# Equivariant dynamical systems

Post-publication activity

Curator: Edgar Knobloch

Equivariant dynamical systems are dynamical systems that have symmetries. A symmetry of a dynamical system is a transformation that takes solutions to solutions. The equations describing a physical or biological system may have symmetries as a result of the system geometry, modeling assumptions, and/or simplifying normal form transformations.

## Group Theory

The natural language for describing symmetry properties of a dynamical system is that of group theory, which we briefly review. A group $$\Gamma$$ is a set of elements with an operation $$\cdot$$ which satisfies

• closure$\gamma_1 \cdot \gamma_2 \in \Gamma$ for all $$\gamma_1 \in \Gamma$$ and $$\gamma_2 \in \Gamma\ ,$$
• associative law$\gamma_1 \cdot (\gamma_2 \cdot \gamma_3) = (\gamma_1 \cdot \gamma_2) \cdot \gamma_3$ for all $$\gamma_1 \in \Gamma\ ,$$ $$\gamma_2 \in \Gamma\ ,$$ and $$\gamma_3 \in \Gamma\ ,$$
• existence of identity element: there exists an element $$e$$ such that $$e \cdot \gamma = \gamma \cdot e = \gamma$$ for all $$\gamma \in \Gamma\ ,$$
• existence of inverses: for every $$\gamma \in \Gamma\ ,$$ there is a unique inverse $$\gamma^{-1} \in \Gamma$$ such that $$\gamma^{-1} \cdot \gamma = \gamma \cdot \gamma^{-1} = e\ .$$

We note the following:

• The elements of $$\Gamma$$ could be numbers, matrices, transformations such as permutation, rotation, or reflection, or other abstract objects.
• The operation in the definition of a group is often referred to as multiplication, but it does not need to be multiplication in the usual sense. For example, the set of integers is a group under addition. Here zero is the identity element and the inverse of an element is minus one times that element.
• A group is said to be abelian if the group operation is commutative, that is, if $$\gamma_1 \cdot \gamma_2 = \gamma_2 \cdot \gamma_1$$ for all $$\gamma_1 \in \Gamma$$ and $$\gamma_2 \in \Gamma\ .$$ A group is said to be non-abelian if it is not abelian.
• The order of a group $$\Gamma$$ is the number of elements in $$\Gamma\ .$$ If this is finite, $$\Gamma$$ is called a finite group. If this is infinite, $$\Gamma$$ is called an infinite group.
• A Lie group is a group whose elements have the topology of an $$m$$-dimensional smooth manifold, and whose group operation is a smooth function of the elements. When $$m>0$$ Lie groups are useful for describing continuous symmetries.
• A group $$\Gamma$$ is said to be generated by a subset $$S$$ of elements of $$\Gamma$$ if every element of $$\Gamma$$ can be expressed as the product of finitely many elements of $$S$$ and their inverses. Notationally, if, for example, $$\Gamma$$ is generated by $$\gamma_1$$ and $$\gamma_2\ ,$$ we write $$\Gamma = \langle \gamma_1,\gamma_2 \rangle.$$
• A subgroup $$H$$ of a group $$\Gamma$$ is a subset of $$\Gamma$$ which is itself a group under the same operation. In particular, $$H$$ must satisfy the closure property under the operation, contain the identity element $$e\ ,$$ and contain inverses of all its elements.
• A homomorphism between groups $$\Gamma$$ and $$G$$ is a map $$h:\Gamma \rightarrow G$$ such that $$h(\gamma_1 \cdot \gamma_2) = h(\gamma_1) \cdot h(\gamma_2)$$ for all $$\gamma_1, \gamma_2 \in \Gamma\ .$$ An isomorphism is a homomorphism $$h:\Gamma \rightarrow G$$ such that for every $$g \in G$$ there is exactly one $$\gamma \in \Gamma$$ such that $$h(\gamma) = g\ .$$
• An $$n$$-dimensional representation of a group $$\Gamma$$ is a homomorphism from $$\Gamma$$ to the group of $$n \times n$$ matrices defined on a field $$\Bbb F\ .$$ Typically $$\Bbb F$$ is $$\Bbb R$$ or $$\Bbb C\ ,$$ that is, the matrices have either real or complex entries. A representation is faithful if this mapping is an isomorphism onto a subgroup of $$n \times n$$ matrices. A representation of $$\Gamma$$ is irreducible if the only proper subspace of $$\Bbb F^n$$ left invariant by all elements of $$\Gamma$$ is the origin. A representation is absolutely irreducible on $$\Bbb F^n$$ if all linear maps $$A$$ on $$\Bbb F^n$$ that commute with all $$\gamma \in G$$ are scalar multiples of the identity matrix.

