Type Of Hopf Bifurcation Research Articles

Time-modulated oscillatory structures in phase-separating reactive mixtures.

We investigate the effect of temporal modulation on the spatiotemporal patterns produced by phase separation in chemically reactive ternary mixtures. The temporal modulation is introduced by making one of the reaction rates periodic in time. Our main concern is the stability of traveling waves, which appear above a Hopf-type bifurcation at a finite wave number and are stable in the absence of external modulation. It is shown by computer simulations in two dimensions that when the external modulation is present, various types of coherent standing waves emerge near the special point in the parameter space, where the Turing-type bifurcation line and the Hopf-type bifurcation line meet each other. We carry out a theoretical analysis to understand the phase diagram of the synchronized oscillatory structures obtained numerically.

PARALLEL FLOW STABILITY UNDER A UNIFORM HEAT FLUX: EFFECT OF THE PRANDTL NUMBER

Stability of free convection flow is examined in a horizontal cavity, insulated on the sides and heated from the bottom by a constant uniform heat flux. Base flow solutions are derived under the parallel flow assumption, for which linear stability analysis reveals a Hopf type bifurcation at the critical Rayleigh number, indicating oscillating thermal instability modes over the entire range of Prandtl numbers considered

Global Nonlinear Exponential Stability of the Conduction-Diffusion Solution for Schmidt Numbers Greater than Prandtl Numbers

The nonlinear exponential stability of the conduction-diffusion solution of a binary fluid mixture heated and salted from below is studied in the case of a horizontal layer when the Schmidt numbers are bigger than the Prandtl numbers (i.e., when the linear theory does not exclude Hopf-type bifurcations at the onset of convection). For any boundary condition (rigid or stress-free), the coincidence of the critical linear R L 2 and nonlinear R E 2 Rayleigh numbers is shown when the Rayleigh numbers for the concentration C 2 are small. This result is obtained using a Lyapunov function equivalent to the classical energy and choosing in an optimal way the Lyapunov parameters. Critical nonlinear Rayleigh numbers close to the linear ones are also obtained for large Rayleigh numbers for the solute concentration.

Theory of a multimode semiconductor laser with optical feedback

We discuss the derivation of multimode rate equations for the description of a semiconductor laser with external cavity. We adopt a formulation where the complex field amplitudes are coupled to the nonlinear gains. For N lasing modes, this leads to $2N$ equations that display in-phased and antiphased time-dependent solutions. A simplified reference model is obtained by assuming that the key parameters are frequency independent. A general linear stability analysis leads to the prediction of two types of Hopf bifurcations. A nondegenerate Hopf bifurcation occurs with the relaxation oscillation frequency as the characteristic bifurcation frequency. A $(N\ensuremath{-}1)$-degenerate Hopf bifurcation occurs with a lower characteristic frequency. To assess the nature and stability of the solutions emerging from the Hopf bifurcations, we perform a nonlinear stability analysis on a reduced model obtained in the limit of large linewidth enhancement factor. In this asymptotic limit, the steady state is always destabilized in favor of a stable periodic inphased or antiphased state. A numerical analysis yields a bifurcation diagram of the multimode equations, which confirms the analytic results and reveals further complex regimes (quasiperiodic and chaotic, in-phased and antiphased) as the amplitude of the field fed back into the laser is increased.

A predator-prey model with disease in the prey

After the seminal models of Vito Volterra and Alfred James Lotka in the mid 1920s for predator-prey interactions, mutualist and competitive mechanisms have been studied extensively in the recent years by researchers. There are so many references in this context, we have just cited here some books (e.g. see, [14, 16{18, ?] and the refer- ences therein). Similarly, after the pioneering work of Kermack{McKendrick on SIRS (susceptible-infective-removal-susceptible) epidemiological models have also received much attention from scientists. Relevant references in this context are also vast and we shall again mention here some books (see [1, 2, 4], to mention a few). But little attention has been paid so far to merge these two important areas of research (see [7, 21]). In this paper, we shall put emphasis in such an eco-epidemiological system. We consider a three species eco-epidemiological system, namely, sound prey (suscep- tible), infected prey (infective) and predator. We consider the case when the predator mainly eats the infected prey. This is in accordance with a previous model by Hadeler and Freedman [7] which describes a predator-prey model where the prey is infected by a parasite, and the prey in turn infects the predator with that parasite. The infec- tion weakens the prey and increases its susceptibility to predation, while no predator impairing elect is accounted for. While the paper is mainly theoretical and does not address any specific situation, the reader may and several examples in [7]. We derive persistence and extinction conditions of the populations and we also determine conditions for which the system enters a Hopf-type bifurcation. Moreover, we observe that the bifurcated branches are supercritical in some parametric region space in a special case when the predator response function is a Holling-type II function.

