scispace - formally typeset
Search or ask a question

Showing papers on "Bifurcation diagram published in 1977"


Journal ArticleDOI
TL;DR: One of the simplest topological variations of the phase space of a one-parameter family of differential equations (vector fields, flows) is the creation of periodic orbits from equilibria as the parameter crosses a critical value as mentioned in this paper.

226 citations



Journal ArticleDOI
TL;DR: In the last few years there has been considerable interest in the asymptotic behavior of maps of the interval into itself under iteration as discussed by the authors, and some of the interest has been generated by population biology.
Abstract: In the last few years there has been considerable interest in the asymptotic behavior of maps of the interval into itself under iteration. Some of this interest has come from the theory of dynamical systems (where most authors have studied maps of the circle), and some of the interest has been generated by population biology. In population biology, maps of the unit interval have been used as models for the dynamics of populations with discrete generations. One of the questions of most interest in the theory has been the determination of the limit sets of points for a map f : I -* I. Here I=[0 , 1]. The limit set ofxs t is the set of limit points of the sequence {f~(x)}. The superscript denotes repeated composition. Of particular interest are periodic orbits: points x such that f~(x)= x for some i>0. Even greater interest focuses upon attracting periodic orbits: if

132 citations


Journal ArticleDOI
TL;DR: A general bifurcation theorem for potential operators is proved in this article, which describes the possible behavior of the set of solutions of an operator equation as a function of the eigenvalue parameter in a neighborhood of the bifurlcation point.

116 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the Hopf bifurcation in the Volterra model and showed that there is a stable periodic solution in the small, but for parameters such that a long time delay is required to make the equilibrium point locally unstable there is no such stable solution.
Abstract: The model studied is the Volterra prey-predator model with prey population limited to lie below the carrying capacity, and with time delay treated as in paper I by adding an extra ordinary differential equation. This makes it possible to examine periodic solutions in the small by means of the theory of the Hopf bifurcation. It is found that for parameters such that short time delay makes the equilibrium point locally unstable there is a stable periodic solution in the small, but for parameters such that a long time delay is required to make the equilibrium point locally unstable there is no such stable solution.

50 citations


Journal ArticleDOI
TL;DR: In this paper, an iterative algorithm was developed for the numerical solution of bifurcation problems involving ordinary differential equations, based on a new method for computing generalized inverses, and the classical Liapunov-Schmidt theory.
Abstract: An iterative algorithm has been developed for the numerical solution of bifurcation problems involving ordinary differential equations. It is based on a new method for computing generalized inverses, and the classical Liapunov-Schmidt theory. A novel formulation of the Hopf bifurcation theorem makes possible the treatment of both simple bifurcation and Hopf bifurcation by the same iterative approach. The method has been applied to the Euler rod buckling problem as an example of simple bifurcation, and to a biological feedback-inhibition control system, as an example of Hopf bifurcation. The results exhibit accuracy and efficiency in a sizable neighborhood of the bifurcation point, and in the second example reveal interesting behaviour not shown by the Poincare-Hopf asymptotic solution or by previous numerical solutions.

45 citations



Journal ArticleDOI
TL;DR: In this paper, an asymptotic method was used to show that secondary bifurcation occurs in a model biochemical reaction studied as a simplified example of morphogenesis, and the secondary branches were presented.
Abstract: We use an asymptotic method to show that secondary bifurcation occurs in a model biochemical reaction studied as a simplified example of morphogenesis. The secondary bifurcation points are computed, and asymptotic expansions for the secondary branches are presented.

41 citations


Journal ArticleDOI
TL;DR: In this paper, the authors extended the theory of elastic stability to include the imperfection-sensitivity of twofold compound branching points with symmetry of the potential function in one of the critical modes (semi-symmetric points of bifurcation).
Abstract: The general theory of elastic stability is extended to include the imperfection-sensitivity of twofold compound branching points with symmetry of the potential function in one of the critical modes (semi-symmetric points of bifurcation). Three very different forms of imperfection-sensitivity can result, so a subclassification into monoclinal, anticlinal and homeoclinal semi-symmetric branching is introduced. Relating this bifurcation theory to Rene Thom's catastrophe theory, it is found that the anticlinal point of bifurcation generates an elliptic umbilic catastrophe, while the monoclinal and homeoclinal points of bifurcation lead to differing forms of the hyperbolic umbilic catastrophe. Practical structural systems which can exhibit this form of branching include an optimum stiffened plate with free edges loaded longitudinally, and an analysis of this problem is presented leading to a complete description of the imperfection-sensitivity. The paper concludes with some general remarks concerning the nature of the optimization process in design as a generator of symmetries, instabilities and possible compound bifurcations.

39 citations





Journal ArticleDOI
TL;DR: In this article, the steady state solutions of a non-linear reaction diffusion system are evaluated exactly, and the bifurcation diagram as well as their stability is discussed, respectively.