We list some examples of groups:

• The cyclic group $$Z_n$$ is the group of $$n$$ elements generated by a single element $$\gamma_1\ ,$$ that is, $$Z_n = \langle \gamma_1 \rangle\ ,$$ where $$\gamma_1^n = e\ .$$
• The dihedral group $$D_n$$ is the group of $$2 n$$ elements generated by $$\gamma_1$$ and $$\gamma_2\ ,$$ that is, $$D_n = \langle \gamma_1, \gamma_2 \rangle\ ,$$ where $$\gamma_1^2 = e, \gamma_2^n = e\ ,$$ and $$\gamma_2 \cdot \gamma_1 \cdot \gamma_2 = \gamma_1\ .$$ It is sometimes useful to think of the dihedral group $$D_n$$ as being the set of transformations of a regular polygon with $$n$$ sides into itself; here $$\gamma_1$$ is a reflection of the polygon about an appropriate symmetry axis passing through the center of the polygon, and $$\gamma_2$$ is the rotation of the polygon through the angle $$2 \pi/n\ .$$
• The permutation group $$S_n$$ is the group of $$n!$$ elements corresponding to all possible permutations of $$n$$ objects. Every element of the permutation group can be written as a product of simple transpositions in which two objects are swapped.
• The orthogonal group $$O(n)$$ is the group of $$n \times n$$ orthogonal matrices under the group operation of matrix multiplication. Recall that a matrix $$A$$ is said to be orthogonal if $$A A^T = Id\ ,$$ where $$A^T$$ is the transpose of $$A$$ and $$Id$$ is the $$n \times n$$ identity matrix. $$O(n)$$ is an infinite group.
• The special orthogonal group $$SO(n)$$ is the subgroup of $$O(n)$$ for which the matrices have unit determinant. The group $$SO(2)$$ and the circle group $$S^1$$ (the group of all complex numbers with unit modulus under the operation of multiplication) are isomorphic. Sometimes $$S^1$$ is written as the 1-torus $$T^1\ .$$ $$SO(n)$$ is an infinite group and a Lie group.

There are many books which cover group theory at different levels of detail and abstraction, such as Hamermesh (1989), Lomont (1993), Sternberg (2004), and Chapter 3 of Hoyle (2006).

## Equivariant Dynamical Systems

Consider the ordinary differential equation $\frac{dx}{dt}=f(x),$ where $$x$$ is in a manifold $$M\ ,$$ and let $$\Gamma$$ be a group acting on $$M\ .$$ This equation is said to be $$\Gamma$$-equivariant if $$f$$ commutes with the group action of $$\Gamma\ ,$$ that is $f(\gamma \cdot x)= \hat{\gamma} \cdot f(x)$ for all $$\gamma \in \Gamma$$ and $$x \in M\ ,$$ where $$\hat{\gamma}$$ acts on the tangent space $$TM\ .$$ When $$M$$ is a Euclidean space $$\mathbb R^N$$ or $$\mathbb C^N\ ,$$ $$\hat{\gamma} = \gamma.$$ An important consequence of $$\Gamma$$-equivariance is that if a solution $$x(t)$$ solves the ordinary differential equation, then so does $$\gamma \cdot x(t)$$ for all $$\gamma \in \Gamma\ .$$ The set $$\Gamma \cdot x(t) = \{\gamma \cdot x(t) | \gamma \in \Gamma \}$$ is called the group orbit of $$x(t)\ ;$$ thus, if we find one solution to an equivariant ordinary differential equation, the whole group orbit of this solution will also exist as solutions.

