Showing papers on "Bifurcation diagram published in 1989"

The slow passage through a Hopf bifurcation: delay, memory effects, and resonance

Abstract: This paper explores analytically and numerically, in the context of the FitzHugh–Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the system's response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described.The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the system's response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset o...

Kuramoto-Sivashinsky dynamics on the center-unstable manifold

Abstract: This paper studies the dynamical behavior of solutions of the Kuramoto–Sivashinsky partial differential equation with periodic boundary conditions on a spatial interval $[ 0,h ]$. The length h is the bifurcation parameter and reduction is made to a two-(complex-)dimensional system on a local center-unstable manifold near the second bifurcation point $h_2 $ from the trivial solution. The resulting $O( 2 )$-equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point. In particular, bifurcation sequences to stable traveling waves, unstable modulated traveling waves, and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within $1\%$ in the parameter range $h_2 \pm 20\% $. A clear understanding of the global dynamical behavior in this region is thus obtained.

Hopf bifurcation for functional differential equations of mixed type

Abstract: We prove Hopf bifurcation and center manifold theorems for functional differential equations of mixed type. An application to the dynamic behavior of a competitive economy (business cycle) is provided.

Bifurcations in a forced softening duffing oscillator

Abstract: The response of a damped Duffing oscillator of the softening type to a harmonic excitation is analyzed in a two-parameter space consisting of the frequency and amplitude of the excitation. An approximate procedure is developed for the generation of the bifurcation diagram in the parameter space of interest. It is a combination of second-order perturbation solutions of the system in the neighborhood of its non-linear resonances and Floquet analysis. The results show that the proposed scheme is capable of predicting symmetry-breaking and period-doubling bifurcations as well as Jumps to either bounded or unbounded motions. The results obtained are validated using analogand digital-computer simulations, which show chaos and unbounded motions, among other behaviors.

Degenerate Hopf bifurcation formulas and Hilbert's 16th problem

Abstract: This paper presents explicit formulas for the solution of degenerate Hopf bifurcation problems for general systems of differential equations of dimension $n \geqq 2$ , with smooth vector fields. The main new result is the general solution of the problem for a weak focus of order 3. For bifurcation problems with a distinguished parameter, we present the formulas for the defining conditions of all cases with codimension $ \leqq 3$. The formulas have been applied to Hilbert’s 16th problem, yielding a new proof of Bautin’s theorem, and correcting an error in Bautin’s formula for the third focal value. The approach used is the Lyapunov–Schmidt method. A review of five other approaches is given, along with literature references and comparisons to the present work.

On the structure of the Mandelbar set

Abstract: The authors study the bifurcation diagram for iterates of the non-analytic maps z to fc(z)=z2+c. The set which they call the Mandelbar set, displays many similarities to the Mandelbrot set. However, bifurcations in MBAR can take place across boundary arcs rather than through boundary points.

Nonlinear dynamics and chaos in parametrically excited surface waves

Abstract: Surface waves in a closed container subject to vertical oscillation are studied Nonlinear dynamical equations of two nearly degenerate subharmonic modes responding to the external forcing are derived, using the averaged Lagrangian method for slowly varying amplitudes Stability and bifurcation diagrams are shown for the system with linear damping Period-doubling bifurcation and chaotic solutions with one positive Lyapunov characteristic exponent are obtained numerically It is shown that some of the period-doubling bifurcations are related to the symmetry of the dynamical system

Equivariant bifurcation theory and symmetry breaking

Abstract: A study is made of the failure of the Maximal Isotropy Subgroup Conjecture for the Weyl group seriesW(D) k . As part of the investigation, a general genericity and stability theorem is proved for bifurcation diagrams in equivariant bifurcation theory. As well, a concept of determinacy for equivariant bifurcation theory is introduced and it is shown that, for all compact Lie groupsG and absolutely irreducibleG-representationsV, G-equivariant bifurcation problems onV are finitely determined.

A detailed study of a forced chemical oscillator: Arnol'd tongues and bifurcation sets

Abstract: We investigate in detail the dynamics of a time‐periodically forced chemical oscillator in the parameter plane of forcing amplitude and forcing period. In particular, we present computed bifurcation sets for two typical cases of a forced, autonomously oscillating continuous stirred tank reactor system. The total mass flow rate j is used as the forcing variable by varying it sinusoidally in time about the autonomous system’s value. We find a wide variety of new nonlinear phenomena, including a global bifurcation structure—the skeletal bifurcation structure—that is common to the two cases presented and to other forced oscillator systems. The skeletal bifurcation structure is periodic along the forcing period axis and is mainly composed of the boundaries of Arnol’d tongues, which terminate at finite forcing amplitudes. In one of the cases studied, the invariant torus is destroyed between two critical curves and cascades of period doubling occur within the Arnol’d tongues; we relate this destruction of the t...