Book ChapterDOI
01 Jan 1977
TL;DR: In this paper, the authors present a version of the exchange of stability in the context of Hopf bifurcation, i.e., the bifurbation of a family of time periodic solutions from a series of equilibrium solutions, and present the results for the time periodic case first in the simpler context of hopf's work and then for an evolution equation in a Banach space.
Abstract: Publisher Summary This chapter focuses on the principle of exchange of stability. The principle of exchange of stability is generally employed in the context of a family of evolution equations for which bifurcation occurs. It refers to a qualitative relationship between the shape of the bifurcating curve of solutions and their stability. This chapter presents a version of the principle of exchange of stability in the context of Hopf bifurcation, that is, the bifurcation of a family of time periodic solutions from a family of equilibrium solutions. It discusses some preliminaries on bifurcation. The chapter presents the results for the time periodic case, first in the simpler context of Hopf's work and then for an evolution equation in a Banach space.

Journal ArticleDOI
01 Feb 1977
TL;DR: In this article, the authors present a bifurcation analysis on the existence of a global solution for a semilinear parabolic equation and characterize the local stability and the instability of the corresponding steady-state solutions.
Abstract: The aim of this paper is to present a bifurcation analysis on the existence of the nonexistence of a global solution for a semilinear parabolic equation and to characterize the local stability and the instability of the corresponding steady-state solutions. The bifurcation result can be described either by a parameter X for a fixed spatial domain a or by varying S for a fixed X. The stability analysis gives a result which can be used to determine the stability or instability problem when the system possesses nonintersecting multiple steady-state solutions.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of bifurcating subharmonic solutions, nTperiodic solutions, of T-periodic non-autonomous systems and established a factorization theorem for the stability of periodic solutions.
Abstract: In this paper I prove theorems about the stability of bifurcating solutions without restricting the study to small amplitudes. 1 do not even always require that the solutions which I call "bifurcating" form connected branches; they may be isolated branches which are not easily described, or not described at all, by conventional types of analysis of bifurcation. I begin in w I with a simple theory of bifurcation and stability of equilibrium solutions of evolution problems in R 1. This theory gives complete and rigorous results for stability and repeated branching which are in an extensive analogy to results which hold for the stability and repeated branching of steady solutions in Banach space. In the more general problem, I have in mind steady equilibrium solutions of nonlinear evolution equations possessing different patterns of spatial symmetry. These solutions are points in a Banach space and the families with different symmetries may be projected as plane curves (bifurcation curves). In R1 the projections and the solutions coincide and the theory simplifies enormously. In w I consider the problem of bifurcating subharmonic solutions, nTperiodic solutions, of T-periodic nonautonomous systems. On each such solution branch there is a factorization theorem which relates the stability and subsequent bifurcation of that branch to its shape. I show that such factorization theorems hold even at eigenvalues with non-zero Jordan chains of generalized eigenvectors. In w I show how factorization theorems may be used to characterize points of secondary and repeated n T-periodic bifurcation at a simple eigenvalue. In w I establish a factorization theorem for the stability of periodic solutions of autonomous systems. This theorem generalizes my earlier work (JOSEPH, 1976; JOSEPH & NIELD, 1976) on factorization theorems for the stability of the Hopf bifurcation. As in the non-autonomous problem treated in w 3, the factorization theorem implies a necessary condition for repeated bifurcation at a simple eigenvalue; that is, at a simple zero Floquet exponent. This exponent (or the equivalent unit multiplier) always has a multiplicity greater than one and is typically a double eigenvalue with one proper eigenvector and one generalized eigenvector (a two-link Jordan chain).

Journal ArticleDOI
01 Nov 1977
TL;DR: The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter as discussed by the authors.
Abstract: The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear and singular bifurcation problem is studied to illustrate to what extent the singularity given by a pole can influence the behavior of a pole.


Book ChapterDOI
01 Jan 1977
TL;DR: In this article, the authors consider model equations of increasing complexity which exhibit bifurcation in a continuum of points and show that, under certain simple geometrical conditions, there is a selection principle yielding only a finite number of stable solutions.
Abstract: In this contribution we consider model equations of increasing complexity which exhibit bifurcation in a continuum of points. In spite of the wealth of solutions present there is - under certain simple geometrical conditions - a selection principle yielding only a finite number of stable solutions. The geometrical conditions mentioned guarantee essentially “supercritical” bifurcation for the continuum as an entity and thus reflect at this level the now classical result for simple eigenvalue bifurcation [2], [7].


Journal ArticleDOI
Jack K. Hale1
TL;DR: In this article, it was shown that one can systematically determine the bifurcation surfaces by elementary scaling techniques and the implicit function theorem under certain generic hypotheses on M. The results have applications of the buckling theory of plates and shells under the effect of external forces, imperfections, curvature and variations in shape.
Abstract: : Suppose Lambda, X, Z are Banach spaces, M: Lambda x X yields Z is a mapping continuous together with derivatives up through some order r. A Bifurcation surface for the equation (1) M(lambda,x) = 0 is a surface in parameter space Lambda for which the number of solutions x of (1) changes as lambda crosses this surface. Under certain generic hypotheses on M, it is shown that one can systematically determine the bifurcation surfaces by elementary scaling techniques and the implicit function theorem. This document gives a summary of these results for the case of bifurcation near an isolated solution or families of solutions of the equation M(lambda sub 0, x) = 0. The results have applications of the buckling theory of plates and shells under the effect of external forces, imperfections, curvature and variations in shape. The results on bifurcation near families has applications in nonlinear oscillations and the theory of homoclinic orbits.