A similar definition of equivariance holds for maps.

References that discuss equivariant dynamical systems include Golubitsky, Stewart, and Schaeffer (1988), Crawford and Knobloch (1991), Chossat and Lauterbach (2000), Golubitsky and Stewart (2002), and Hoyle (2006).

### Example 1

Suppose $\frac{dx}{dt} = a_1 x + a_3 x^3 \equiv f(x),$ where $$x \in \mathbb R\ .$$ One shows that this is equivariant with respect to the group $$Z_2 = \langle \gamma \rangle\ ,$$ where $\gamma: x \rightarrow -x,$ as follows. First, $f(\gamma \cdot x) = f(-x) = a_1 (-x) + a_3 (-x)^3 = -(a_1 x + a_3 x^3).$ Next, notice that $$x \in \mathbb R$$ so that $$\hat{\gamma} = \gamma\ .$$ Now, $\gamma \cdot f(x) = -f(x) = -(a_1 x + a_3 x^3),$ so $$f(\gamma \cdot x) = \gamma \cdot f(x)\ .$$

By inspection there is an equilibrium at $$x=\sqrt{-a_1/a_3} \equiv x_e\ ,$$ provided the argument in the square root is nonnegative. We therefore expect that there will also be an equilibrium at $$\gamma \cdot x_e = -\sqrt{-a_1/a_3}\ ,$$ as may be readily verified. There is also an equilibrium at $$x \equiv x_0 =0\ ;$$ we verify that $$\gamma \cdot x_0 = 0 = x_0$$ is (trivially) also an equilibrium.

### Example 2: Rayleigh-Bénard Convection

Consider two-dimensional Rayleigh-Bénard convection in which a layer of viscous fluid is confined between rigid horizontal plates separated by a distance $$H=1\ ,$$ with the top and bottom plates maintained at fixed temperatures $$T_1$$ and $$T_0$$ ($$T_0>T_1$$), respectively. Gravity acts in the negative $$z$$-direction with acceleration $$g$$ (see Figure 1). If the temperature difference across the layer $$\Delta T = T_0 - T_1$$ is below a critical value $$\Delta T_c\ ,$$ the fluid is motionless with heat transferred by conduction. At $$\Delta T = \Delta T_c\ ,$$ the buoyancy force overcomes the stabilizing effects of thermal diffusion and viscous damping, and the conduction state loses stability to a convecting state characterized by fluid motion.

In the Boussinesq approximation the nondimensional evolution equations for the fluid velocity (written in terms of the stream function $$\psi$$) $u \equiv -\frac{\partial \psi}{\partial z} \hat{x} + \frac{\partial \psi}{\partial x} \hat{z},$ and the perturbation $$\theta$$ to the conduction state temperature profile are $\frac{\partial \; \nabla^2 \psi}{\partial t} + \frac{\partial \psi}{\partial x} \frac{\partial \nabla^2 \psi}{\partial z} - \frac{\partial \psi}{\partial z} \frac{\partial \nabla^2 \psi}{\partial x} = R P \frac{\partial \theta}{\partial x} + P \nabla^4 \psi,$ $\frac{\partial \theta}{\partial t} + \frac{\partial \psi}{\partial x} \frac{\partial \theta}{\partial z} - \frac{\partial \psi}{\partial z} \frac{\partial \theta}{\partial x} = \frac{\partial \psi}{\partial x} + \nabla^2 \theta.$ Here $$R$$ and $$P$$ are nondimensional parameters known as the Rayleigh number (which is proportional to $$\Delta T$$) and the Prandtl number, respectively; see, e.g., Hirschberg and Knobloch (1997).

