Equivariant bifurcation theory

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Dynamical systems with appropriate symmetry properties are called equivariant dynamical systems. The theory of bifurcations for dynamical systems with symmetry is called equivariant bifurcation theory.

Terminology

We first recall how symmetries of solutions can be characterized; see Equivariant Dynamical Systems for more discussion, or the following references that discuss equivariant bifurcation theory: Golubitsky, Stewart, and Schaeffer (1988), Crawford and Knobloch (1991), Chossat and Lauterbach (2000), Golubitsky and Stewart (2002), and Hoyle (2006).

For \(x \in \mathbb R^n\) and a parameter \(\lambda \in \mathbb R\ ,\) a dynamical system \[ \frac{dx}{dt} = f(x,\lambda) \] is said to be equivariant with respect to the group \(\Gamma\) if \[ f(\gamma \cdot x,\lambda) = \gamma \cdot f(x,\lambda) \] for all \(\gamma \in \Gamma\ .\) The symmetry of a solution \(x_0(t)\) for this dynamical system is characterized by the isotropy subgroup \[ \Sigma_{x_0(t)} = \{\gamma \in \Gamma: \gamma \cdot x_0(t) = x_0(t) \} \] for a spatial symmetry, or \[ \Sigma_{x_0(t)} = \{(\gamma,\theta) \in \Gamma \times S^1 : \gamma \cdot x_0(t + \theta) = x_0(t) \} \] for a spatiotemporal symmetry. Associated with an isotropy subgroup is the fixed point subspace \[ {\rm Fix}[\Sigma_{x_0(t)}] = \{x(t) \in \mathbb R^n: \sigma \cdot x(t) = x(t) \; {\rm for} \; {\rm all} \; \sigma \in \Sigma_{x_0(t)} \}. \]

To understand the statement of the equivariant branching lemma and the equivariant Hopf bifurcation theorem below, we recall definitions related to representations of groups as matrices defined on a field \(\mathbb F;\) see Equivariant Dynamical Systems for more discussion. A representation of \(\Gamma\) is irreducible if the only proper subspace of \(\mathbb F^n\) left invariant by all elements of \(\Gamma\) is the origin. A representation is absolutely irreducible on \(\mathbb F^n\) if all linear maps \(A\) on \(\mathbb F\) that commute with all \(\gamma \in \Gamma\) are multiples of the identity matrix. Finally, a representation of \(\Gamma\) is said to act \(\Gamma\)-simply on\({\mathbb R^n}\) if it satisfies one of the following two conditions: i) it is irreducible, but not absolutely irreducible, or ii) it is isomorphic to the direct sum of two copies of an absolutely irreducible representation. Generically for a bifurcation involving imaginary eigenvalues as in the Equivariant Hopf Bifurcation Theorem discussed below, \(\Gamma\) acts \(\Gamma\)-simply on the center eigenspace; see Golubitsky, Stewart, and Schaeffer (1988).

The Equivariant Branching Lemma

One of the most important results in equivariant bifurcation theory is the Equivariant Branching Lemma. Without loss of generality, we assume that the bifurcation occurs at \(\lambda=0\) and involves an equilibrium at \(x=0\ .\)

Equivariant Branching Lemma

Let \(\Gamma\) be a finite group or a compact Lie group acting absolutely irreducibly on \(\mathbb R^n\ ,\) and let

\[ \frac{dx}{dt} = f(x;\lambda) \ :\]

be a \(\Gamma\)-equivariant bifurcation problem such that as \(\lambda\) passes through \(0\ ,\) real eigenvalues (of multiplicity \(n\)) of \({\rm D}f(0;0)\) pass through zero with nonzero speed, i.e., the derivative of the real part of these eigenvalues with respect to \(\lambda\) is nonzero. Let \(\Sigma\) be an isotropy subgroup satisfying

\[ {\rm dim} ({\rm Fix}(\Sigma)) = 1. \] Then there exists a unique equilibrium solution branch bifurcating from \(x=0\) such that the isotropy subgroup of solutions on the branch is \(\Sigma\ .\)

Before giving an example, we note the following:

The equivariant branching lemma does not tell us if the equilibrium solution branches bifurcate into positive or negative values of \(\lambda\ .\)

The equivariant branching lemma does not address the stability properties of the equilibria. However, symmetry can constrain the form of the Jacobian evaluated at an equilibrium, which can simplify stability calculations. See, for example, the discussion of the isotypic decomposition in Golubitsky, Stewart, and Schaeffer (1988) or Golubitsky and Stewart (2002).