Order and chaos in a third-order RC ladder network with nonlinear feedback

Abstract: The influence of nonlinear characteristics on the performance of a very simple RC oscillator consisting of a third-order RC-ladder phase-shift network (quadrupole) and a nonlinear amplifier in the feedback path is studied. Proof for the existence of a periodic solution is given, and bounds for the nonlinear characteristic ensuring the existence of a unique periodic solution have been calculated. Bifurcation phenomena associated with the changes of slopes of chosen piecewise-linear characteristics are studied in detail both numerically and experimentally. A complete bifurcation diagram has been constructed revealing several interesting phenomena in the circuit, namely, creation of periodic orbits and symmetry-breaking bifurcation giving birth to two stable periodic orbits, which separately undergo period-doubling bifurcation leading to chaotic behavior. Two coexisting chaotic attractors born via period-doubling sequence finally merge together. Several periodic windows within the chaos range can be detected. >

Bifurcation from the essential spectrum for some non-compact non-linearities

Abstract: Keywords: non-compact nonlinearities ; essential spectrum ; bifurcation ; semilinear elliptic equation Reference ANA-ARTICLE-1989-001doi:10.1002/mma.1670110408View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

Analysis of voltage control systems exhibiting Hopf bifurcation

Abstract: Stability at critical load levels is analyzed using Hopf bifurcation theory. The authors consider the detailed two-axis model and show that a pair of complex eigenvalues associated with the excitation system undergo Hopf bifurcation near the critical load level. A center manifold is constructed, and the stability of the periodic solution so obtained is examined. >

Instabilities of a boost converter system under large parameter variations

Abstract: The instabilities of the boost converter system, caused by large variations of system parameters, are explored by using multiple-parameter bifurcation theory oriented approaches. It is revealed that, depending on the eigenvalues of the Jacobian of the system, both static bifurcation (jump phenomenon) and dynamic bifurcation (onset of limit cycles) can occur at critical points. The conditions for stability for both precritical and postcritical behavior of the system have been obtained with the support of experimental verification. >

Numerical determination of an emanating branch of Hopf bifurcation points in a two-parameter problem

Abstract: In a two-parameter problem a branch of Hopf bifurcation points can bifurcate from a branch of simple turning points of the steady state problem, at a point for which the Frechet derivative has a double eigenvalue zero with a one-dimensional nullspace. It is indicated how the origin of a branch of Hopf points can be detected during the continuation of a branch of simple turning points.Further, an augmented system of equations is presented, for which this “origin for Hopf bifurcation” is an isolated solution. If the steady state problem is described by a system of algebraic equations, Newton's method for the solution of the augmented system can be implemented very efficiently. The authors also discuss switching to the branch of Hopf points.Results are given for the one-dimensional “Brusselator” model, a system of four partial differential equations.

Hopf bifurcation and transition to chaos in Lotka-Volterra equation

Abstract: It is shown that in a suitable class of Lotka-Volterra systems it is possible to characterize the centre-critical case of the Hopf bifurcation of the multipopulation equilibrium. Moreover, for three populations, it is shown that, in the non-critical case, Hopf bifurcation is supercritical. Numerical evidence of transition to chaotic dynamics, via period-doubling cascades, from the limit cycle is reported.

Use of bifurcation diagrams as fingerprints of chemical mechanisms

Abstract: Various chemical reagents were fed continuously into a continuously stirred tank reactor to perturb the Belousov-Zhabotinskii system. The resulting bifurcation diagrams each contain multiple curves separating regions with different types of dynamical behavior. These very complex diagrams can be used as fingerprints of the perturbing chemical mechanism. Essentially the same bifurcation structures were observed under the addition of formaldehyde and sodium bromite, indicating the same mechanism. The effect of bromomalonic and hypobromous acids was also found to be nearly identical; the slight differences between their fingerprints is explained by the effect of bromine, which contaminates the HOBr. Finally, the effect of added bromide is shown in another bifurcation diagram.

Hopf bifurcation from nonperiodic solutions of differential equations. I. Linear theory

Abstract: In this and succeeding papers we consider some foundational elements of a theory of Hopf bifurcation from non-periodic solutions of ordinary differential equations.

Modes of nonlinear couplers: building blocks for physical insight.