Spatio-temporal dynamics of forced periodic flows in a confined domain

Nonlinear dynamics of a localized linear array of vortices is investigated by numerical simulations. The setup consists of a thin fluid layer (electrolyte) enclosed in a rectangular box and driven by the injection of homogeneous electric currents in an alternating magnetic field. The spectral model simulates the Navier–Stokes equations in two dimensions with steady forcing and linear bottom friction. The model provides an accurate representation of the evolution of flow pattern. Fourier decomposition of the streamfunction shows that a subharmonic instability occurs in the same symmetry subspace as that of a basic flow in the primary instability regime and a shear flow mode appears in a different symmetry subspace in the secondary instability regime. The exploration of temporal behavior shows that the system produces Hopf-type bifurcations. Chaos and frequency locking due to the mutual interaction between the unstable modes are observed. A general scenario of the dynamics of forced periodic flows is discussed by the amplitude equations, which are modeled using formal group theoretical techniques.

Criticality and bifurcation in the gravitational collapse of a self-coupled scalar field

We examine the gravitational collapse of a non-linear sigma model in spherical symmetry. There exists a family of continuously self-similar solutions parameterized by the coupling constant of the theory. These solutions are calculated together with the critical exponents for black hole formation of these collapse models. We also find that the sequence of solutions exhibits a Hopf-type bifurcation as the continuously self-similar solutions become unstable to perturbations away from self-similarity.

THE STEELPAN AS A SYSTEM OF NON-LINEAR MODE-LOCALIZED OSCILLATORS, I: THEORY, SIMULATIONS, EXPERIMENTS AND BIFURCATIONS

By combining the method of multiple scales with the short-time Fourier transform, the vibrational motions of notes on the steelpan have been investigated analytically and experimentally. The motions on this musical instrument display many of the characteristics of non-linear dynamical systems, in particular, the partials are generated by internal resonances which lead to the exchange of energy between modes. In addition the results show the jump phenomenon and Hopf-type bifurcation in response to sinusoidal excitation, and dependence of the amplitude and spectral structures on the delivery of the stick impact when the instrument is played. Vibrational motions are localized within elliptical regions on the pan face. These aspects, together with the resulting coupling of mode frequency to amplitude, result in a musical instrument that is unique.

Delay induced periodicity in a neural netlet of excitation and inhibition

The dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated. It is shown that in the absence of delay, the netlet relaxes to the trivial resting state. If the delay is of sufficient magnitude, the network is excited to a temporally periodic cyclic behaviour. The analytical mechanism for the onset of cyclic behaviour is through a Hopf-type bifurcation. Approximate solusions to the periodic output of the netlet is calculated; stability of the temporally periodic cycle is investigated. It is shown that the bifurcation is supercritical. A related discrete version of the continuous time system is formulated. It is found that the discrete system also displays a cyclic behaviour. Results of a number of computer simulations are displayed graphically; the article concludes with a brief neurobiological discussion.

Large-scale flows and resonances in 2-D thermal convection

Recent experiments of thermal convection in finite containers of intermediate and large aspect ratios have shown the presence of flows spanning the largest dimension of the container [R. Krishnamurti and L. N. Howard, Proc. Natl. Acad. Sci. 78, 1985 (1981); J. Fluid Mech. 170, 385 (1986)]. Large-scale flows of this kind computed from two-dimensional (2-D) numerical simulations are presented. The marginal stability curves for the bifurcations are computed in the range of aspect ratios L=1,...,6 and for Prandtl number σ =10. The nonlinear dynamics of the bifurcated solution is explored for containers with aspect ratios L=1,2,4. By increasing the Rayleigh number from criticality the system produces different sequences of symmetry breaking, Hopf-type bifurcations, which finally result in large scale flows, oscillatory net mass flux and chaos. The bifurcation involves different mode resonances with vertical and horizontal couplings, which are modeled using formal group theoretical techniques.

Hopf bifurcations and hysteresis in resonant tunneling diode circuits

Conditions for Hopf bifurcations and associated current oscillations are presented for a standard model of a resonant tunneling diode circuit. We derive a simple analytic formula for the Hopf bifurcation type (subcritical or supercritical) and discuss how this affects circuit operation, e.g., whether or not hysteresis and bistability are observed in current-voltage curves.

A probabilistic automata network epidemic model with births and deaths exhibiting cyclic behaviour

A probabilistic automata network model for the spread of an infectious disease in a population of moving individuals is studied. The local rule consists of two subrules. The first one, applied synchronously, models infection, birth and death processes. It is a probabilistic cellular automaton rule. The second, applied sequentially, describes the motion of the individuals. The model contains six parameters: the probabilities p for a susceptible to become infected by contact with an infective; the respective birth rates bs, and bi of the susceptibles from either a susceptible or an infective parent; the respective death rates ds, and di of susceptibles and infectives; and a parameter m characterizing the motion of the individuals. The model has three fixed points. The first is trivial, it describes a stationary state with no living individuals. The second corresponds to a disease-free state with no infectives. The third and last one characterizes an endemic state with non-zero densities of susceptibles and infectives. Moreover, the model may exhibit oscillatory behaviour of the susceptible and infective densities as functions of time through a Hopf-type bifurcation. The influence of the different parameters on the stability of all these states is studied with a particular emphasis on the influence of motion which has been found to be a stabilizing factor of the cyclic behaviour.

