W. Güttinger

Imperfection sensitivity of interacting hopf and steady-state bifurcations and their classification☆

Abstract The bifurcation problem of interacting time-periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory. A complete classification, up to symmetry-covariant contact equivalence and codimension three, of generic perturbations of interacting Hopf and steady-state bifurcations is presented. The sensitivity of the bifurcation diagrams to imperfections is analyzed. Normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined. A variety of phenomena, such as gaps in Hopf branches, periodic motions not stably connected to steady states and the formation of islands, is discovered, which one can expect to find in perturbed evolution equations on pure geometric grounds. Implications for physical systems are discussed.

Synchronized patterns in hierarchical networks of neuronal oscillators with D 3 × D 3 symmetry

Abstract The spatiotemporal patterns generated by systems of nine coupled nonlinear oscillators which are equivariant under the permutation symmetry group D 3 × D 3 are determined. This system can be interpreted as a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators. We determine generic synchronized oscillation patterns and transitions between these analytically, by numerical simulations, and experimentally with an electronic analog-network. In the theoretical analysis the representative nonlinear ordinary differential equations are reduced to the normal form equations for coupled Hopf bifurcations in an eight-dimensional center eigenspace, whose generic states have been classified previously. The results are applied to a specific model system in which the network is formed by a class of oscillators, each composed of two asymmetrically coupled Hopfield neurons. Experiments performed on an analog-electronic network of such nonlinear oscillators show that most of the states predicted by the theory of the Hopf bifurcation with D 3 × D 3 - symmetry appear in a stable way. We find a great variety of periodic and quasiperiodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two-level pattern hierarchy. In addition to these states we find in simulations homoclinic cycles within the same isotropy class as well as heteroclinic switchings between such cycles.

Geometrical Principles of Pattern Formation and Pattern Recognition

In spite of its complexity our world is not chaos but a ceaseless creation and destruction of forms or structures which are endowed with a degree of stability: they take up some part of space and last for some period of time. One of the central problems of science and, in particular, of synergetics is to explain this change of form and, if possible, to predict it. The formation of spatio-temporal patterns and modes of behavior is first of all a topological problem. Thus, in our quest for understanding how forms are generated in nature, the relationship between physical forces and the stable geometries they can shape is obviously of fundamental importance. This relationship and the ensuing universal character of structure-forming processes are the major themes underlying this paper. Its objective is to set up, in terms of invariant bifurcation theory, a unifying topological framework for spontaneous pattern formation in nonlinear systems and to show that the same geometrical concepts also govern the inverse problem of pattern recognition.

Semiclassical Path Integrals in Terms of Catastrophes

Using a splitting procedure for action functionals, a semiclassical approximation of path integrals, remaining finite on caustics, is constructed. It is governed by oscillatory generalized Airy integrals involving catastrophe polynomials, whose bifurcation properties reflect those of real and complex classical paths.

Bifurcation Theory in Physics: Recent Trends and Problems

The application of bifurcation theory to nonlinear physical problems is reviewed and an outlook is given on future developments. It is shown, in terms of representative examples, that, at the macroscopic and microscopic levels, the topological singularities and bifurcation processes deriving from the principle of structural stability determine the dominating phenomena and features observed in both structure formation and structure recognition. Because of their universality and classifiability, these bifurcation processes also provide a unifying topological framework for our understanding of the analogies that have been discovered in the critical behavior of nonlinear systems of quite different genesis. After a survey on the basic concepts of bifurcation theory, some new developments are outlined. These include bifurcation phenomena in pattern recognition and remote sensing, in fluid dynamics and nonlinear optics, in condensed matter physics and materials sciences, and in high energy physics and cosmology. We conclude by briefly exploring the frontiers and limits of present bifurcation concepts in dealing with instabilities producing fractal patterns and phase transitions in complex networks.

Bifurcation Geometry in Physics

The application of bifurcation theory to nonlinear physical problems is reviewed. It is shown that the topological singularities and bifurcation processes deriving from the concept of structural stability determine the most significant phenomena observed in both structure formation and structure recognition. From this emerges a unifying geometrical framework for the description of nonlinear physical systems which, when passing through instabilities, exhibit analogous critical behavior both at the microscopic and macroscopic levels. After a survey on the basic concepts of singularity and bifurcation theory some new developments are outlined. These include nonlinear conservation laws in various physical fields, the relation between analytical and topological singularities in the inverse scattering problem and in phonon focusing, interacting Hopf and steady-state bifurcations in nonlinear evolution equations and applications to optical bistability and neuronal activity.

Variational Principles in Pattern Theory

The understanding of pattern formation and its dual, pattern recognition, is one of the most exciting areas of present research. It is the question of how complex systems can generate coherent global structures and how systems are designed which, by means of sensory and perceptional mechanisms, can construct internal representations of patterns in the outside world. The field represents a remarkable confluence of several different strands of thought.

Interactions of Stationary Modes in Systems with Two and Three Spatial Degrees of Freedom

This paper deals with steady state mode-interactions in three-dimensional directional solidification and in two-dimensional thermohaline convection with variable aspect ratios. We show that a solidifying material can exhibit three coupled stationary cellular modes which occur at codimension-three bifurcation points and discuss the associated bifurcation diagrams. The thermohaline convection problem gives rise to a double tricritical point identifiable with a codimension-four bifurcation point. To obtain structurally stable configurations, non-Boussinesq effects have to be taken into account which give rise to temporally oscillating states. Mixed mode patterns and hysteretic behaviour between different unstable modes are modeled and simulated using cellular automata concepts.

Nonlinear Phonon Focusing

Acoustic phonon propagation in a cold anisotropic crystal is dominated by focusing which typically occurs on structurally stable caustics [1]. By applying singularity theory, the forms of these caustics and the associated high-intensity diffraction patterns can be classified into a few topological types [2], [3]. Suppose a monochromatic point source of frequency ω generates phonons with wave vectors k that propagate ballistically in a crystal whose anisotropy is described by a dispersion relation ω=Ω(k). Then only those k contribute to the phonon field u(r,ω) at a point r in space which make up the constant-frequency surface S:ω=Ω(k)=const, i.e.,