Bifurcation analysis and control of chaos for a hybrid ratio-dependent three species food chain

In this paper, a hybrid ratio-dependent three species food chain model with time delay is studied by using the theory of functional differential equation and Hopf bifurcation, the condition on which positive equilibrium exists and the quality of Hopf bifurcation are given. Chaotic solutions are observed and are controlled by delay parameter. Finally, we indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.     

一类非线性系统的周期扰动Hopf分支

研究了小周期扰动对一类存在Hopf分支的非线性系统的影响.特别是应用平均法讨论了扰动频率与Hopf分支固有频率在共振及二阶次调和共振的情形周期解分支的存在性.表明了在某些参数区域内,系统存在调和解分支和次调和解分支,并进一步讨论了二阶次调和分支周期解的稳定性.     

Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations

We revisit Nicholson?s blowflies model with natural death rate incorporated into the delay feedback. We consider the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from a positive equilibrium. We show that the model has only a finite number of Hopf bifurcation values and we describe how branches of Hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals. The bifurcation analysis and the Matlab package DDE-BIFTOOL developed by Engelborghs et al. guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions.     

Bifurcations of periodic solutions and chaos in Duffing-van der Pol equation with one external forcing

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.     

Hopf bifurcation in a control system for the Washout filter-based delayed neural equation

Washout filter is a simple filter that can be designed easily. In this paper, a system for controlling a neural equation with discrete time delay based on Washout filter is presented. The transcendental equation of the corresponding linearized system is analyzed. In this control system, it is found that Hopf bifurcation occurs when the control parameters are chosen properly and that a chaotic orbit can be controlled to a stable periodic solution. The stability condition for bifurcating periodic solutions and the direction of Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Some numerical results are also presented to illustrate the correctness of our results.     

Bifurcation analysis and chaos control of the modified Chua’s circuit system

From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.     

Voltage transformer ferroresonance analysis using multiple scales method and chaos theory

In this article, the Multiple Scales Method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes subharmonic, quasi‐periodic, and also chaotic oscillations. In this article, the chaotic behavior and various ferroresonant oscillations modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as Period Doubling Bifurcation (PDB), Saddle Node Bifurcation (SNB), Hopf Bifurcation (HB) and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via Multiple Scales Method obtaining Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system. © 2013 Wiley Periodicals, Inc. Complexity 18: 34‐45, 2013     

Stability and Hopf bifurcation in a class of nonlocal delay differential equation with the zero‐flux boundary condition

The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time.     

Stability and bifurcations for the chronic state in Marchuk's model of an immune system

In the paper we present known and new results concerning stability and the Hopf bifurcation for the positive steady state describing a chronic disease in Marchuk's model of an immune system. We describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system. We compare these results with the results known in the literature. We show that the positive steady state can be stable only for very specific parameter values. Stability analysis is illustrated by Mikhailov's hodographs and numerical simulations. Conditions for the Hopf bifurcation and stability of arising periodic orbit are also studied. These conditions are checked for arbitrary chosen realistic parameter values. Numerical examples of arising due to the Hopf bifurcation periodic solutions are presented.     

Countable infinite sequence of attractors'' families for the simplest known equivariant chaotic flow

More than 30 new families of periodic and strange attractors for the simplest known equivariant chaotic jerk equation are studied. Inside each family, both periodic and chaotic attractors result from the subharmonic cascade of a limit cycle born by saddle-node bifurcation. Our numerical results provide clear evidence for the following conjecture: the 35 observed families are the first members of a countable infinite sequence. Furthermore, our simulations point out four power laws relating to the initial infinite sequence of saddle-node bifurcations.     

Synchronized Hopf bifurcation analysis in a neural network model with delays

We consider the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling. Employing the center manifold theorem and normal form method introduced by Hassard et al., we give an algorithm for determining the Hopf bifurcation properties. Using the global Hopf bifurcation theorem for FDE due to Wu and Bendixson's criterion for high-dimensional ODE due to Li and Muldowney, we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system

This paper reports a four-dimension (4D) fast-slow hyperchaotic system with the structure of two time scales by adding a slow state variable w into a three-dimension (3D) chaotic dynamical system, studies the stability and Hopf bifurcation of origin point. Furthermore, based on the fast-slow dynamical bifurcation analysis and the phase planes analysis, different bursting phenomena, symmetric fold/fold bursting, symmetric sub-Hopf/sub-Hopf bursting and chaotic bursting, as well as chaotic and periodic spiking, are observed in the fast-slow hyperchaotic system. Numerical simulations are presented to show these results.     

Oscillations in three-reaction quadratic mass-action systems

It is known that rank-two bimolecular mass-action systems do not admit limit cycles. With a view to understanding which small mass-action systems admit oscillation, in this paper we study rank-two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greater generality. One key finding is that an isolated periodic orbit cannot occur in a three-reaction, trimolecular, mass-action system with bimolecular sources. In fact, we characterize all networks in this class that admit a periodic orbit; in every case, all nearby orbits are periodic too. Apart from the well-known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov–Hopf bifurcation. Furthermore, we characterize all two-species, three-reaction, bimolecular-sourced networks that admit an Andronov–Hopf bifurcation with mass-action kinetics. These include two families of networks that admit a supercritical Andronov–Hopf bifurcation and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that are tetramolecular.     

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.     

Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy

In this paper, by using theories and methods of ecology and ODE, a two-prey one-predator system with Watt-type functional response and impulsive perturbations on the predator is established. The system is affected by impulse which can be considered as a control. Conditions for the permanence of the system are obtained. The numerical analysis is carried out to study the effects of perturbation varying parameters of the system. The system shows the rich dynamic behavior including quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and crises, etc.     

Complex dynamics in Chen’s system

This paper characterizes some complex dynamics of Chen’s system. Some conditions of existence for pitchfork bifurcation and Hopf bifurcation are derived by using bifurcation theory and the center manifold theorem. Numerical simulation results not only show consistence with the theoretical analysis but also display some new and interesting dynamical behaviors including homoclinic bifurcation and the coexistence of two stable limit cycles and one chaotic attractor as well as some periodic solutions emerging from Hopf bifurcation but ending in homoclinic bifurcation, which are different from those reported in the literature before. All these show that Chen’s system has very rich nonlinear dynamics.     

Stability and Hopf Bifurcation of a Type of Protein Synthesis System with Time Delay and Negative Feedback

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results.