# Bifurcation

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Curator: John Guckenheimer

A bifurcation of a dynamical system is a qualitative change in its dynamics produced by varying parameters.

## Definition

Consider an autonomous system of ordinary differential equations (ODEs) $\tag{1} \dot{x}=f(x,\lambda),\ \ \ x \in {\mathbb R}^n, \ \ \ \lambda \in {\mathbb R}^p$

where $$f$$ is smooth. A bifurcation occurs at parameter $$\lambda = \lambda_0$$ if there are parameter values $$\lambda_1$$ arbitrarily close to $$\lambda_0$$ with dynamics topologically inequivalent from those at $$\lambda_0\ .$$ For example, the number or stability of equilibria or periodic orbits of $$f$$ may change with perturbations of $$\lambda$$ from $$\lambda_0\ .$$ One goal of bifurcation theory is to produce parameter space maps or bifurcation diagrams that divide the $$\lambda$$ parameter space into regions of topologically equivalent systems. Bifurcations occur at points that do not lie in the interior of one of these regions.

## Bifurcation theory

Bifurcation theory provides a strategy for investigating the bifurcations that occur within a family. It does so by identifying ubiquitous patterns of bifurcations. Each bifurcation type or singularity is given a name; for example, Andronov-Hopf bifurcation. No distinction has been made in the literature between "bifurcation" and "bifurcation type," both being called "bifurcations."

Associated with each bifurcation type are

• defining equations that locate bifurcations of that type in a family $$\dot{x} = f(x,\lambda)$$
• normal forms that give model systems exemplifying the bifurcation type

Inequalities called non-degeneracy conditions are part of the specification of a bifurcation type. The bifurcation types and their normal forms serve as templates that facilitate construction of parameter space maps. Bifurcation theory analyzes the bifurcations within the normal forms and investigates the similarity of the dynamics within systems having a given bifurcation type. The "gold standard" for similarity of systems used by the theory is topological equivalence. In some cases, bifurcation theory proves structural stability of a family. One of the principal objectives of bifurcation theory is to prove the structural stability of normal forms. Note, however, that there are bifurcation types for which structurally stable normal forms do not exist. An important aspect of the definition of structural stability in the context of bifurcation theory is the specification of which perturbations of a family are allowed. For example, bifurcation types of systems possessing specified symmetries have been studied extensively (Equivariant Bifurcation Theory).

## Classification of bifurcations

One can view bifurcations as a failure of structural stability within a family. A starting point for classifying bifurcation types is the Kupka-Smale theorem that lists three generic properties of vector fields:

Different ways that these Kupka-Smale conditions fail lead to different bifurcation types. Bifurcation theory constructs a layered graph of bifurcation types in which successive layers consist of types whose defining equations specify more failure modes. These layers can be organized by the codimension of the bifurcation types, defined as the minimal number of parameters of families in which that bifurcation type occurs. Equivalently, the codimension is the number of equality conditions that characterize a bifurcation.

Codimension one bifurcations comprise the top level of bifurcation types. Single failures of the Kupka-Smale properties yield the following types of codimension one bifurcations:

This is not a comprehensive list of codimension one bifurcations. Additional types can be found in systems with quasiperiodic oscillations or chaotic dynamics. Moreover, there are subcases in the list above that deal with such issues as whether an Andronov-Hopf bifurcation is sub-critical or super-critical, and the implications of eigenvalue magnitudes for homoclinic bifurcation.

The classification of bifurcation types becomes more complex as their codimension increases. There are five types of "local" codimension two bifurcations of equilibria:

## Numerical Methods

One of the principal uses of bifurcation theory is to analyze the bifurcations that occur in specific families of dynamical systems. Investigations commonly identify the types of bifurcations in parameter space maps either by comparison of simulation results with normal forms or by solving defining equations for those bifurcation types in the systems under investigation and computing coefficients of the normal forms. Several software packages (AUTO, CONTENT, MATCONT, XPPAUT, PyDSTool) give implementations of algorithms that perform the latter type of analysis. The numerical core of these packages consist of

• Regular implementations of defining equations for the bifurcation types
• equation solvers such as Newton's method
• Numerical continuation methods for differential equations
• Computation of normal forms
• initial and
• boundary value solvers for differential equations.

The continuation methods compute curves of solutions to regular systems of $$N$$ equations in $$N+1$$ variables. The bifurcation analysis of a system implemented to varying degrees in the packages listed above is based upon the following strategy:

• An initial equilibrium or periodic orbit is located.
• Numerical continuation is used to follow this special orbit as a single active parameter varies.
• Defining equations for codimension one bifurcations detect and locate bifurcations that occur on this branch of solutions.
• Starting at one of the located codimension one bifurcations,

two parameters are designated to be active and the continuation methods are used to compute a curve of codimension one bifurcations.

• Defining equations for codimension two bifurcations detect and locate bifurcations that occur on this branch of solutions.
• Starting at one of the located codimension two bifurcations,

three parameters are designated to be active and the continuation methods are used to compute a curve of codimension two bifurcations.

This process can be continued as long as one has regular defining equations for bifurcations of increasing codimension, but these hardly exist beyond codimension three. Moreover, the dynamic behaviour near bifurcations with codimension higher than three is usually so poorly understood that the computation of such points is hardly worthwhile. In many cases, bifurcation analysis identifies additional curves of codimension k bifurcations that meet at a codimension k+1 bifurcation. Continuation methods can be started at one of these codimension k bifurcations to find curves of this type of bifurcation with k+1 active parameters. Switching to the continuation of a periodic orbit at an Andronov-Hopf bifurcation or to the continuation of a saddle homoclinic bifurcation curve from the Bogdanov-Takens bifurcation are examples of such starting techniques based on normal form computations.

## Bifurcation Theory of Chaotic and Quasiperiodic Systems

Bifurcation theory has intensively investigated varied topics that bear on chaotic and quasiperiodic dynamics. Much of this theory has been developed in the context of discrete time dynamical systems defined by iteration of mappings. The bifurcation theory described above has analogous results for this setting. In some areas, bifurcation theory of discrete systems goes farther than that for continuous time systems. In particular, an extensive, deep theory describing the properties of iterations of one dimensional mappings was developed over the last quarter of the twentieth century. This theory characterizes universal sequences of bifurcations and the existence of chaotic attractors. Some of this theory carries over to the setting of invertible mappings in higher dimensions and to continuous time dynamical systems via Poincar\'e maps. There are also results that are specific to continuous time systems, especially those that apply to homoclinic orbits of equilibrium points. Early results in this area include the theory of the Lorenz Attractor and Silnikov's analysis of systems with a homoclinic orbit of a saddle-focus in three dimensional systems. Methods originating in KAM (Kolmogorov-Arnold-Moser) theory describe how quasiperiodic invariant sets arise naturally in families of vector fields. Sophisticated numerical methods have been developed based upon this theory to compute invariant tori with (quasi)periodic motion in families of vector fields.