In the simplest case the temperature at the top and bottom plates is held constant, and there is no tangential stress, corresponding to the boundary conditions $\psi = \frac{\partial^2 \psi}{\partial z^2} = \theta = 0, \qquad z = 0,1.$ In the horizontal direction we choose no-slip and perfectly insulating lateral boundary conditions: $\psi = \frac{\partial \psi}{\partial x} = \frac{\partial \theta}{\partial x} = 0, \qquad x =\pm L.$ These equations are solved by the conduction solution $$(\psi,\theta) = (0,0)\ ;$$ linearization about the conduction solution shows that it loses stability as the Rayleigh number increases past a critical value $$R_c\ ,$$ which depends on $$L\ .$$ The eigenfunctions at onset of convection for two different $$L$$ values are shown in the Figure 2. For more detail on these eigenfunctions, see Drazin (1975), Hirschberg and Knobloch (1997). Figure 2: Eigenfunctions for Rayleigh-Bénard convection.

For Rayleigh numbers $$R = R_c (1 + \epsilon^2 \alpha)$$ with $$\epsilon \ll 1$$ and $$\alpha>0\ ,$$ that is, just beyond the onset of convection, the solution to the evolution equations resembles the eigenfunction of the linear stability problem: $\psi(x,z,\tau) = \epsilon A(\tau) f(x) \sin \pi z + O(\epsilon^2)$ $\theta(x,z,\tau) = \epsilon A(\tau) g(x) \sin \pi z + O(\epsilon^2),$ where the $$O(1)$$ convection amplitude $$A$$ evolves on the slow time scale $$\tau = \epsilon^2 t\ ,$$ and the $$O(\epsilon^2)$$ terms are slaved to $$A(\tau)\ .$$ Asymptotic analysis now leads to an evolution equation (an amplitude equation) for $$A\ .$$ Instead of deriving this amplitude equation explicitly, we discuss the form it must take based on symmetry considerations.

The evolution equations and boundary conditions have a reflection symmetry about $$z=1/2\ ;$$ specifically, if $$(\psi(x,z,t),\theta(x,z,t))$$ is a solution, so is the solution obtained by taking $$z \rightarrow 1-z\ ,$$ namely $$(-\psi(x,1-z,t),-\theta(x,1-z,t))\ .$$ Given the above expansions, we see that $-\psi(x,1-z,\tau) = -\epsilon A(\tau) f(x) \sin \pi z + O(\epsilon^2),$ $-\theta(x,1-z,\tau) = -\epsilon A(\tau) g(x) \sin \pi z + O(\epsilon^2).$ For the amplitude equation, this implies that if $$A(\tau)$$ is a solution, so is $$-A(\tau)\ .$$

The evolution equations and boundary conditions also have a reflection symmetry about $$x=0\ ;$$ specifically, if $$(\psi(x,z,t),\theta(x,z,t))$$ is a solution, so is the solution obtained by taking $$x \rightarrow -x\ ,$$ namely $$(-\psi(-x,z,t),\theta(-x,z,t))\ .$$ Given the above expansions, we see that $-\psi(-x,z,\tau) = -\epsilon A(\tau) f(-x) \sin \pi z + O(\epsilon^2)$ $\theta(-x,z,\tau) = \epsilon A(\tau) g(-x) \sin \pi z + O(\epsilon^2),$ For $$L = 1\ ,$$ Figure 2 shows that $f(-x) = f(x), \qquad g(-x) = -g(x).$ In this case if $$A(\tau)$$ is a solution to the amplitude equation, then so is $$-A(\tau)\ .$$ In contrast, for $$L = 1.5\ ,$$ $f(-x) = -f(x), \qquad g(-x) = g(x).$ In this case $$-\psi(-x,z,\tau) = \psi(x,z,\tau)\ ,$$ $$\theta(-x,z,\tau) = \theta(x,z,\tau)\ ,$$ and the solution is unchanged by reflection in $$x\ .$$ Effectively, the reflection takes $$A \rightarrow A\ .$$ In neither case does the $$x$$ reflection impose any additional requirements on the amplitude equation.

