In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the lin...In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.展开更多
The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation ar...The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.展开更多
In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a dif...In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.展开更多
One method to change the bifurcation characteristics of chaotic systems is anti-control,which can either delay or advance bifur-cation and transform an unstable state into a stable one.The chaotic system with infinite...One method to change the bifurcation characteristics of chaotic systems is anti-control,which can either delay or advance bifur-cation and transform an unstable state into a stable one.The chaotic system with infinite equilibria exhibits complex bifurcation characteris-tics.The Hopf bifurcation and hidden attractors with symmetric coexistence of the system are analyzed.An improved dynamic state feed-back control method is adopted to reduce the tedious calculation process to prevent the Hopf bifurcation from being controlled.A hybrid controller that includes both nonlinear and linear controllers is set up for the system.With the method,the delay and stability of the Hopf bifurcation of the system are studied and the goal of anti-control is achieved.Numerical analysis verified the correctness.展开更多
We study the limit cycle bifurcations perturbing a class of quartic linear-like Hamiltonian systems having an elementary center at the origin. First, using methods of the qualitative theory, all possible phase portrai...We study the limit cycle bifurcations perturbing a class of quartic linear-like Hamiltonian systems having an elementary center at the origin. First, using methods of the qualitative theory, all possible phase portraits of the unperturbed system are found. Then, using the first order Melnikov function, Hopf bifurcation problem of the perturbed system is investigated, and an upper bound for the function is obtained near the origin.展开更多
A stroboscopic map for voltage-controlled single ended primary inductor converter （SEPIC） with pulse width modulation （PWM） is presented, where low-frequency oscillating phenomena such as quasi-periodic and interm...A stroboscopic map for voltage-controlled single ended primary inductor converter （SEPIC） with pulse width modulation （PWM） is presented, where low-frequency oscillating phenomena such as quasi-periodic and intermittent quasi-periodic bifurcations occurring in the system are captured by numerical and experimental methods. According to bifurcation diagrams and nonlinear dynamical theory, the characteristics of the low-frequency oscillation and the mechanism for the appearance of the low-frequency oscillation are investigated. It is shown that as the controller parameter varies, the change in the conduction mode takes place from the continuous conduction mode （CCM） under the originally stable period one and high periodic orbits to the intermittent changes between CCM and discontinuous conduction mode （DCM）, which may be related to the losing stability of the system and brought the system to exhibiting low-frequency oscillating behaviour in the time domain. Moreover, the occurrence of the intermittent quasi-periodic oscillation reflects that the system undergoes a Neimark-Sacker bifurcation.展开更多
As a typical biochemical reaction, the process of continuous fermentation of ethanol is studied in this paper. An improved model is set forward and in agreement with experiments. Nonlinear oscillations of the process ...As a typical biochemical reaction, the process of continuous fermentation of ethanol is studied in this paper. An improved model is set forward and in agreement with experiments. Nonlinear oscillations of the process are analyzed with analytical and numerical methods. The Hopf bifurcation region is fixed and further analyses are given.展开更多
In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynam...In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate <em>r</em> and searching efficiency <em>a</em>. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for <em>b≠a</em> where <em>a,b</em> are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.展开更多
The transition boundaries of period doubling on the physical parameter plane of a Duffing system are obtained by the general Newton′s method, and the motion on different areas divided by transition boundaries is stu...The transition boundaries of period doubling on the physical parameter plane of a Duffing system are obtained by the general Newton′s method, and the motion on different areas divided by transition boundaries is studied in this paper. When the physical parameters transpass the boundaries, the solutions of period T =2π/ω will lose their stability, and the solutions of period T =2π/ω take place. Continuous period doubling bifurcations lead to chaos.展开更多
The dynamical equations of a thin rectangle plate subjected to the friction support boundary and its plane force are established in this paper. The local bifurcation of this system is investigated by using L S method...The dynamical equations of a thin rectangle plate subjected to the friction support boundary and its plane force are established in this paper. The local bifurcation of this system is investigated by using L S method and the singularity theory. The Z 2 bifurcation in non degenerate case is discussed. The local bifurcation diagrams of the unfolding parameters and the bifurcation response characters referred to the physical parameters of the system are obtained by numerical simulation. The results of the computer simulation are coincident with the theoretical analysis and experimental results.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61272069,61272114,61073026,61170031,and 61100076)
文摘In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.