Since \(\gamma \cdot 0 = 0\) for all \(\gamma \in \Gamma\ ,\) the equilibrium at \(x=0\) must be an element of all fixed point subspaces. The restriction to a one-dimensional fixed point subspace thus leads to a one-dimensional bifurcation problem in which \(x=0\) is always an equilibrium. The generic bifurcations for this restriction are either a transcritical bifurcation, or a pitchfork bifurcation if the isotropy \(\Sigma_x\) contains an element that acts by -1. The statement that the branch of equilibria is unique assumes that the symmetry-related equilibria created in a pitchfork bifurcation are identified.

The equivariant branching lemma guarantees that certain solutions will bifurcate from \(x=0\ .\) However, other solutions not guaranteed by this lemma may also bifurcate from \(x=0\ .\)

Example: Steady Bifurcation with \(D_4\) Symmetry

Generalizing Example 4 from Equivariant Dynamical Systems to include a parameter \(\lambda \in \mathbb R\ ,\) the dynamical system for \((x_1,x_2) \in \mathbb R^2\) given by \[ \left( \begin{array}{c} \frac{dx_1}{dt} \\ \frac{dx_2}{dt} \end{array} \right) = \left( \begin{array}{c} g(x_1^2, x_2^2,\lambda) x_1 \\ g(x_2^2,x_1^2,\lambda) x_2 \end{array} \right) \] is equivariant with respect to \(D_4 = \langle \gamma_1,\gamma_2 \rangle\ ,\) where \[ \gamma_1: (x_1,x_2) \rightarrow (x_1,-x_2), \qquad \gamma_2: (x_1,x_2) \rightarrow (-x_2,x_1). \] Notice that the origin \((x_1,x_2) = (0,0)\) is an equilibrium. This system has the following one-dimensional fixed point subspaces associated with the given isotropy subgroups:

Isotropy Subgroup	Fixed Point Subspace
\(Z_2(\gamma_1)\)	\(\{(a,0) \| a \in \mathbb R \}\)
\(Z_2(\gamma_2 \cdot \gamma_1)\)	\(\{(a,a) \| a \in \mathbb R \}\)

Here \(Z_2(\gamma_1) = \langle \gamma_1 \rangle\) and \(Z_2(\gamma_2 \cdot \gamma_1) = \langle \gamma_2 \cdot \gamma_1 \rangle.\) If its hypotheses are satisfied, the equivariant branching lemma thus guarantees that branches with these isotropy subgroups will bifurcate from the origin. Notice that the first fixed point subspace in this table has a symmetry-related fixed point subspace \(\{(0,a)|a \in \mathbb R\},\) and the second has a symmetry-related fixed point subspace \(\{(a,-a)|a \in \mathbb R\};\) see Example 4 of Equivariant Dynamical Systems. Typically in applying the equivariant branching lemma, one lists only fixed point subspaces and equilibrium solution branches that are not related by symmetry.

It is instructive to consider the following vector field which undergoes a steady bifurcation with \(D_4\) symmetry: \[ \frac{dx_1}{dt} = \lambda x_1 + A_R (x_1^2 + x_2^2) x_1 + B_R x_1^3, \] \[ \frac{dx_2}{dt} = \lambda x_2 + A_R (x_1^2 + x_2^2) x_2 + B_R x_2^3. \] This vector field is obtained as a third order truncation of Example 7 of Equivariant Dynamical Systems. We assume the nondegeneracy conditions \(A_R+B_R \neq 0, 2 A_R + B_R \neq 0\ ,\) and \(B_R \neq 0\ .\) It is readily shown that the following equilibria exist with associated eigenvalues determining their stability:

