Showing papers on "Bifurcation diagram published in 1979"

Group theoretic methods in bifurcation theory

Abstract: Physical examples of bifurcation.- Mathematical preliminaries.- Stability and bifurcation.- Bifurcation at multiple eigenvalues.- Elements of group representation theory.- Applications.- Appendix: How to find the symmetry group of a differential equation.

A Theory for Imperfect Bifurcation via Singularity Theory.

Abstract: : This paper applies the theory of singularities of differentiable mappings - specifically the unfolding theorem - to study the effect of imperfections in a system subject to bifurcation. In a number of special cases we have classified (up to a suitable equivalence) all the possible perturbations of the bifurcation equations by a finite number of imperfection parameters. These cases include both bifurcation from a double eigenvalue and from a simple eigenvalue degenerate in the sense of Crandall-Rabinowitz.

Non-normality and bifurcation in plane strain tension and compression

Abstract: T he bifurcations of a rectangular block subject to plane strain tension or compression are investigated. The block material is taken to be incompressible and is characterized by an incrementally linear constitutive law for which “normality” does not necessarily hold. The consequences of non-normality regarding bifurcation are given primary emphasis here. The characteristic regimes of the governing equations (elliptic, parabolic and hyperbolic) are detennined. In each of these regimes both symmetric and antisymmetric diffuse bifurcation modes are available. Additionally, in the hyperbolic and parabolic regimes, bifurcation into a localized shear band mode is also possible. Particular attention is given to the limiting cases of long wavelength and soon wavelength diffuse bifurcation modes. The range of parameter values is identified for which bifurcation into some localized mode may precede bifurcation into a long wavelength diffuse mode. Some difficulties associated with employing a linear incremental solid in a bifurcation analysis, when primary interest is in the bifurcation of an underlying elastic-plastic solid, are also discussed.

Bifurcation of higher dimensional tori

Abstract: Bifurcation theory is concerned with the study of differential equations which depend upon a real parameter e. For certain critical values of the parameter one finds a structural change in the behavior of the solutions of the differential equation. Such a change is oftentimes called a bifurcation. The first systematic investigation of bifurcation phenomena in an n-dimensional space (n>2) was probably in the paper of E. HOPF [13] in 1942. Specifically HOPF studied a family of ordinary differential equations in R"

Spatial patterns for an interaction-diffusion equation in morphogenesis.

Abstract: A certain interaction-diffusion equation occurring in morphogenesis is considered. This equation is proposed by Gierer and Meinhardt, which is introduced by Child's gradient theory and Turing's idea about diffusion driven instability. It is shown that slightly asymmetric gradients in the tissue produce stable striking patterns depending on its asymmetry, starting from uniform distribution of morphogens. The tool is the perturbed bifurcation theory. Moreover, from a mathematical point of view, the global existence of steady state solutions with respect to some parameters is discussed.

On the bifurcation diagram of discrete analogues for ordinary bifurcation problems

Abstract: Ordinary bifurcation problems of the form (1) typically have at most one nontrivial, nonnegative solution for λ > 0. The paper shows that this is in general not true for discrete analogues to (1) no matter how small the step width h > 0 is chosen.

On minimal nonlinearities which permit bifurcation from the continuous spectrum

Abstract: Bifurcation from the continuous spectrum of a linearized operator is of interest in many physical problems. For example it occurs in the nonlinear Klein-Gordon equation and in nonlinear integrodifferential equations as the Choquard problem; it further appears in nonlinear integral equations of the convolution type. A general theory enclosing all these problems is not yet known. To understand the basic phenomena, we therefore consider monotone differential operators whose linearisations have a purely continuous spectrum. It is shown that in fact the lowest point of the continuous spectrum is a bifurcation point, if the nonlinearity grows sufficiently strong.

Asymptotic Methods for Bifurcation and Stability Problems

Abstract: A method of asymptotically determining the bifurcating solutions of a nonlinear eigenvalue problem is described. The method is based on the smallness of a parameter which is different from the usual parameter used in the LyapunovSchmidt procedure. A discussion of the stability and evolution of bifurcating solutions is included. It is shown how the method may be useful for determining secondary bifurcations and turning points on bifurcation curves.

The bifurcation space of the lorenz attractor

Abstract: In a joint paper3 John Guckenheimer and the present author showed that the space of the title is two-dimensional, in that the pair (kl, k,) of kneading sequences is a complete conjugacy class invariant for the flows near the three-dimensional Lorenz system.498 But there are reasons for obtaining more complete knowledge of the space 3C of all such pairs, k. Thus, one can use the \"Parry coordinates\" (A, c) (see below) but a priori, one doesn't know these all correspond to Lorenz models-i.e., to differential equations in lR3. Secondly, a recurring problem in bifurcation theory is just how do the periodic orbits change, as one changes a bifurcation parameter? Thus we shall present a formula for computing the periodic orbits in terms of the parameter k, i.e., the kneading sequences.

Secondary Bifurcation and Multiple Eigenvalues

Abstract: Secondary bifurcation of general nonlinear operator equations is studied in the neighborhood of multiple eigenvalues of the linearized problem. Necessary conditions for the existence of secondary bifurcation are derived and the location of points of secondary bifurcation are found. The geometrical significance of these conditions is discussed and conditions on primary bifurcation at a multiple eigenvalue are found which guarantee the existence of points of secondary bifurcation.

Bifurcation analysis of the sixteen-vertex model

Abstract: We shall construct a hierarchy of subclasses of the 16-vertex model having qualitatively different symmetry properties. We determine the bifurcation points in the parameter space of the model where new symmetry elements are added to the invariance group of the partition function. In this paper we restrict ourselves to the study of site-dependent transformations converting a homogeneous 16-vertex model into a different homogeneous model. Apart from a trivial transformation, resulting in a change of sign of all vertex weights, such site-dependent transformations exist only for those points in parameter space where particular relations are satisfied. The solution of these relations gives rise to three 6-parameter families of models, two of which are equivalent to the general 8-vertex model, and two families of 4-parameter models. The primary bifurcation models depending on six parameters contain three different types of secondary bifurcation models, depending on 4 parameters, one of which is equivalent to Baxter's symmetric 8-vertex model.

Hopf‐Bifurkation bei Lasern

Abstract: The semiclassical equations describing a ring laser show two successive bifurcations, one stationary and one Hopf bifurcation. This phenomenon is analyzed mathematically. The initial value problem for the laser equations and the stability of the stationary solutions are discussed in detail. The transition to ultrashort laser pulses is shown to be a Hopf bifurcation. The direction of the bifurcation is determined for a numerical example. It turns out that it depends on the parameters of the system.

Synergetics and bifurcation theory

Abstract: Synergetics'-6 is a rather new field of interdisciplinary research related to mathematics, physics, astrophysics, electrical and mechanical engineering, chemistry, biology, ecology, and possibly to other disciplines. It studies the selforganized behavior of complex systems (composed of many subsystems) and focuses its attention to those phenomena where dramatic changes of macroscopic patterns or functions occur owing to the cooperation of subsystems. Some examples are exhibited in FIGURE 1. In spite of this rather general scope, synergetics has been able to unearth astounding analogies between entirely different systems. In the course of this research program it more and more transpired that bifurcation theory plays a crucial role. As long as the systems we are encountered with can be described by mathematical models we very often deal with the following problem. A system is represented by a set of time-dependent variables