Classification in bifurcation theory and reaction-diffusion systems (original) (raw)

Symmetry Breaking in Dynamical Systems

1996

Symmetry breaking bifurcations and dynamical systems have obtained a lot of attention over the last years. This has several reasons: real world applications give rise to systems with symmetry, steady state solutions and periodic orbits may have interesting patterns, symmetry changes the notion of structural stability and introduces degeneracies into the systems as well as geometric simplifications. Therefore symmetric systems are attractive to those who study specific applications ns as well as to those who are interested in a the abstract theory of dynamical systems. Dynamical systems fall into two classes, those with continuous time and those with discrete time. In this paper we study only the continuous case, although the discrete case is as interesting as the continuous one. Many global results were obtained for the discrete case. Our emphasis are heteroclinic cycles and some mechanisms to create them. We do not pursue the question of stability. Of course many studies have been made to give conditions which imply the existence and stability of such cycles. In contrast to systems without symmetry heteroclinic cycles can be structurally stable in the symmetric case. Sometimes the various solutions on the cycle get mapped onto each other by group elements. Then this cycle will reduce to a homoclinic orbit if we project the equation onto the orbit ~pace. Therefore techniques to study homoclinic bifurcations become available. In recent years some efforts have been made to understand the behaviour of dynamical sys~ems near points wher.e the symmetry of the system was perturbed by outside influences. This can lead to very complicated dynamical behaviour, as was pointed out by several authors. We will discuss some of the technical difficulties which arise in these problems. Then we will review some recent results on a geometric approach to this problem near steady state bifurcation points.

Bifurcation of relative equilibria in mechanical systems with symmetry

Advances in Applied Mathematics, 2003

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [LS98] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example is presented to illustrate the use of this approach.

Some global bifurcations related to the appearance of closed invariant curves

Mathematics and Computers in Simulation, 2005

In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark-Hopf (N-H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark-Hopf bifurcations. We shall see that the mechanism is not unique, and that it may be related to homoclinic connections of a saddle cycle, that is to a closed invariant curve formed by the merging of a branch of the stable set of the saddle with a branch of the unstable set of the same saddle. This will be shown by analyzing the bifurcations arising inside a periodicity tongue, i.e., a region of the parameter space in which an attracting cycle exists. A. Agliari et al. / Mathematics and Computers in Simulation 68 (2005) 201-219 suppliers, oligopoly, many suppliers, polypoly, to perfect competition. In the last case, each supplier is so small that it cannot in any way influence market price. In the opposite case, the monopolist deliberately limits supply so as to be able to charge a high market price to the end of obtaining a maximum monopoly profit. The cases of duopoly and oligopoly are the most complicated, because each competitor has to take account not only of consumer demand, as reflected in the demand function, but also of the expected retaliations of the competitors. Oligopoly theory is one of the oldest branches of mathematical economics, created in 1838 by the mathematician Augustin Cournot .

False bifurcations in chemical systems: canards

Philosophical transactions, 1991

A canard is a false bifurcation in which the am plitude of an oscillatory system m ay change by orders of m agnitude while th e qu alitativ e dynam ical features rem ain unchanged. Recent theoretical considerations suggest th a t canards are characteristic of fast-slow dynam ical system s and are associated w ith the stable and unstable manifolds of th e phase plane. An altern ativ e characterization of canard behaviour is proposed involving the crossing of an inflection line by a lim it cycle growing out from an unstable statio n ary state. The inflection line comprises th e locus of points a t which th e curvature of any phase plane trajecto ry is zero. The role of the inflection line in the onset of canard behaviour as well as in the continuity of the tran sitio n is exam ined in a tw o-variable model for th e oscillatory EO E reaction, the Autocatalator, and the tw o-variable O regonator. The approach is also applied to the van der Pol oscillator, the system in which canard behaviour was first exam ined.

Versal matrix families, normal forms and higher order bifurcations in dynamic chemical systems

Chemical Engineering Science, 1985

Characterization of the possible dynamic phenomena which arise when a steady state loses stability due to parametric variations is important in process and control system design and is useful for dynamic model identification and evaluation. The linearized system Jacobian Jordan block structure at bifurcation implies corresponding possibilities for nonlinear dynamic phenomena near bifurcation. The linearized system characteristic equation and the theory of versa1 representation of matrix families are applied to identify the number of system parameters which must be varied simultaneously to achieve different eigenvalue configurations. The theory of normal forms is used to illustrate the topological equivalence, near bifurcation, of the original system and the normal form representation which contains a relatively small number of nonlinear differential equations. This theory is used to organize and interpret studies of dynamics in two chemical reaction systems: (i) consecutive-competitive reactions in an isothermal CSTR with multivariable proportional feedback control; and (ii) coupled oscillations in two interacting CSTR's with autocatalytic reactions.

Aspects of Bifurcation Theory for Nonsmooth Dynamical Systems

Invariant manifolds play an important role in the study of Dynamical Systems, since they help to reduce the essential dynamics to lower dimensional objects. In that way, a bifurcation analysis can easily be performed. In the classical approach, the reduction to invariant manifolds requires smoothness of the system which is typically not given for nonsmooth systems. For that reason, techniques have been developed to extend such a reduction procedure to nonsmooth systems. In the present paper, we present such an approach for systems involving sliding motion. In addition, an analysis of the reduced equation shows that the generation of periodic orbits through nonlinear perturbations which is usually related to Hopf bifurcation follows a different type of bifurcation if nonsmooth elements are present, since generically symmetry is broken by the nonsmooth terms. Keywords: Invariant manifold, Sliding motion, Nonlinear piecewise dynamical systems, Non-smooth systems, Invariant cones, Perio...

Bistability and oscillations in chemical reaction networks

Journal of Mathematical Biology, 2009

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461–3470, 1987; Clarke and Jiang in J Chem Phys 99:4464–4476, 1993; Gatermann et al. in J Symb Comput 40:1361–1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks.

Symmetry-breaking bifurcation in the non-linear

2010

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of doublewell potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter. We employ a novel technique combining concentration-compactness and spectral properties of linearized Schrödinger type operators to show that the symmetric ground states can either be uniquely continued for the entire interval of the eigenvalue parameter or they undergo a symmetry-breaking pitchfork bifurcation due to the second eigenvalue of the linearized operator crossing zero. In addition we prove the appropriate scaling for the L q , 2 ≤ q ≤ ∞ and H 1 norms of any stationary states in the limit of large values of the eigenvalue parameter. The scaling and our novel technique imply that all ground states at large eigenvalues must be localized near a critical point of the potential and bifurcate from the soliton of the focusing NLS equation without potential localized at the same point. The theoretical results are illustrated numerically for a double-well potential obtained after the splitting of a single-well potential. We compare the cases before and after the splitting, and numerically investigate bifurcation and stability properties of the ground states which are beyond the reach of our theoretical tools.