Together the above symmetry arguments imply that if $$A(\tau)$$ is a solution to the amplitude equation so is $$-A(\tau)\ .$$ In other words, $\frac{d A}{d \tau} = f(A) \qquad \Rightarrow \qquad \frac{d (-A)}{d \tau} = f(-A).$ Thus the amplitude equation is equivariant under the group generated by the action $$A \rightarrow -A\ .$$ In the Taylor expansion $f(A) = \sum_{j=1}^{\infty} a_j A^j,$ it is therefore necessary that $$(-1)^j = -1\ ,$$ that is, that $$j$$ is odd. Thus, $f(A) = h(A^2) A$ for some function $$h\ .$$ Truncation at cubic order leads to $\frac{d A}{d \tau} = a_1 A + a_3 A^3,$ where $$a_1$$ and $$a_3$$ are real, and hence the same equation as in Example 1. Thus the transition to convection is described by a pitchfork bifurcation as $$a_1$$ crosses through zero. For the convection problem, one finds that $$a_1 \sim \alpha \sim R - R_c\ ,$$ and that $$a_3<0\ .$$ The fixed point with $$A=0$$ corresponds to the conduction state, and is stable for $$\alpha<0\ .$$ For $$\alpha>0\ ,$$ the conduction state is unstable, and there are two stable symmetry-related fixed points with $$A \neq 0\ ,$$ corresponding to symmetry-related convection states (see Figure 3). Figure 3: Velocity fields just above onset for the convection states related by the reflection symmetry about $$z=1/2\ .$$

### Example 3: The Lorenz Equations

Consider the evolution equations for Rayleigh-Bénard convection given in the previous example, with the ansatz $\psi(x,z,t) = 3 X(t) \sin k_c x \sin \pi z,$ $\theta(x,z,t) = \frac{27}{2 \sqrt{2}} \frac{\pi^3}{R} Y(t) \cos k_c x \sin \pi z - \frac{R_c}{\pi R} Z(t) \sin 2 \pi z,$ where $$k_c = \pi/\sqrt{2}, R_c = 27 \pi^4 / 4\ .$$ Note that $$\psi$$ and $$\theta$$ satisfy different lateral boundary conditions from the previous example, namely $\psi = \frac{\partial^2 \psi}{\partial x^2} = \frac{\partial \theta}{\partial x} = 0, \qquad x = 0,\frac{2 \pi}{k_c}.$

Performing a Galerkin projection, we obtain the Lorenz equations (Lorenz (1963)): $\dot{X} = -P X + P Y,$ $\dot{Y} = -X Z + r X - Y,$ $\dot{Z} = X Y - b Z,$ where the dot refers to differentiation with respect to $$3 \pi^2 t/2\ ,$$ $$r = R/R_c$$ with $$R$$ being the Rayleigh number, $$P$$ is the Prandtl number (following Lorenz, one typically takes $$P=10$$), and $$b = 8/3\ .$$

It is readily shown that the Lorenz equations are equivariant with respect to the group $$Z_2$$ generated by the action $\rho: (X,Y,Z) \rightarrow (-X,-Y,Z).$ This reflection symmetry is responsible for the presence of a pitchfork bifurcation at $$r=1\ .$$ Furthermore, there are two homoclinic trajectories to the origin at $$r=13.926$$ which are related by this symmetry: the symmetry implies that if the homoclinic orbit $$(X_h(t),Y_h(t),Z_h(t))$$ is a solution, then so is $$\rho \cdot (X_h(t),Y_h(t),Z_h(t)) = (-X_h(t),-Y_h(t),Z_h(t))\ .$$ For these parameter values this fact is responsible for a homoclinic explosion in this system, which is in turn responsible for the presence of chaos (Sparrow (1982)).

