Showing papers on "Infinite-period bifurcation published in 2010"

The codimension-two bifurcation for the recent proposed SD oscillator

Abstract: In this paper, the complicated nonlinear dynamics at the equilibria of SD oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter α, are investigated. It is found that SD oscillator admits codimension-two bifurcation at the trivial equilibrium when α=1. The universal unfolding for the codimension-two bifurcation is also found to be equivalent to the damped SD oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped SD oscillator as the perturbation parameters vary.

Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure

Abstract: In this paper, we deal with an SIRS epidemic model with stage structure and time delays. We perform some bifurcation analysis to the model. The delay τ is used as the bifurcation parameter. We show that the positive equilibrium is locally asymptotically stable when the time delay is suitable small, while change of stability of positive equilibrium will cause a bifurcating periodic solution as the time delay τ passes through a sequence of critical values. Applying the normal form theory and center manifold argument, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.

Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms*

Abstract: Dynamical phenomena are studied near a Hopf-saddle-node bifurcation of fixed points of 3D-diffeomorphisms. The interest lies in the neighbourhood of weak resonances of the complex conjugate eigenvalues. The 1?:?5 case is chosen here because it has the lowest order among the weak resonances, and therefore it is likely to have a most visible influence on the bifurcation diagram. A model map is obtained by a natural construction, through perturbation of the flow of a Poincar??Takens normal form vector field. Global bifurcations arise in connection with a pair of saddle-focus fixed points: homoclinic tangencies occur near a sphere-like heteroclinic structure formed by the 2D stable and unstable manifolds of the saddle points. Strange attractors occur for nearby parameter values and three routes are described. One route involves a sequence of quasi-periodic period doublings of an invariant circle where loss of reducibility also takes place during the process. A second route involves intermittency due to a quasi-periodic saddle-node bifurcation of an invariant circle. Finally a route involving heteroclinic phenomena is discussed. Multistability occurs in several parameter subdomains: we analyse the structure of the basins for a case of coexistence of a strange and a quasi-periodic attractor and for coexistence of two strange attractors. By construction, the phenomenology of the model map is expected in generic families of 3D diffeomorphisms.

Period Doubling Bifurcation and Center Manifold Reduction in a Time-periodic and Time-delayed Model of Machining

Abstract: A closed-form calculation is presented for the analysis of the period-doubling bifurcation in the time-periodic delay-differential equation model of interrupted machining processes such as milling where the nonlinearity is essentially nonsymmetric We prove the subcritical sense of this period-doubling bifurcation and approximate the emerging period-two oscillations by the Lyapunov—Perron method for computing the center manifold and by calculating the Poincare—Lyapunov constant of the bifurcation analytically at certain characteristic parameter values The existence of the unstable period-two oscillations around the stable stationary cutting is confirmed using a numerical continuation algorithm developed for time-periodic delay-differential equations

Bifurcation Analysis of the Full Velocity Difference Model

Abstract: Bifurcation is investigated with the full velocity difference traffic model. Applying the Hopf theorem, an analytical Hopf bifurcation calculation is performed and the critical road length is determined for arbitrary numbers of vehicles. It is found that the Hopf bifurcation critical points locate on the boundary of the linear instability region. Crossing the boundary, the uniform traffic flow loses linear stability via Hopf bifurcation and the oscillations appear.

Stability and Hopf bifurcation of a delay eco‐epidemiological model with nonlinear incidence rate*

Abstract: In this paper, a three‐dimensional eco‐epidemiological model with delay is considered. The stability of the two equilibria, the existence of Hopf bifurcation and the permanence are investigated. It is found that Hopf bifurcation occurs when the delay τ passes a sequence of critical values. Moreover, by applying Nyquist criterion, the length of delay is estimated for which the stability continues to hold. Numerical simulation with a hypothetical set of data has been done to support the analytical results.

Delayed Hopf bifurcation in time-delayed slow-fast systems

Abstract: 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.