Abstract: The physics of nonlinear couplers is dictated by its normal modes, modes that are found from linear (axially uniform) couplers. Elementary power-flow arguments establish whether the mode is stable or unstable. These facts provide the bifurcation diagram that fully characterizes nonlinear coupling.

Degenerate Hopf bifurcation and nerve impulse, part II

Abstract: The bifurcation from equilibrium of periodic solutions of the Hodgkin and Huxley equations for the nerve impulse is studied. In earlier work singularity theory techniques were used to establish that these equations have a branch of periodic solutions undergoing two Hopf bifurcations, and the equations were conjectured to be equivalent to a member of a one-parameter family of generalized Hopf bifurcation problems. Here the invariants for equivalence to this family and the value of the modal parameter are computed (see [W. W. Farr et al., “Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem,” SIAM J. Math. Anal., 20 (1989), pp. 13–30]). The value of this parameter determines the type of bifurcation, and in this way it is decided which of the proposed bifurcation diagrams are actually to be found. Thus a topological description of periodic orbits of the Hodgkin and Huxley equations near the equilibrium solution is obtained. In this way, a periodic solution branch is found that does not arise thro...

Plastic bifurcation and postbifurcation analysis for generalized standard continua

Abstract: THE PRESENT work is concerned with the bifurcation and postbifurcatio~ analysis of a class of rate independent plasticity models obeying Hill’s maximum dissipation principle. A variational inequality approach. which differs from the classical formulation of the plastic bifurcation problem. is employed. The rate n bifurcation problem is formulated and sufficient conditions for uniqueness of the corresponding boundary value problem are given. A connection is made with Hill’s nonbifurcation criterion. In addition. the issue of the postbifurcation behavior of the solid is addressed in this more general context showing the possibility of angular as well as smooth bifurcations of rate n > 1. Finally an example, capab% of exhibiting both an angular as well as a smooth bifurcation is analysed using the general fo~uIation derived in this work. The presentation is concluded with some comments and comparisons of the present methodology with the classical approach.

Bifurcation analysis of nonlinear retinal horizontal cell models. I. Properties of isolated cells

Abstract: 1. Bifurcation theory is used to study properties of nonlinear analytical and computational models of isolated retinal horizontal cells. The analytical model is based on the published data of Shingai and Christensen describing steady-state I-V characteristics of horizontal cells isolated from catfish (Ictalurus punctatus) retina. The computational model is based on I-V characteristics of distinct macroscopic membrane currents observed in horizontal cells isolated from goldfish (Carassius auratus) retina. Slow-model dynamics are analyzed assuming that excitatory processes occur rapidly with respect to the time course of inactivation of the inward Ca2+ and outward K+ currents. 2. A global bifurcation diagram plotting the location and stability properties of critical points as a function of photoreceptor-evoked horizontal-cell postsynaptic membrane conductance Gsyn is derived for the analytical model. The automated bifurcation analysis software AUTO is used to compute global bifurcation diagrams for the computational model. Bifurcation diagrams exhibit a bistable regime at small Gsyn values characterized by two stable and one unstable critical point and a monostable regime at larger Gsyn values characterized by a single globally attracting stable critical point. The transition between bistable and monostable behavior occurs at a Gsyn value of roughly 0.9 nS for the computational model and 1.7 nS for the analytical model. Estimates of horizontal-cell glutamate-channel conductance suggest that this transition corresponds to the activation of as few as 400-700 glutamate channels. Dark-evoked release of neurotransmitter from photoreceptors may therefore set horizontal-cell synaptic conductance Gsyn to a value within the monostable regime. 3. Photoreceptor-evoked horizontal-cell membrane conductance, total Ca2+ channel conductance, and inactivation of the inward Ca2+ current are shown to be the major factors controlling the bifurcation structure of the computational model. Inactivation of the inward Ca2+ current is required to account for the dark resting potential of horizontal cells as well as light-evoked hyperpolarizing responses. Inactivation of the outward K+ current has little effect on model properties. 4. Isolated horizontal cells generate Ca2+ action potentials whereas cells in the intact retina normally do not. Simple procedures for modeling the slow dynamics of isolated horizontal-cell Ca2+ action potentials are described.(ABSTRACT TRUNCATED AT 400 WORDS)

Bifurcation Spectrum in a Delay-Differential System Related to Optical Bistability

Abstract: A delay-differential system related to or device is discussed in this paper. We obtained the instability conditions for the steady state and the critical relations among the parameters and for the modes by solving the characteristic equation of the system. Based on the numerical result the chaos and bifurcation "spectrum" in the plane was shown for a simple model equation when higher modes were neglected. We discovered the coexistence of two different kinds of attractors for this equation in a certain region of the parameter plane.