Complex dynamics in a linear impulsive system

The dynamical behavior of a linear impulsive system is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the equilibrium and period-one solution, the discontinuous jumps of eigenvalues, and the conditions for system possessing infinite period-two, period-three, and period-six solutions. By using discrete maps, the conditions of existence for Neimark–Sacker bifurcation are derived. In particular, chaotic behavior in the sense of Marotto’s definition of chaos is rigorously proven. Moreover, some detailed numerical results of the phase portraits, the periodic solutions, the bifurcation diagram, and the chaotic attractors, which are illustrated by some interesting examples, are in good agreement with the theoretical analysis.     

一类二阶奇点附近的分支解及其数值计算方法

§1.引言 设X,Y是Banach空间,R是实数域;D和A分别表示X和R中的开集.F:D×A→Y是c~3算子,满足F(x~*,λ~*)=0.本文将讨论在(x~*,λ~*)附近方程     

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.     

Dynamics of a Predator-Prey Model with Allee Effect and Herd Behavior

This paper deals with dynamics of a predator-prey model with Allee effect and herd behavior. We first study the stability of non-negative constant solutions for such system. We also establish the existence of Hopf bifurcation solutions for such predator-prey model. The stability and bifurcation direction of Hopf bifurcation solution in the case of spatial homogeneity are further discussed. At the same time, several examples are given by MATLAB. Finally, the numerical simulations of the system are carried out through MATLAB, which intuitively verifies and supplements the theoretical analysis results.     

Bifurcation analysis of delay-induced periodic oscillations

In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude and a constant frequency; the number of solutions increases with the size of the delay. Indeed, for many physical applications in which oscillatory instabilities are induced by a delayed response or feedback mechanism, the system under consideration forms the underlying backbone for a mathematical model. Our study showcases the effectiveness of performing a numerical bifurcation analysis, alongside the use of analytical and geometrical arguments, in investigating systems with delay. We identify curves of codimension-one bifurcations of periodic solutions. We show how these curves interact via codimension-two bifurcation points: double singularities which organise the bifurcations and dynamics in their local vicinity.     

Bifurcation of travelling wave solutions for the modified dispersive water wave equation

The method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. In different regions, different types of travelling solutions including solitary wave solutions, shock wave solutions and periodic wave solutions are simulated. Furthermore, with a generalized projective Riccati equation method, several new explicit exact solutions are obtained.     

Hopf bifurcation for neutral functional differential equations

In this paper, we extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al. [N.D. Kazarinoff, P. van den Driessche, Y.H. Wan, Hopf bifurcation and stability of periodic solutions of differential–difference and integro-differential equations, J. Inst. Math. Appl. 21 (1978) 461–477] to a kind of neutral functional differential equation (NFDE). As an example, a neutral delay logistic differential equation is considered, and the explicit formulas for determining the direction of bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out to support the analytic results.     

奇点处分岔解支的数目问题*

本文在孤立奇点的假设下证明了带参数非线性方程组解的孤立性,同时指出在分岔点处必有而且仅有有限条解支分岔出来．这一结论是分岔问题数值方法的一个理论基础．     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm

This paper focuses on the delay induced Hopf bifurcation in a dual model of Internet congestion control algorithms which can be modeled as a time-delay system described by a one-order delay differential equation (DDE). By choosing communication delay as the bifurcation parameter, we demonstrate that the system loses its stability and a Hopf bifurcation occurs when communication delay passes through a critical value. Moreover, the bifurcating periodic solution of the system is calculated by means of the perturbation method. Discussion of stability of the periodic solutions involves the computation of Floquet exponents by considering the corresponding Poincaré–Lindstedt series expansion. Finally, numerical simulations for verifying the theoretical analysis are provided.     

Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators

This paper is concerned with a predator–prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation. Using a global Hopf bifurcation result of Wu [Wu JH. Symmetric functional differential equations and neural networks with memory, Trans Amer Math Soc 1998;350:4799–4838] for functional differential equations, we may show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given.     

Local and global bifurcations in an SIRS epidemic model

This paper studies the existence and stability of the disease-free equilibrium and endemic equilibria for the SIRS epidemic model with the saturated incidence rate, considering the factor of population dynamics such as the disease-related, the natural mortality and the constant recruitment of population. Analytical techniques are used to show, for some parameter values, the periodic solutions can arise through the Hopf bifurcation, which is important to carry different strategies for the controlling disease. Then the codimension-two bifurcation, i.e. BT bifurcation, is investigated by using a global qualitative method and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium. Moreover, several numerical simulations are given to support the theoretical analysis.     

Two new approaches to computing Hopf bifurcation problems

We consider the computation of Hopf bifurcation for ordinary differential equations. Two new extended systems are given for the calculation of Hopf bifurcation problems: the first is composed of differential-algebraic equations with index 1, the other consists of differential equations by using a symmetry inherited from the autonomous system of ordinary differential equations. Both methods are especially suitable for calculating bifurcating periodic solutions since they transform the Hopf bifurcation problem into regular nonlinear boundary value problems which are very easy to implement. The bifurcation solutions become isolated solutions of the extended system so that our methods work both in the subcritical and supercritical case. The extended systems are based on an additional parameter ε; practical experience shows that one gets convergence for ε sufficiently large so that a substantial part of the bifurcating branch can be computed. The two methods are illustrated by numerical examples and compared with other procedures.     

Local and global behaviour of steady-state solutions of the Sel'kov model

In this paper we discuss steady-state solutions of the systemof reaction-diffusion equations known as the Sel'kov model.This model has been the subject of much discussion; in particular,analytical and numerical results have been discussed by Lopez-Gomezet al. (1992, IMA J. Num. Anal. 12, 405–28). We show thata simple analysis of the bifurcation function associated withthe system can explain many of the numerical observations, suchas the formation and development of loops of nontrivial solutions,in a simpler and more complete manner than the analysis of Lopez-Gomezet al. This allows for a clearer understanding of the qualitativebehaviour of the set of nontrivial solutions and hence of thebifurcation diagram.     

Stability and Hopf bifurcation in a delayed competition system

In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.     

Quasi-periodic solutions and sub-harmonic bifurcation of Duffing’s equations with quasi-periodic perturbation

The quasi-periodic perturbation for the Duffing’s equation with two external forcing terms has been discussed. The second order averaging method and sub-harmonic Melnikov’s method through the medium of the averaging mrthod have been applied to detect the existence of quasiperiodic solutions and sub-harmonic bifurcation for the system. Sub-harmonic bifurcation curves are given by using numerical computation for sub-harmonic Melnikov’s function.     

Hopf bifurcation in a predator-prey system with Holling type III functional response and time delays

This paper is concerned with a delayed predator-prey system with modified Leslie-Gower and Holling type III schemes. By analyzing the associated characteristic equation, its local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. Based on the normal form method and center manifold theorem, the formulaes for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations to illustrate the theoretical analysis are also carried out.     

Hopf-Bifurkation bei Lasern

The semiclassical equations describing a ring laser show two successive bifurcations, one stationary and one Hopf bifurcation. This phenomenon is analyzed mathematically. The initial value problem for the laser equations and the stability of the stationary solutions are discussed in detail. The transition to ultrashort laser pulses is shown to be a Hopf bifurcation. The direction of the bifurcation is determined for a numerical example. It turns out that it depends on the parameters of the system.     

Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Hopf bifurcations in general systems of Brusselator type

This paper is concerned with general models of Brusselator type subject to the homogeneous Neumann boundary condition. The existence of Hopf bifurcation for the ODE and PDE models is obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of bifurcating periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results.