The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf...The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.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
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 positive connection between the total individual fitness and population density is called the demographic Allee effect.A demographic Allee effect with a critical population size or density is strong Allee effect.I...The positive connection between the total individual fitness and population density is called the demographic Allee effect.A demographic Allee effect with a critical population size or density is strong Allee effect.In this paper,discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects.The steady states of the model,the existence and local stability are examined.Moreover,proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations.This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method(IRK)with very small step size.The comparison yields that discrete-time model has sensitive dependence on initial conditions.By implementing center manifold theorem and bifurcation theory,we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations.Moreover,numerical simulations are provided to validate the theoretical results.展开更多
Based on the Silnikov criterion,this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions.Using the Cardano formula,it obtains the exact range of the value of the para...Based on the Silnikov criterion,this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions.Using the Cardano formula,it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory.By calculating the manifold near the equilibrium point,the series expression of the homoclinic orbit is also obtained.The space trajectory and Lyapunov exponent are investigated via numerical simulation,which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system.The results obtained here mean that chaos occurred in the exact range given in this paper.Numerical simulations also verify the analytical results.展开更多
Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new m...Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.展开更多
文摘The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.
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.
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 positive connection between the total individual fitness and population density is called the demographic Allee effect.A demographic Allee effect with a critical population size or density is strong Allee effect.In this paper,discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects.The steady states of the model,the existence and local stability are examined.Moreover,proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations.This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method(IRK)with very small step size.The comparison yields that discrete-time model has sensitive dependence on initial conditions.By implementing center manifold theorem and bifurcation theory,we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations.Moreover,numerical simulations are provided to validate the theoretical results.
基金Project supported by the National Natural Science Foundation of China (Grant No.10872141)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20060056005)
文摘Based on the Silnikov criterion,this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions.Using the Cardano formula,it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory.By calculating the manifold near the equilibrium point,the series expression of the homoclinic orbit is also obtained.The space trajectory and Lyapunov exponent are investigated via numerical simulation,which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system.The results obtained here mean that chaos occurred in the exact range given in this paper.Numerical simulations also verify the analytical results.
文摘Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.