Bifurcations of limit cyclesin a reversible quadratic system with a center, a saddle and two nodes

Abstract: We study the bifurcations of limit cycles in a class of planar reversible quadratic systems whose critical points are a center, a saddle and two nodes, under small quadratic perturbations. By using the properties of related complete elliptic integrals and the geometry of some planar curves defined by them, we prove that at most two limit cycles bifurcate from the period annulus around the center. This bound is exact.

Bifurcation of limit cycles from a two-dimensional center inside Rn

Abstract: We study the bifurcation of limit cycles from the periodic orbits of a linear differential system in R n perturbed inside a class of piecewise linear differential systems, which appears in a natural way in control theory. Our main result shows that at most one limit cycle can bifurcate up to first-order expansion of the displacement function with respect to the small parameter. This upper bound is reached. For proving this result we use the averaging theory in a form where the differentiability of the system is not needed.

Switching Exponent Scaling near Bifurcation Points for Non-Gaussian Noise

Abstract: We study noise-induced switching of a system close to bifurcation parameter values where the number of stable states changes. For non-Gaussian noise, the switching exponent, which gives the logarithm of the switching rate, displays a non-power-law dependence on the distance to the bifurcation point. This dependence is found for Poisson noise. Even weak additional Gaussian noise dominates switching sufficiently close to the bifurcation point, leading to a crossover in the behavior of the switching exponent.

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry.

Abstract: In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chua’s oscillator.

Bifurcations of Limit Cycles in a Z4-Equivariant Quintic Planar Vector Field

Abstract: In this paper, a Z 4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert’s Problem.

Bifurcations of Limit Cycles in a Z4-Equivariant

Abstract: In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert's Problem.

Bifurcation to fronts due to delay.

Abstract: The stability of a steady-state front (kink) subject to a time-delayed feedback control (TDFC) is examined in detail. TDFC is based on the use of the difference between system variables at the current moment of time and their values at some time in the past. We first show that there exists a bifurcation to a moving front. We then investigate the limit of large delays but weak feedback and obtain a global bifurcation diagram for the propagation speed. Finally, we examine the case of a two-dimensional front with radial symmetry and determine the critical radius above which propagation is possible.

Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays

Abstract: The congestion control algorithm, which has dynamic adaptations at both user ends and link ends, with heterogeneous delays is considered and analyzed. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as the bifurcation parameter, and when the delay exceeds a critical value, a limit cycle emerges via a Hopf bifurcation. Resonant double Hopf bifurcation is also found to occur in this model. An efficient perturbation-incremental method is presented to study the delay-induced resonant double Hopf bifurcation. For the bifurcation parameter close to a double Hopf point, the approximate expressions of the periodic solutions are updated iteratively by use of the perturbation-incremental method. Simulation results have verified and demonstrated the correctness of the theoretical results.

Two-coupled pendulum system: bifurcation, chaos and the potential landscape approach

Abstract: In this paper, we have explored the bifurcation behavior and chaos of a two-coupled pendulum system with a coupling energy of the form: κ(θ1 - θ2)2. Using fixed point analysis, we have determined the bifurcation map, which provides a pictorial view of the number and stability properties of the fixed points with respect to the coupling parameter. The bifurcation map shows that the two-coupled pendulum system can exhibit three forms of bifurcation: pitchfork; saddle-node; and a new bifurcation in which a fixed point of the mixed type changes to a center, while a second fixed point of mixed type is born. In order to analyze the chaotic dynamics of the two-coupled pendulum system, we have introduced an equivalent model to the system. This model enables us to investigate the system dynamics in terms of the motion of a particle interacting with a potential landscape. Through analyzing the geometry of the landscape, we are able to determine the dynamical transition points from regular to locally chaotic, and the...

