Chapter 4 - Numerical Continuation, and Computation of Normal Forms

https://doi.org/10.1016/S1874-575X(02)80025-XGet rights and content

Section snippets

Contents

  • 1.

    Introduction 151

    • 1.1

      Notation 152

  • 2.

    Continuation of stationary and periodic solutions 152

    • 2.1

      Parameter continuation 153

    • 2.2

      Pseudo-arclength continuation 154

    • 2.3

      Periodic solution continuation 155

  • 3.

    Locating codimension-1 bifurcations 157

    • 3.1.

      Test functions 157

    • 3.2

      Locating codimension-1 equilibrium bifurcations 158

    • 3.3

      Locating codimension-I bifurcations of periodic solutions 161

    • 3.4

      Test functions defined by bordering techniques 162

  • 4.

    Branch switching 164

    • 4.1

      The algebraic branching equation 164

    • 4.2

      Branch

Continuation of stationary and periodic solutions

Here we consider the computation of one-parameter families of equilibria of (1), that is, solutions of f(x,α)=0.Such a continuum of solutions is often referred to as a solution branch. With α ∈ ℝ and X = (x, α) the above equation can be written as f(X)=0,where f :ℝn+1 → ℝn. A solution X0 of f (X) = 0 is called regular if fX0 has maximal rank, that is, if Rank (fX0)=n. Near a regular solution one has a unique solution branch, as we now make precise.

Locating codimension-1 bifurcations

An equilibrium without eigenvalues having zero real part, or a periodic solution without nontrivial Floquet multipliers of unit modulus, is called hyperbolic. When following an stationary or periodic solution branch, one can encounter bifurcation points, where the solution loses hyperbolicity. In the case of equilibria, the Jacobian matrix fx has at least one eigenvalue with zero real part at such a point, while for the case of a periodic solution there is at least one nontrivial Floquet

Branch switching

In this section we consider the computation of solution branches that emanate from certain bifurcation points. Specifically, we consider stationary solutions near simple branch points, and periodic solutions near Hopf and near period-doubling points.

Continuation of codimension-1 bifurcations

Suppose Equation (1) has a codimension-1 equilibrium bifurcation at α = α0. Generically, there is a curve α = α(s), with α ∈ℝ2 and s ∈ℝ1, along which the equation has an equilibrium with the given bifurcation. The bifurcation curve, say, B, can be computed as a projection of a certain curve Γ in a space of larger dimension onto the α-plane. Thus, we have to specify a continuation problem for Γ, that is, we shall define functions determining the curve in a certain higher-dimensional space.

Continuation of codimension-1 homoclinic orbits

A heteroclinic solution x(t) connecting two equilibria x- and x+ of an ODE system (1) satisfies limtx(t)=x,limt+x(t)=x+.We shall concentrate on the special case where x+ = x- ≡ x0, called homoclinic solutions, as they have particular importance in global bifurcation theory. The approach is easily extended to the case where x+ ≠ x provided a careful count is taken of the codimension of the connecting orbit, and the consequent number of free parameters α required.

For the case of a

Locating codimension-2 equilibrium bifurcations

When investigating a two-parameter problem, one usually encounters higher-order degeneracies along codimension-1 bifurcation curves. Some of these degeneracies are determined by the Jacobian matrix, while others can only be detected taking into account nonlinear terms. For this reason we start this section with the nonlinear normal forms for codimension-1 equilibrium bifurcations, namely the fold and Hopf. Appropriate coefficients in these normal forms play the role of test functions for

Locating codimension-2 homoclinic bifurcations

In Section 6 we considered numerical methods for the two-parameter continuation of homoclinic orbits to equilibria. Suppose that we continue a branch of regular codimension-1 homoclinic orbits to (1) in two parameters, i.e., α ∈ ℝ2, so that a homoclinic loop to the equilibrium xe(s) exist whenever α = α(s) with s ∈ ℝ1, see Section 2.2 and Section 6. Here the one-dimensional parameter s is typically Keller's pseudo-arclength. We refer to these homoclinic solutions as the primary homoclinic

Continuation of codimension-2 equilibrium bifurcations

In this section we give regular defining systems based on bordering techniques for continuing codimension-2 equilibrium bifurcations of (1) in three parameters. Detailed proofs of regularity can be found in Govaerts [32]. Test functions to detect codimension−3 bifurcations due to linear terms are also given. All defining and test functions are implemented in CONTENT (Kuznetsov and Levitin [52]). Earlier methods based on minimally augmented systems with determinants were proposed by Khibnik [45]

Normal forms for codimension-2 equilibrium bifurcations

Bifurcations of phase portraits of (1) are determined by the normal form coefficients at critical parameter values. For example, depending on the values of certain coefficients for the fold-Hopf and double-Hopf bifurcation, the system may exhibit quasi-periodic and “chaotic” behavior. In this section we show how such coefficients can be computed numerically, while the final section deals with the nondegeneracy of parametric unfolding. If the critical parameter values and the equilibrium

Branch switching at codimension-2 bifurcations

Suppose that the system (1) has a codimension-2 equilibrium x = 0 at α = 0. Generically, one expects curves of codimension-1 bifurcations to emerge from the origin in the α-plane. In this section we describe methods to start the continuation of such curves based on the information available at the singularity. As we mentioned in Section 10, there are not only codimension-1 equilibria that emanate from codimension-2 points but, depending on the type, there may also be codimension-1 families of

First page preview

First page preview
Click to open first page preview

References (0)

Cited by (139)

View all citing articles on Scopus
View full text