USE OF SVD FOR POWER SYSTEM STABILIZER SIGNAL SELECTION

Abstract A detailed model of a single machine infinite busbar system is considered. The stator transients are included, the rotor is represented by four blocks, and the transmission line connecting the machine to the infinite busbar is assumed to be long with series compensation. The model exhibits three types of Hopf bifurcations in different regions in the parameter space. Two approaches based on linear algebra are used for the systematic evaluation of the effectiveness of the supplementary signals used for feedback in the excitation loop to stabilize the system. The first approach is based on using the minimum singular value of a complex matrix as a measure of observability of a mode in a signal. The second approach is based on the determination of the angle between the row vector representing the signal and the nullspace of a complex matrix (A–λI). The right singular vectors are used to find a basis for the nullspace.

Local and global bifurcations in a model of the economic long wave

Combining mathematical analysis with simulations, we obtain a bifurcation diagram for a simple (two state variables) system dynamics model of the economic long wave. Previous linear analysis of the onset of oscillatory behavior (Hopf bifurcation) is extended to include nonlinear effects. It is shown that both sub- and supercritical Hopf bifurcation can occur. In the case of subcritical Hopf bifurcation, the system has two coexisting stable solutions, one stationary and one periodic, in a parameter interval below the bifurcation point. The importance of distinguishing between the two types of Hopf bifurcation is discussed. The model also exhibits homoclinic bifurcation to infinity for some parameter combinations: a limit cycle explodes and disappears. Finally, we discuss how the results may apply to other system dynamics models.

Explosive route to chaos through a fractal torus in a generalized lotka-volterra model

The behavior of a model that generalizes the Lotka-Volterra problem into three dimensions is presented. The results show the analytic derivation of stability diagrams that describe the system's qualitative features. In particular, we show that for a certain value of the bifurcation parameter the system instantly jumps out of a steady state solution into a chaotic solution that portrays a fractal torus in the three-dimensional phase space. This scenario, is referred to as the explosive route to chaos and is attributed to the non-transversal saddle connection type bifurcation. The stability diagrams also present a region in which the Hopf type bifurcation leads to periodic and chaotic solutions. In addition, the bifurcation diagrams reveal a qualitative similarity to the data obtained in the Texas and Bordeaux experiments on the Belousov-Zhabotinskii chemical reaction. The paper is concluded by showing that the model can be useful for representing dynamics associated with biological and chemical phenomena.

Hopf-type bifurcation in the presence of multiple critical eigenvalues

The authors show that, assuming a 'weak' form of symmetry in the equations, one can have Hopf-type bifurcation of periodic solutions even in the presence of doubly degenerate critical imaginary eigenvalues.

Some criticism concerning the Brusselator model of an oscillating chemical reaction

The Brusselator is well known to exhibit an instability of the Hopf-bifurcation type leading to the appearance of limit-cycle oscillations. Here we show that this behaviour disappears entirely when detailed balancing and conservation of mass are strictly applied. Violation of both of these principles simultaneously is necessary for the instability.

Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation

Poincaré observed that for a differential equation x′ = ƒ(x, α) depending on a parameter α, each periodic orbit generally lies in a connected family of orbits in ( x, α)-space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index φ and a “center” index ▪ defined at certain stationary points. We show that genetically there are two types of Hopf bifurcation, those we call “sources” ( ▪ = 1 ) and “sinks” ( ▪ = −1 ). Generically if the set Q is bounded in ( x, α)-space, and if there is an upper bound for periods of the orbits in Q, then Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter “snake” of orbits. A “snake” is a maximal path of orbits that contains no orbits whose orbit index is 0. See Fig. 1.1.

On Spinning Propagation of Combustion Waves

We consider a system of reaction-diffusion equations describing the propagation of combustion waves along a thermally insulated cylindrical sample of solid fuel. Uniform propagation of a plane combustion wave is subjected to a linear stability analysis. It is shown that, if the activation energy is sufficiently high and the diameter of the sample sufficiently large, then the experimentally observed spinning propagation of combustion waves appears as a Hopf-type bifurcation of the solution corresponding to a plane wave.The possibility of similar phenomena in gas combustion is discussed.

Dynamics of a discrete prey-predator model, and chaos

Even if the usual differential equations occurring in ecology show simple dynamical behavior, the corresponding equations modified by a time-delay mechanism or remodelled in a discrete version sometimes exhibit very complicated dynamics. For example, the dynamical behavior of a discrete prey-predator model in two dimensional system is considered. This model exhibits such complicated dynamical behavior as (1) a stable equilibrium point, (2) bifurcating cycles of period 2n (n=1, 2, ...), (3) Hopf-type bifurcations and subsequent bifurcations, (4) chaotic behavior.