Showing papers on "Bifurcation diagram published in 1987"

Stability and Folds

Abstract: It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordinate. The results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds. The theory is illustrated with applications to simple models of thermal selfignition and of a chemical reactor, both of which systems are of Emden-Fowler type. An analysis concerning an elastic rod is also presented.

Chaos via torus breakdown

Abstract: Chaos has been observed from a four-element autonomous circuit whose only nonlinear element is a two-terminal resistor characterized by a three-segment piecewise-linear \upsilon-i characteristic. Both laboratory measurements and computer simulations have confirmed the chaotic behavior to have resulted from the breakdown of a "quasi-periodic" attractor (torus) into a "folded torus." A two-parameter bifurcation diagram is carefully constructed to predict and explain various observed bifurcation phenomena, such as rotation number, devil's staircase, and Arnold tongue.

Bifurcation Properties of a Stratospheric Vacillation Model

Abstract: Nonlinear properties of a stratospheric vacillation model are investigated numerically in the light of bifurcation theory. The model is exactly the same as that used by Holton and Mass, which describes the wave-zonal flow interaction in a β-channel under a nonconservative constraint with zonal-flow forcing and wave dissipation. A set of 81 nonlinear ordinary differential equations with variables depending on time is obtained by a severe truncation and vertical differencing. All of the external parameters are fixed in time. The amplitude of the wave forcing or the intensity of zonal wind forcing at the bottom boundary is changed as a bifurcation parameter. Three branches of the steady solutions are obtained by use of Powell's hybrid method and the pseudo-arclength continuation method. Linear stability of these solution branches is investigated by solving an eigenvalue problem in the linearized system. In some range of the bifurcation parameter, there exists a multiplicity of stable steady solution...

Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

Abstract: Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.

Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit

Abstract: A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Gruner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.

Hopf Bifurcation in the Double-Glazing Problem With Conducting Boundaries

Abstract: Oscillatory convection has been observed in recent experiments in a square, air-filled cavity with differentially heated sidewalls and conducting horizontal surfaces. The author shows that the onset of the oscillatory convection occurs at a Hopf bifurcation in the steady-state equations for free convection in the Boussinesq approximation. The location of the bifurcation point is found by solving an extended system of steady-state equations. The predicted critical Rayleigh number and frequency at the onset of oscillations are in excellent agreement with the values measured recently and with those of a time-dependent simulation. Four other Hopf bifurcation points are found near the critical point and their presence supports a conjectured resonance between traveling waves in the boundary layers and interior gravity waves in the stratified core.

Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser

Abstract: We consider the effect of periodically modulated cavity losses on the bifurcation diagram of the laser rate equations. The double limit of very slow atomic inversion relaxation and small modulation amplitude is investigated. Our control parameter lambda is the ratio of the damped oscillation frequency of the rate equations (at zero modulation amplitude) to the forcing frequency. We concentrate on the case of pure resonance (lambda = 1) and the first subharmonic resonance (lambda = (1/2)) because they are most representative of the effects of the periodic control. We first reformulate the laser problem as a weakly perturbed conservative system and construct harmonic and subharmonic large-amplitude periodic solutions by a regular perturbation analysis. The conditions for the existence of these solutions are analyzed and evaluated numerically. We show that harmonic and subharmonic solutions may coexist. We then determine a small-amplitude periodic solution oscillating at the forcing frequency. We show that perturbations of this basic state lead to slowly decaying quasiperiodic oscillations except in the vicinity of the critical points lambda = (1/2) and 1. In the first case, subharmonic bifurcation may occur and the period of the oscillations suddenly doubles. In the second case, bistability of periodic solutions is observed.

Multiple steady states, complex oscillations, and the devil’s staircase in the peroxidase–oxidase reaction

Abstract: The steady and oscillatory states of the Olsen and Degn model for the peroxidase–oxidase oscillator are found The stability of the steady states is determined, and a bifurcation diagram for the oscillatory regime is found A complex sequence of multiply periodic oscillations is observed and found to follow an orderly pattern when described via the firing number, the number of small oscillations divided by the total number of large and small oscillations per period A plot of the firing number vs one of the system parameters is found to have a stairstep relationship and it is shown that this stairstep relationship is a complete devil’s staircase, an infinite self‐similar staircase with chaotic properties and a fractal dimension

Periodically perturbed Hopf bifurcation

Abstract: In this paper the influence of small periodic perturbations on systems exhibiting Hopf bifurcation is studied in detail. In two-dimensional systems that ordinarily exhibit Hopf bifurcation, the addition of small periodic parametric excitation gives rise to interesting “secondary phenomena.” Explicit results for various primary and secondary bifurcations, along with their stabilities, are obtained. In this work, the ideas related to method of averaging, Poincare–Birkhoff normal forms, and center manifold theorem are used appropriately at different stages. It is found that various results obtained using these techniques agree in their common regions of validity.

