A new approach is proposed to compute Hopf bifurcation points. The method could produce small extended systems and therefore could reduce the computational effort and storage. One numerical example is presented to dem...A new approach is proposed to compute Hopf bifurcation points. The method could produce small extended systems and therefore could reduce the computational effort and storage. One numerical example is presented to demonstrate that the method is efficient.展开更多
The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase spac...The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase space. The qualitative property and stability of equilibrium points were analysed. The conditions under which the positive equilibrium point exists and becomes and O+ attractor are obtained. The problems on Hopf bifurcation are discussed in detail when small perturbation occurs.展开更多
In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter va...In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.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
Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth deg...Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous （PWSC） systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.展开更多
文摘A new approach is proposed to compute Hopf bifurcation points. The method could produce small extended systems and therefore could reduce the computational effort and storage. One numerical example is presented to demonstrate that the method is efficient.
文摘The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase space. The qualitative property and stability of equilibrium points were analysed. The conditions under which the positive equilibrium point exists and becomes and O+ attractor are obtained. The problems on Hopf bifurcation are discussed in detail when small perturbation occurs.
基金supported by the National Natural Science Foundation of China(Grant Nos.10772135 and 60874028)the Young Scientists Fund of the National Natural Science Foundation of China(Grant No.11202148)+2 种基金the Incentive Funding of the National Research Foundation of South Africa(GrantNo.IFR2009090800049)the Eskom Tertiary Education Support Programme of South Africathe Research Foundation of Tianjin University of Science and Technology
文摘In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.
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supported by the National Natural Science Foundation of China(Nos.U1204106,11372282,11272024,and 11371046)the National Basic Research Program of China(973 Program)(Nos.2012CB821200 and 2012CB821202)
文摘Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous （PWSC） systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.