Stability and bifurcation of a two-neuron network with distributed time delays

Abstract: In this paper we study the stability and bifurcation of the trivial solution of a two-neuron network model with distributed time delays. This model consists of two identical neurons, each possessing nonlinear instantaneous self-feedback and connected to the other neuron with continuously distributed time delays. We first examine the local asymptotic stability of the trivial solution by studying the roots of the corresponding characteristic equation, and then describe the stability and instability regions in the parameter space consisting of the self-feedback strength and the product of the connection strengths between the neurons. It is further shown that the trivial solution may lose its stability via a certain type of bifurcation such as a Hopf bifurcation or a pitchfork bifurcation. In addition, the criticality of Hopf bifurcation is investigated by means of the normal form theory. We also provide numerical evidence to support our theoretical analyses.

Analytics of heteroclinic bifurcation in a 3:1 subharmonic resonance

Abstract: Analytical approximation of heteroclinic bifurcation in a 3:1 subharmonic resonance is given in this paper. The system we consider that produces this bifurcation is a harmonically forced and self-excited nonlinear oscillator. This bifurcation mechanism, resulting from the disappearance of a stable slow flow limit cycle at the bifurcation point, gives rise to a synchronization phenomenon near the 3:1 resonance. The analytical approach used in this study is based on the collision criterion between the slow flow limit cycle and the three saddles involved in the bifurcation. The amplitudes of the 3:1 subharmonic response and of the slow flow limit cycle are approximated and the collision criterion is applied leading to an explicit analytical condition of heteroclinic connection. Numerical simulations are performed and compared to the analytical finding for validation.

On the Relationship between Equilibrium Bifurcations and Ideal MHD Instabilities for Line-Tied Coronal Loops

Abstract: For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated imposing line-tied boundary conditions. Using a well-studied example based on the Gold-Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the GradShafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials the first bifurcation point of the Grad-Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.

Zero-Hopf bifurcation analysis on power system dynamic stability

Abstract: Under parametric variations, the phase portraits of a dynamical system such as a power system undergoes qualitative changes at bifurcation points Several global codimension-two bifurcation points such as Zero-Hopf, generalized Hopf, Bogdanov-Takens, among others, can move the system much close to its instability limit, and lead to chaos Due to this fact, there has been major effort in understanding these instability phenomena, in order to design control and preventive actions for power systems In this paper, the dynamics of a straightforward system which includes a single machine-infinite bus power system (SMIB) is analyzed The system is forced to operate under several conditions in order to study its behavior close to codimension-two bifurcation points This paper is specifically oriented to analyze the Zero-Hopf and the Bogdanov Takens bifurcations, which contributes significantly to the system dynamics

Hopf bifurcation control in a coupled nonlinear relative rotation dynamical system

Abstract: A coupled nonlinear relative-rotation system is studied, and the Hopf bifurcation is analyzed under the condition of primary resonance and 1:1 internal resonance. In order to control the Hopf bifurcation point, the stability and amplitude of limit cycle, a nonlinear feedback controller is proposed, and numerical calculation can confirm the validity of the method.

Range parameter induced bifurcation in a single neuron model with delay-dependent parameters

Abstract: This paper deals with the single neuron model involving delay-dependent parameters proposed by Xu et al [Phys Lett A, 354, 126-136, 2006] The dynamics of this model are still largely undetermined, and in this paper, we perform some bifurcation analysis to the model Unlike the article [Phys Lett A, 354, 126-136, 2006], where the delay is used as the bifurcation parameter, here we will use range parameter as bifurcation parameter Based on the linear stability approach and bifurcation theory, sufficient conditions for the bifurcated periodic solution are derived, and critical values of Hopf bifurcation are assessed The amplitude of oscillations always increases as the range parameter increases; the robustness of period against change in the range parameter occurs.

The stability and Hopf bifurcation for a predator-prey system with discrete and distributed delays

Abstract: In this paper, we investigate a predator-prey model with discrete and distributed delays. By choosing the discrete delay as a bifurcation parameter, we show that Hopf bifurcation occur as delay crosses a critical value. Then using normal form theorem and central manifold argument, we obtain the property of the bifurcation periodic solution. Finally, we give an example to support our theoretical analysis.

On the supercriticality of the first hopf bifurcation in a delay boundary problem

Abstract: The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.