Equilibria	Eigenvalues
\((0,0)\)	\(\lambda, \lambda\)
\(\left( \pm \sqrt{-\frac{\lambda}{A_R+B_R}},0 \right)\)	\(-2 \lambda, \frac{\lambda B_R}{A_R + B_R}\)
\(\left( 0,\pm \sqrt{-\frac{\lambda}{A_R+B_R}} \right)\)	\(-2 \lambda, \frac{\lambda B_R}{A_R + B_R}\)
\(\left( \pm \sqrt{-\frac{\lambda}{2 A_R+B_R}},\pm \sqrt{-\frac{\lambda}{2 A_R+B_R}} \right)\)	\(-2 \lambda, \frac{-2 \lambda B_R}{2A_R+B_R}\)

For the equilibria not at the origin, the first eigenvalue indicates stability to perturbations within the appropriate fixed point subspace, and the second indicates stability transverse to the subspace. These equilibria exist only when the arguments of the square roots are nonnegative; this requirement determines if the equilibrium solution branches bifurcate into positive or negative values of \(\lambda\ .\) In this example the only equilibria near \(\lambda=0\) are those guaranteed by the equivariant branching lemma.

The Equivariant Hopf Bifurcation Theorem

Another important result in equivariant bifurcation theory is the Equivariant Hopf Bifurcation Theorem. Without loss of generality, we assume that the bifurcation occurs at \(\lambda =0\) and involves an equilibrium at \(x = 0\ .\)

Equivariant Hopf Bifurcation Theorem

Let \(\Gamma\) be a finite group or a compact Lie group acting \(\Gamma\)-simply on \(\mathbb R^n\ ,\) where \(n\) is even. Let

\[ \frac{dx}{dt} = f(x;\lambda) \ :\]

be a \(\Gamma\)-equivariant bifurcation problem such that as \(\lambda\) passes through \(0\ ,\) complex conjugate eigenvalues (each of multiplicity \(n/2\)) of \({\rm D}f(0,0)\) pass through the imaginary axis with nonzero speed. Let \(\Sigma \subset \Gamma \times S^1\) be an isotropy subgroup satisfying

\[ {\rm dim} ({\rm Fix}(\Sigma)) = 2. \ :\]

Then there exists a unique branch of periodic orbits bifurcating from \(x=0\) such that the isotropy subgroup of each solution is \(\Sigma\ .\)

Before giving an example, we note the following

Like the equivariant branching lemma, the equivariant Hopf bifurcation theorem does not tell us if the solution branches bifurcate into positive or negative values of \(\lambda\ ,\) nor does it address the stability of the periodic orbits.

The \(S^1\) symmetry in the equivariant Hopf bifurcation theorem is associated with writing the vector field into normal form, and its action on a periodic orbit may be interpreted as a time shift.

The equivariant Hopf bifurcation theorem guarantees that certain solutions will bifurcate from \(x=0\ .\) However, other solutions not guaranteed by the theorem may also bifurcate from \(x=0\ .\)

Example: The Hopf Bifurcation with \(O(2)\) Symmetry

We consider a dynamical system for \((z_1,z_2) \in \mathbb C^2 \cong \mathbb R^4\) which satisfies the hypotheses of the equivariant Hopf bifurcation theorem, and which is equivariant with respect to \(O(2) \times S^1\ ,\) where \(O(2) = \langle T_\phi,\kappa \rangle\) with \[ T_\phi:(z_1,z_2) \rightarrow (e^{i \phi} z_1,e^{-i \phi} z_2), \;\; \phi \in [0, 2\pi), \qquad \kappa: (z_1,z_2) \rightarrow (z_2,z_1), \] and \(S^1\) is the normal form symmetry with action \[ N_\sigma:(z_1,z_2) \rightarrow (e^{i \sigma} z_1,e^{i \sigma} z_2), \qquad \sigma \in [0,2 \pi). \] The isotropy subgroups and fixed point subspaces for this problem can be shown (see e.g. Golubitsky, Stewart, and Schaeffer 1988) to be given by

Name	Isotropy Subgroup	Fixed Point Subspace	Dimension of Fixed Point Subspace
trivial state	\(O(2) \times S^1\)	\(z_1 = z_2 = 0\)	\(0\)
SW	\(Z_2(\kappa) \times Z_2^c\)	\(z_1 = z_2\)	\(2\)
TW	\(\widehat{SO}(2)\)	\(z_2 = 0\)	\(2\)
nonsymmetric	\(Z_2^c\)	\(\mathbb C^2\)	\(4\)

