero, the dissipative term of the class-B laser system is balanced with external interference, and the quasi-periodic flows with conservative phase volume appear. And when the injected signal is not zero, the stable state in the system is broken, and the attractors(period, quasi-period, and chaos) with contractive phase volume are generated. The numerical simulation finds that the system has not only one attractor, but also coexisting phenomena(period and period, period and quasi-period) in special cases. When the injected signal passes the critical value,the class-B laser system has a fold-Hopf bifurcation and exists torus “blow-up” phenomenon, which will be proved by theoretical analysis and numerical simulation.展开更多
The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cel...The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.10071022)the Shanghai Priority Academic Discipline.
文摘The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.
基金Supported by the National Science Foundations of China(10671063 and 61571052)
文摘In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.
基金Project supported by the National Natural Science Foundation of China(Nos.12272150,12072132,12372093)。
文摘Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the previous work.The present paper aims to explore composite fast-slow dynamics when the frequency ratio is variable.As a result,a novel route to composite fast-slow dynamics is obtained.We find that,when presented with variable frequency ratios in a 1:n fashion,the sliding fast-slow oscillations may turn into the ones characterized by the fact that the clusters of large-amplitude oscillations of relaxational type are exhibited in each period of the oscillations,and hence the mixedmode fast-slow oscillations.Depending on whether the transition of the trajectory is from the upper subsystem via the fold bifurcation or not,these interesting oscillations are divided into two classes,both of which are investigated numerically.Our study shows that,when the frequency ratio n is increased from n=3,newly created boundary equilibrium bifurcation points may appear on the original sliding boundary line,which is divided into smaller parts,showing sliding and downward crossing dynamical characteristics.This is the root cause of the clusters,showing large-amplitude oscillations of relaxational type,resulting in the formation of mixed-mode fast-slow oscillations.Thus,a novel route to composite fast-slow dynamics by frequency switching is explained.Besides,the effects of the forcing on the mixed-mode fast-slow oscillations are explored.The magnitude of the forcing frequency may have some effects on the number of large-amplitude oscillations in the clusters.The magnitude of the forcing amplitude determines whether the fast-slow characteristics can be produced.
基金Supported by the National Natural Science Foundation of China(71303181)the Science Foundation of Ministry of Education of China(11YJC790004)+3 种基金China Postdoctoral Science Foundation Funded Project(2014M550864)the Fundamental Research Funds for the Central Universities of China(K50511060001)supported by China Scholarship Council(201406965014) while the first author was visiting George Mason UniversityResearch Funds for Research Base of Modern Service Industry in Hunan Province(16jdmszd01)
文摘The purpose of this paper is to investigate the macroeconomic effect of pollution emission in an overlapping generation system. This is done through constructing an endogenous economy which generates pollution and environment deterioration. It shows how different level of pollution emission could cause complexity such as fold bifurcation in this system. In addition, numerical simulations are provided to show the emergency of complexity.
基金supported by the National Natural Science Foundation of China(Grant No.61973175)the Natural Science Foundation of Tianjin(Grant Nos.20JCYBJC01060 and 20JCQNJC01450).
文摘Dynamical behaviors of a class-B laser system with dissipative strength are analyzed for a model in which the polarization is adiabatically eliminated. The results show that the injected signal has an important effect on the dynamical behaviors of the system. When the injected signal is zero, the dissipative term of the class-B laser system is balanced with external interference, and the quasi-periodic flows with conservative phase volume appear. And when the injected signal is not zero, the stable state in the system is broken, and the attractors(period, quasi-period, and chaos) with contractive phase volume are generated. The numerical simulation finds that the system has not only one attractor, but also coexisting phenomena(period and period, period and quasi-period) in special cases. When the injected signal passes the critical value,the class-B laser system has a fold-Hopf bifurcation and exists torus “blow-up” phenomenon, which will be proved by theoretical analysis and numerical simulation.
文摘The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.