One parameter bifurcation diagram for Chua's circuit

Abstract: The objective of this letter is to report on the usefulness of a 1-dimensional map for characterizing the dynamics of a third-order piecewise-linear circuit. The map is used to reproduce period-doubling birfureations and particularly to compute Feigenbaum's number. The 1-dimensional map gives qualitatively identical results as the circuit equations and proved to be far more computationally efficient.

Simulation studies of the dynamic behavior of semiconductor lasers with Auger recombination.

Abstract: We apply the phase portrait analysis to semiconductor lasers with Auger recombination and extend the analysis to high modulation frequency ωm. On the two‐dimensional bifurcation diagram of modulation depth and modulation frequency, there are seven regions: digital pulsing regions, analog modulation region, period doubling regions, chaos regions, and one multiloop region. It is found that Auger recombination tends to suppress chaos for ωm ωr, chaotic behavior becomes prominent. Furthermore, in the pulsing region, the maximum pulsation frequency is limited to a value around ωr even though ωm may be twice or three times ωr. A normalized two‐dimensional bifurcation diagram defining the digital pulse region and the analog modulation region is presented for the purpose of locating the suitable region for analog and digital operation of semiconductor lasers.

Bifurcation of quasi-periodic solutions from equilibrium points of reversible dynamical systems

Abstract: It is well known that a hyperbolic equilibrium point of a finite dimensional dynamical system (the eigenvalues of the Jacobian are not on the imaginary axis) is an isolated solution in the space of bounded functions (cf. [8], [10]). Such an equilibrium point is "structurally stable", and bifurcations can occur only at those equilibrium points for which the eigenvalues have vanishing real part. Moreover, the natural question of bifurcation of oscillating solutions has been considered almost exclusively for the periodic case, very little being known about bifurcation of quasi-periodic solutions. In the linear case, quasi-periodic solutions can be obtained simply by superposition of periodic solutions with rationally independent frequencies. In the general non-linear case, however, their construction leads necessarily to the socalled problem of small divisors. In connection with the stability of an equilibrium point MOSER [17, pp. 21] and BIBIKOV [3, pp. 91] have treated analytic systems in R 2" which are either reversible or Hamiltonian. They assume that the origin is an elliptic equilibrium point (the Jacobian has only imaginary eigenvalues). Under certain conditions on the terms of order less than four of the Birkhoff normal form, they show that in every neighborhood of the origin there are quasiperiodic solutions with n independent frequencies. This set of solutions is in general not connected, but the relative measure of the points of these trajectories tends to one when the neighborhood shrinks to zero. In [1] the existence of such solutions has been extended to the more general case where eigenvalues with nonvanishing real part are allowed. In recent papers BROER [5], and BRAAKSMA and BROER [4], have constructed quasi-periodic bifurcating solutions for divergence free C l vector fields in R 3 and R 4 involving a single real parameter. We consider here reversible analytic systems in R q, which depend on a pdimensional parameter r/

A geometric framework for the numerical study of singular points

Abstract: While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter-dependent equation $F(z,\lambda ) = 0$. The presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points.

