Global Hopf Bifurcation in Reaction Diffusion Systems with Symmetry

Bifurcation and chaotic behavior of a discrete-time predator–prey system

Bifurcations of periodic orbits and band zones of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is first considered without any dissipation or external forcing. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in the in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling simultaneously occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution and to detect the bifurcation location. Although the trajectory of the period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced away from the bifurcation point. Pre-compression is found to affect the periodic orbit frequencies, which in turn result in changes in the wave propagation band zones. These changes are illustrated by studying vibration transmission through a granular chain at different frequencies to ascertain the band zone limits.

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

In this paper we present an experimental analysis of the effect of RF modulation injection current in DFB and FP lasers nearthe first period doubling bifurcation (PDB). By increasing the amplitude of the modulation frequency at the onset of PDB thefirst results of a virtual Hopf phenomenon like behavior has been observed in both types of lasers. We characterized thisobservation by increasing the amplitude of the current modulation where two peaks appear in the spectrum, approaching oneto the other until fmally merging the onset of period doubling bifurcation as predicted by theory. The same virtual Hopfbehavior was observed when we set the amplitude modulation at a fixed value and we sweep the frequency modulation at theonset of the PDB point.Keywords: Laser Diode, Nonlinear Dynamics, Period Doubling Bifurcations. 1. INTRODUCTION Nonlinear dynamics in diode lasers has been the subject of many theoretical and experimental investigations in the lastdecade as they show a rich variety of irregular behavior in their route to chaos1. A single frequency diode laser is commonlystudied by a rate equation model which describes the time evolution of the photon and carrier densities. In these equations,the inclusion of an additional degree of freedom as modulation of the injection current, external light injection or opticalfeedback is needed for nonlinear dynamics to appear2.To our knowledge, the nonlinear behavior in semiconductor diode lasers has been extensively studied by many authors usingdelayed optical feedback and strong current modulation. In the former, the observed route to chaos was through successivesubharmonic bifurcations or by quasi periodic route3'4. In the strong current modulation case the period sequence wastruncated by period tripling before the onset of chaos5'6.On the other hand, the onset of a period doubling bifurcation point in many physical systems brings to nonlinear phenomenasuch as: 1) small signal amplification, 2) shift and suppression of the bifurcation point and 3) the emergence of a closelyspaced set of peaks in the frequency response7.In semiconductor lasers the parametric amplification was observed by Harth and co-workers8'9 which was analyzed byWiesenfeld few years after'°. In the study developed by Wiesenfeld in ref. 10, semiconductor lasers were proposed as goodcandidate to study the rich variety of nonlinear phenomena near period doubling bifurcations. A recent study has shown thatthe effect of Langevin noise in period doubling semiconductor lasers has a virtual Hopf ur12 previously predicted byWiesenfeld'34 as a virtual Hopf phenomenon in systems undergoing a period doubling bifurcations in the presence ofexternal noise.In the present study we have experimentally investigated the nonlinear dynamics of diode lasers in the period doublingbifurcation point. We study the influence of the amplitude modulation in Distributed Feedback (DFB) lasers near perioddoubling bifurcations. In Fabry-Perot (FP) lasers we performed as well the same study as DFB lasers and the analysis of thefrequency modulation near period doubling bifurcation. So in section II we described the experimental procedure for theonset of the first period doubling bifurcation performed in commercial DFB and FP InGaAsP diode lasers at 1,3 tm emission

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O (1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

We investigate the topological types and bifurcations of periodic orbits in the gravitational field of irregular bodies by the well-known two parameter analysis method. Results show that the topological types of periodic orbits are determined by the locations of these two parameters and that the bifurcation types correspond to their variation paths in the plane. Several new paths corresponding to doubling period bifurcations, tangent bifurcations and Neimark–Sacker bifurcations are discovered. Then, applications in detecting bifurcations of periodic orbits near asteroids 101955 Bennu and 2867 Steins are presented. It is found that tangent bifurcations may occur three times when continuing the vertical orbits near the equilibrium points of 101955 Bennu. The continuation stops as the Jacobi energy reaches a local maximum. However, while continuing the vertical orbits near the equilibrium points of 2867 Steins, the tangent bifurcation and pseudo period-doubling bifurcation occur. The continuation can always go on, and the orbit ultimately becomes nearly circular.

Bifurcations of periodic motion in a three-degree-of-freedom vibro-impact system with clearance

Bifurcation and global periodic solutions in a delayed facultative mutualism system

Bifurcations of periodic orbits and chaos in the Brusselator chemical reaction with different shape of external periodic forces are studied numerically. The external periodic forces considered are sine wave, square-wave, rectified sine wave, symmetric saw-tooth wave, asymmetric saw-tooth wave and modulus of sine wave. Period doubling bifurcations, chaos, reverse period doubling bifurcations, quasiperiodic bifurcations are found to occur due to the applied forces. Parametric regimes where suppression of chaos occurs are predicted. Numerical investigations including bifurcation diagram, maximal Lyapunov exponent, Poincare map, Phase portrait and time series plot are used to detect chaos, quasiperiodic and periodic orbits.

This paper is concerned with a predator-prey model with Leslie–Gower functional response and stage structure for prey. By regarding time delay as the bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are given. Some explicit formulas determining the properties of Hopf bifurcation are established by using the normal form method and center manifold theorem. The global continuation of periodic solutions bifurcating from the positive equilibrium is given due to a global Hopf bifurcation result for functional differential equations. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

Global Hopf bifurcation for differential equations with state-dependent delay

Global Hopf bifurcation of a general predator-prey system with diffusion and stage structures

In this work we study a wide class of symmetric control systems that has the Chua's circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-three global bifurcations that act as organizing centres of the complex dynamics Chua's circuit exhibits in the parameter range considered. A detailed (although partial) bifurcation set in a three'parameter space is presented in this paper. We show relations between several high-codimension bifurcations of equilibria, periodic orbits and global connections. Some of the global bifurcations found have been neither analytically nor numerically treated in the literature.

Classical reaction–diffusion systems exhibit diverse dynamical behaviors, which become more complex particularly in the context of bifurcation analysis and normal form theory when chemotactic terms are incorporated. This paper investigates the stability of a positive constant equilibrium and the Turing–Hopf (TH) bifurcation in a diffusive predator–prey model with searching, handling predators, and chemotaxis. A computational method is developed for deriving the normal form of the TH bifurcation on the center manifold in general three-dimensional chemotactic reaction–diffusion systems with Neumann boundary conditions. The method enables an explicit dynamical classification near the bifurcation point. Combined theoretical and numerical analyses reveal that TH interactions significantly enrich spatiotemporal dynamics.

A three-stage-structured prey-predator system with predator density dependent time delay is considered. The positive and uniformly bounded of the solution of the system are obtained, and the conditions for the system occurring global Hopf bifurcation is given. Finally, numerical simulations and brief conclusion are given.