Here \(\widehat{\rm SO}(2) \equiv \{N_\phi \cdot T_{-\phi} : \phi \in [0,2 \pi) \}\) and \(Z_2^c \equiv \langle N_\pi \cdot T_\pi \rangle.\) The names of the solutions come from their interpretation for translation-invariant systems with periodic boundary conditions, where TW stands for traveling waves and SW stands for standing waves; see Crawford and Knobloch (1991). The TW are also called rotating waves; we do not distinguish between the symmetry-related TW solutions that lie in the fixed point subspaces \(z_2 = 0\) and \(z_1 = 0\ .\) Thus, by the equivariant Hopf bifurcation theorem, there are branches of two types of distinct periodic orbits near \(\lambda=0\ .\)

The \(O(2)\) symmetry forces the eigenvalues \(\pm i\omega_0\) of the trivial state to have double (algebraic and geometric) multiplicity at \(\lambda=0\ .\) Near \(\lambda=0\) the normal form equations for the Hopf bifurcation with \(O(2)\) symmetry truncated at cubic order take the form \[ \frac{dz_1}{dt} = (\lambda + i \omega) z_1 + A (|z_1|^2 + |z_2|^2) z_1 + B |z_1|^2 z_1, \] \[ \frac{dz_2}{dt} = (\lambda + i \omega) z_2 + A (|z_1|^2 + |z_2|^2) z_2 + B |z_2|^2 z_2, \] where \(\lambda\) and \(\omega=\omega_0+O(\lambda)\) are real, and \(A=A_R + i A_I\ ,\) \(B=B_R + i B_I\) are complex and independent of \(\lambda\ .\) We assume the nondegeneracy conditions \(A_R + B_R \neq 0\ ,\) \(2 A_R + B_R \neq 0\ ,\) and \(B_R \neq 0\ .\) These equations are most conveniently studied by letting \[ z_1 = r_1 e^{i \theta_1}, \qquad z_2 = r_2 e^{i \theta_2}, \] which gives \[ \frac{dr_1}{dt} = \lambda r_1 + A_R (r_1^2 + r_2^2) r_1 + B_R r_1^3, \] \[ \frac{dr_2}{dt} = \lambda r_2 + A_R (r_1^2 + r_2^2) r_2 + B_R r_2^3, \] \[ \frac{d\theta_1}{dt} = \omega + A_I (r_1^2 + r_2^2) + B_I r_1^2, \] \[ \frac{d\theta_2}{dt} = \omega + A_I (r_1^2 + r_2^2) + B_I r_2^2. \] The equations for \(r_1\) and \(r_2\) decouple from the remaining equations and possess \(D_4\) symmetry; see the Steady Bifurcation with \(D_4\) Symmetry example above. Their equilibrium solutions are given in that example, with the condition that \(r_1\) and \(r_2\) must now be nonnegative. It is readily seen that the TW and SW solutions are periodic orbits in the original equations; indeed, these are the solutions which are guaranteed to bifurcate from the trivial state by the equivariant Hopf bifurcation theorem, and there are no other periodic solutions near \(\lambda=0\ .\) The figure shows the \((A_R,B_R)\) plane together with bifurcation diagrams showing the squared amplitude \(r_1^2+r_2^2\) as a function of \(\lambda\) in each of six distinct regions. The solid (resp., dashed) lines indicate (orbitally) stable (resp., orbitally unstable) solutions, and the signs of the eigenvalues are indicated in the order they appear in the table. The \(-2 \lambda\) eigenvalue for the TW and SW solutions corresponds to amplitude stability, that is, perturbations within the fixed point subspace, while the second eigenvalue indicates stability in the \(r_1^2-r_2^2\) direction. Orbital stability is indicated because the presence of continuous symmetry forces the TW and SW solutions to have neutral stability with respect to perturbations along the group orbit generated by \(T_\phi\) and \(N_\sigma\ .\)

Figure 1: Possible bifurcation diagrams for Hopf bifurcation with \(O(2)\) symmetry.