Thermosolutal convection during directional solidification. II. Flow transitions

Abstract: The effect of thermosolutal convection on the solute segregation in crystals grown by vertical directional solidification of binary metallic alloys or semiconductors is considered. Numerical results are obtained using finite differences in a two‐dimensional, time‐dependent model that assumes a planar crystal–melt interface. The configuration is assumed to be periodic in the horizontal direction with a given period, and the possibility of multiple flow states sharing the same period is examined. The results are summarized in bifurcation diagrams of the nonlinear states associated with the critical points of linear theory. The use of a time‐dependent numerical scheme results in gaps in the bifurcation diagram where presumed unstable states exist that cannot be computed by this procedure. As the solutal Rayleigh number is varied, multiple steady states, time‐periodic states, and quasiperiodic states may occur. This case is compared to the simpler case of thermosolutal convection with linear vertical gradients and stress‐free boundaries, for which a rather complete numerical treatment is possible through the use of a simple spectral representation of the nonlinear solution. Retaining a finite number of terms in the expansion results in a set of coupled nonlinear algebraic equations for the Fourier coefficients, which is solved by a quasi‐Newton method. The linear stability of the nonlinear solutions is also determined from a numerical computation of the eigenvalues of the Jacobian matrix. The resulting bifurcation diagram shows qualitative similarity to the bifurcation diagram computed for the solidification system.

The method of parameter functionalization in the Hopf bifurcation problem

Abstract: depending on a parameter (for an extensive bibliography see [l]). The greater part of these investigations exploit information about not only the linear terms of the right-hand member of the system (l), the linear operator Fl (A, 0), but also information about the terms of higher orders in the Taylor expansion in x of the function F(A, x). With the help of such information one can answer the questions on the number of arising self-oscillations, their stability, their dependence on a parameter, etc. Proofs of corresponding assertions use, as a rule, an analytic technique such as varied forms of the implicit function theorem, the theory of invariant manifolds, and the like. At the same time, in problems arising during the study of complicated physical, technological, etc. processes, often the only rather complete information available is that concerning the linear terms of the right-hand member of the system (1). In such circumstances it is very difficult to apply an analytic technique for studying the problem of generation of self-oscillations. Yet as things turn out [2], in some cases the very fact of generation of small self-oscillations can be picked out of information about the linear terms of the right-hand member of the system (1). Of course, under lack of information about the high order terms in the Taylor expansion in x of F(A, x) one can say next to nothing about properties of arising self-oscillations. This article contains a topological proof of the Hopf theorem. In this proof the method of parameter functionalization [3], introduced by Krasnosel’skii in another situation, has much significance. Employment of topological considerations makes it possible to throw aside usual assumptions of differentiability of the right-hand member of the system (1). This provides an opportunity to investigate systems (l), the right-hand side of which contains, for example, hysteresis-type or relay-type nonlinearities. The proof presented below goes through without changes in the case of functional differential equations with lagging arguments [4]. We do not present here the exact formulations of such assertions because our purpose is the demonstration of the method. In [S, 61 the method of parameter functionalization has been applied to the investigation of bifurcation of long-periodic solutions of differential equations and mappings.

On the Hopf Bifurcation with Broken O(2) Symmetry

Abstract: Translation and reflection symmetries introduce the group 0(2) into bifurcation problems with periodic boundary conditions. The effect on the Hopf bifurcation with 0(2)-symmetry of small terms breaking the translation symmetry is investigated. Two primary branches of standing waves are found. Secondary and tertiary bifurcations involving two different types of modulated waves are analyzed in the neighborhood of secondary Takens-Bogdanov bifurcations. The effects of breaking the phaseshift (in time) and reflection symmetries are briefly considered.

Nonlinear dynamics of tubes carrying a pulsatile flow

Abstract: The nonlinear dynamics of a cantilever tube conveying a pulsatile flow and undergoing planar motions is investigated. The mean flow is near its critical value at which the downward vertical position of the tube gets unstable by flutter and executes limit-cycle oscillations. The pulsations in the flow are assumed to be small and harmonic with frequency nearly twice that of the limit cycle. To study the nonlinear dynamics, the method of averaging is utilized and the governing partial differential equation is reduced to a dynamic system on the plane. These two first-order differential equations depend on three parameters and govern the dynamics of the amplitude of motion of the tube. The planar system is studied for its qualitative behaviour using ideas from the local bifurcation theory and a local bifurcation set in the parameter plane is constructed. Using ideas from codimension-two unfolding of singularities, this bifurcation set is further refined. The resulting partial bifurcation set and the associated...

Computing Bifurcation Diagrams for Large Nonlinear Variational Problems

Abstract: We report on a method to reduce the computational effort of computing bifurcation diagrams for large nonlinear parameterdependent systems of variational type. Included are techniques to find a point on a branch, to trace solution branches, to detect singularities, to compute turning points, simple and double nondegenerate bifurcation points and to calculate emanating directions from bifurcation points. The perfor-mance of the method is demonstrated at two examples.