基金Supported by National Natural Science Foundation of China (No10872141)Doctoral Foundation of Ministry of Education of China (No20060056005)Natural Science Foundation of Tianjin University of Science and Technology (No20070210)
文摘The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.
基金the National Natural Science Foundation of China(Grant No.11871393)the Key Project of the International Science and Technology Cooperation Program of Shaanxi Research&Development Plan(Grant No.2019KWZ-08)the Science and Technology Project founded by the Education Department of Jiangxi Province(Grant No.GJJ14775).
文摘In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.
基金Supported by the Guiding Project of Science and Technology Research Plan of Hubei Provincial Department of Education(B2022458)。
文摘One method to change the bifurcation characteristics of chaotic systems is anti-control,which can either delay or advance bifur-cation and transform an unstable state into a stable one.The chaotic system with infinite equilibria exhibits complex bifurcation characteris-tics.The Hopf bifurcation and hidden attractors with symmetric coexistence of the system are analyzed.An improved dynamic state feed-back control method is adopted to reduce the tedious calculation process to prevent the Hopf bifurcation from being controlled.A hybrid controller that includes both nonlinear and linear controllers is set up for the system.With the method,the delay and stability of the Hopf bifurcation of the system are studied and the goal of anti-control is achieved.Numerical analysis verified the correctness.
基金supported by the National Natural Science Foundations of China(12371171)and the Natural Science Foundation of Jiangsu Province(BK20221339).
文摘We study the limit cycle bifurcations perturbing a class of quartic linear-like Hamiltonian systems having an elementary center at the origin. First, using methods of the qualitative theory, all possible phase portraits of the unperturbed system are found. Then, using the first order Melnikov function, Hopf bifurcation problem of the perturbed system is investigated, and an upper bound for the function is obtained near the origin.
基金Project supported by the Natural Science Foundation of Ningxia Autonomous Region,China(Grant No.NZ0954)
文摘A stroboscopic map for voltage-controlled single ended primary inductor converter （SEPIC） with pulse width modulation （PWM） is presented, where low-frequency oscillating phenomena such as quasi-periodic and intermittent quasi-periodic bifurcations occurring in the system are captured by numerical and experimental methods. According to bifurcation diagrams and nonlinear dynamical theory, the characteristics of the low-frequency oscillation and the mechanism for the appearance of the low-frequency oscillation are investigated. It is shown that as the controller parameter varies, the change in the conduction mode takes place from the continuous conduction mode （CCM） under the originally stable period one and high periodic orbits to the intermittent changes between CCM and discontinuous conduction mode （DCM）, which may be related to the losing stability of the system and brought the system to exhibiting low-frequency oscillating behaviour in the time domain. Moreover, the occurrence of the intermittent quasi-periodic oscillation reflects that the system undergoes a Neimark-Sacker bifurcation.
文摘As a typical biochemical reaction, the process of continuous fermentation of ethanol is studied in this paper. An improved model is set forward and in agreement with experiments. Nonlinear oscillations of the process are analyzed with analytical and numerical methods. The Hopf bifurcation region is fixed and further analyses are given.
文摘In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate <em>r</em> and searching efficiency <em>a</em>. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for <em>b≠a</em> where <em>a,b</em> are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.
文摘The transition boundaries of period doubling on the physical parameter plane of a Duffing system are obtained by the general Newton′s method, and the motion on different areas divided by transition boundaries is studied in this paper. When the physical parameters transpass the boundaries, the solutions of period T =2π/ω will lose their stability, and the solutions of period T =2π/ω take place. Continuous period doubling bifurcations lead to chaos.
文摘The dynamical equations of a thin rectangle plate subjected to the friction support boundary and its plane force are established in this paper. The local bifurcation of this system is investigated by using L S method and the singularity theory. The Z 2 bifurcation in non degenerate case is discussed. The local bifurcation diagrams of the unfolding parameters and the bifurcation response characters referred to the physical parameters of the system are obtained by numerical simulation. The results of the computer simulation are coincident with the theoretical analysis and experimental results.
