Showing papers on "Bifurcation diagram published in 1978"

A Hopf global bifurcation theorem for retarded functional differential equations

Abstract: We prove a result concerning the global nature of the set of periodic solutions of certain retarded functional differential equations. Our main theorem is an analogue, for retarded F.D.E.'s, of a result by J. Alexander and J. Yorke for ordinary differential equations. Introduction. In the past ten or fifteen years there has been considerable interest in the global nature of the set of periodic solutions of certain parametrized families of F.D.E.'s. These equations arise in a variety of applications, for example, mathematical biology [19]. References at the end of this paper give some guidance to the relevant literature. For those equations to which it is applicable, the global bifurcation theorem in [21] appears to provide the sharpest global information. However, there are simple-looking F.D.E.'s for which the results of [21] are not easily applicable. We mention one example; consider the equation (1) x'(t) = [-ax(t 1) cax(t -y)] [ Ix2(t)1, where c and y are positive constants, I 0. Let ao denote the smallest positive a such that the equation (2) z =ae cae-yz has a pair of pure imaginary solutions. For a variety of reasons, it is reasonable to conjecture that for every a > ao, (1) has a "slowly oscillating" (a term we leave undefined) nonconstant periodic solution. Despite remarks made in [15] for the case y = 2, this modest conjecture has still not been proved in general. The cases c = 0 and c = I (for y = 2) treated in [15] are atypical. Thus it seems reasonable to try to obtain a global bifurcation theorem for periodic solutions which would perhaps provide less detailed information than the one in [21] but which would be more broadly applicable. J. Alexander and J. Yorke have established a generalization of the classical Hopf bifurcation theorem [1], and J. Ize [12], [13] has given a considerable Received by the editors June 28, 1976. AMS (MOS) subject classifications (1970). Primary 34K15; Secondary 47H15. Partially supported by a National Science Foundation Grant. ? American Mathematical Society 1978

Topics in bifurcation theory

Abstract: I. Existence and Structure of Bifurcation Branches The problem of bifurcation is formulated as an operator equation in a Banach space, depending on relevant control parameters, say of the form G(u,λ) = 0. If dimN(G_u(u_O,λ_O)) = m the method of Lyapunov-Schmidt reduces the problem to the solution of m algebraic equations. The possible structure of these equations and the various types of solution behaviour are discussed. The equations are normally derived under the assumption that G^O_λeR(G^O_u). It is shown, however, that if G^O_λeR(G^O_u) then bifurcation still may occur and the local structure of such branches is determined. A new and compact proof of the existence of multiple bifurcation is derived. The linearized stability near simple bifurcation and "normal" limit points is then indicated. II. Constructive Techniques for the Generation of Solution Branches A method is described in which the dependence of the solution arc on a naturally occurring parameter is replaced by the dependence on a form of pseudo-arclength. This results in continuation procedures through regular and "normal" limit points. In the neighborhood of bifurcation points, however, the associated linear operator is nearly singular causing difficulty in the convergence of continuation methods. A study of the approach to singularity of this operator yields convergence proofs for an iterative method for determining the solution arc in the neighborhood of a simple bifurcation point. As a result of these considerations, a new constructive proof of bifurcation is determined.

Oscillations, Fluctuations, and the Hopf Bifurcation.

Abstract: : Consider the effects of small random perturbations on deterministic systems of differential equations. The deterministic systems of interest have oscillatory dynamics and may undergo a bifurcation (the Hopf bifurcation). A first exit problem is formulated for experiments beginning near stable and unstable limit cycles. The unstable limit cycle is surrounded by an annulus. Of interest is the probability of first exit from the annulus through a specified boundary, conditioned on initial position. The diffusion approximation is used, so that the conditional probability satisfies a backward diffusion equation. Appropriate solutions on the backward equation are constructed by an asymptotic method. The behavior of the stochastic system in the vicinity of stable and unstable limit cycles is compared. When the deterministic system exhibits the Hopf bifurcation, the above analysis must be modified.

Bifurcation analysis of nonlinear reaction–diffusion systems: Dissipative structures in a sphere

Abstract: The steady state spatial patterns arising in open nonlinear reaction–diffusion systems beyond an instability point of the thermodynamic branch are studied for a simple kinetic scheme derived from glycolysis. Bifurcation theory is used to derive analytic expressions for patterns within a sphere. The results indicate that a gradient of substance may arise spontaneously from a homogeneous distribution within a cell or a blastula, thus establishing a prepattern. Finally, the derived solution scheme is shown to simplify considerably if rotation matrices are introduced into the formalism.

Bifurcation of Crystalline Solutions and Freezing

Abstract: The nonlinear integral equation for the singlet distribution function, deduced from the first equation of the BBGKY hierarchy on the assumption that the difference in short range correlation between fluid and crystalline solid may be ignored, is solved near the bifurcation point for a system of hard spheres. The bifurcation point is the point at which crystalline solutions branch off continuously from the fluid solution, and has been recently obtained by Raveche and Stuart. The branch of the solution with face-centered cubic symmetry is shown to grow in a direction of decreasing density near bifurcation. This is not surprising, because freezing is a first order phase transition. It is argued that the crystalline state for this solution is unstable and the bifurcation point does not represent the metastability limit of the crystalline phase. Cases in which the integral equation has different kernels are also discussed in relation to the fluid instability.

Restricted Generic Bifurcation

Abstract: Publisher Summary This chapter describes the methods for determining the nature of bifurcation when the family of mappings has k ≥ 1 parameters; however, k is generally smaller than the number of parameters necessary to describe the universal unfolding There has been considerable attention devoted to the existence of bifurcation for one-parameter families of mappings Concurrent with this development has been the extensive theory of the universal unfolding of mappings or generic bifurcation for families of mappings which depend on a sufficiently large number of parameters A large quantity of the literature on bifurcation theory is concerned with the existence of bifurcation The existence of a bifurcation is basically a problem in fixed-point theory and one can use all of the existing fixed-point theorems, degree theory, monotone operator theory, and the Luisternik–Schnirelmann theory Moreover, the literature on this subject is extensive

Bifurcation in a reaction diffusion system

Abstract: The bifurcation and nonlinear stability properties of the Meinhardt-Gierer model for biochemical pattern formation are studied. Analyses are carried out in parameter ranges where the linearized system about a trivial solution loses stability through one to three eigenfunctions, yielding both time independent and periodic final states. Solution branches are obtained that exhibit secondary bifurcation and imperfection sensitivity and that appear, disappear, or detach themselves from other branches.

Bifurcation Analysis in Systems with Switch-Type Coefficients

Abstract: In this paper we develop a bifurcation procedure applicable to certain types of nonlinear differential equations involving switching type coefficient functions. An inductive proof is given which establishes conditions sufficient for application of the method, and the technique is illustrated on a problem in nonlinear beam theory.

Recent Progress in Bifurcation Theory

Abstract: Publisher Summary This chapter reviews recent progress in bifurcation theory. To begin with, bifurcation theory deals with the analysis of branch points of nonlinear functional equations in a vector space, usually a Banach space. The subject of bifurcation is an important topic for applied mathematics in as much as it arises naturally in any physical system described by a nonlinear set of equations depending on a set of parameters. When discussing applied problems, stability considerations are unavoidable. In fact, the phenomenon of bifurcation is intimately associated with the loss of stability. A complete resolution of the branching problem therefore requires an analysis of the stability of the bifurcating solutions. When the nonlinear equations are invariant under a transformation group, the invariance is inherited by the bifurcation equations, and this fact may aid considerably in their analysis. Thereafter, the branching problem in the case of higher multiplicities is considerably more complicated. The assumption of some kind of group invariance is a natural one, and the importance of this aspect of bifurcation theory has attracted the attention of a number of workers in recent years. Moreover, the appearance of cellular solutions can be described mathematically as the bifurcation of doubly periodic solutions.

Comments on the role of the diffusion in a model of chemical instabilities

Abstract: The role of the diffusion in the model of chemical instability due to Schlogl (1971) and Nitzan et al. (1974) is discussed. This model leads to the occurrence of a periodic spatial order which is, however, dynamically unstable. The bifurcation diagram for the steady state solutions is obtained and it is shown that the bifurcation parameter is equal to the wavelength of the spatial disturbance, analogously to the hydrodynamical instabilities theory. A few comments on the statistical mechanics of the model are also given.