### Example 4

Suppose $\frac{dx}{dt} \equiv \left( \begin{array}{c} \frac{dx_1}{dt} \\ \frac{dx_2}{dt} \end{array} \right) = \left( \begin{array}{c} g(x_1^2, x_2^2) x_1 \\ g(x_2^2,x_1^2) x_2 \end{array} \right) \equiv \left( \begin{array}{c} f_1(x_1,x_2) \\ f_2(x_1,x_2) \end{array} \right) \equiv f(x),$ where $$x \equiv (x_1,x_2) \in \mathbb R^2\ .$$ We claim that this equation is equivariant with respect to $$D_4 = \langle \gamma_1,\gamma_2 \rangle\ ,$$ where $\gamma_1: (x_1,x_2) \rightarrow (x_1,-x_2),$ $\gamma_2: (x_1,x_2) \rightarrow (-x_2,x_1).$ This is verified as follows. Since $f(\gamma_1 \cdot x) = \left( \begin{array}{c} f_1(x_1,-x_2) \\ f_2(x_1,-x_2) \end{array} \right) = \left( \begin{array}{c} g(x_1^2,x_2^2) x_1 \\ -g(x_1^2,x_2^2) x_2 \end{array} \right)$ and $\gamma_1 \cdot f(x) = \left( \begin{array}{c} f_1(x_1,x_2) \\ -f_2(x_1,x_2) \end{array} \right) = \left( \begin{array}{c} g(x_1^2,x_2^2) x_1 \\ -g(x_1^2,x_2^2) x_2 \end{array} \right),$ we have $$f(\gamma_1 \cdot x) = \gamma_1 \cdot f(x)\ .$$ Furthermore, since $f(\gamma_2 \cdot x) = \left( \begin{array}{c} f_1(-x_2,x_1) \\ f_2(-x_2,x_1) \end{array} \right) = \left( \begin{array}{c} -g(x_2^2,x_1^2) x_2 \\ g(x_1^2,x_2^2) x_1 \end{array} \right)$ and $\gamma_2 \cdot f(x) = \left( \begin{array}{c} -f_2(x_1,x_2) \\ f_1(x_1,x_2) \end{array} \right) = \left( \begin{array}{c} -g(x_2^2,x_1^2) x_2 \\ g(x_1^2,x_2^2) x_1 \end{array} \right),$ we have $$f(\gamma_2 \cdot x) = \gamma_2 \cdot f(x)\ .$$ Now notice that $$f(x)$$ must commute with any product of $$\gamma_1$$ and $$\gamma_2\ ;$$ for example, $$\gamma_1 \cdot \gamma_2 \cdot f(x) = \gamma_1 \cdot f(\gamma_2 \cdot x) = f(\gamma_1 \cdot \gamma_2 \cdot x)\ .$$ Thus, $$f(x)$$ commutes with all elements of $$D_4 = \langle \gamma_1,\gamma_2 \rangle\ .$$ This illustrates the general property that it is only necessary to verify equivariance with respect to the generators of a group in order to show equivariance with respect to the entire group.

Suppose that the point $$(x_1,x_2) = (a,b)$$ is an equilibrium for this vector field, where $$a \neq 0$$ and $$b \neq 0\ .$$ This implies that $$g(a^2,b^2) = g(b^2,a^2) = 0\ .$$ We expect $$\gamma_1 \cdot (a,b) = (a,-b)$$ to also be an equilibrium; this is verified by recognizing that $$g(a^2,(-b)^2) = g(a^2,b^2) = 0$$ and $$g((-b)^2,a^2) = g(b^2,a^2) = 0\ .$$ Similarly, we verify that all points on the group orbit $$\Gamma \cdot (a,b) = \{(a,b),(-a,b),(a,-b),(-a,-b),(b,a),(-b,a),(b,-a),(-b,-a)\}$$ are equilibria.

### Example 5

Suppose $\frac{dz}{dt} = (1 + i \omega) z - |z|^2 z \equiv f(z),$ where $$z \in \mathbb C$$ and $$\omega$$ is a real parameter. This equation is equivariant with respect to $$S^1 = \{\gamma_\theta | \theta \in [0,2 \pi) \},$$ where $$\gamma_\theta : z \rightarrow e^{i \theta} z\ ,$$ since $f(\gamma_\theta \cdot z) = (1 + i \omega) z e^{i \theta} - |z e^{i \theta}|^2 z e^{i \theta} = e^{i \theta} ((1 + i \omega) z - |z|^2 z) = \gamma_\theta \cdot f(z).$

Notice that $$z = e^{i \omega t}$$ is a solution to this differential equation; this is a periodic orbit. We thus expect that $$\gamma_\theta \cdot e^{i \omega t} = e^{i (\omega t + \theta)}$$ is also a solution for all $$\gamma_\theta\ ,$$ as may be verified by direct substitution. We may interpret the action of elements of $$S^1$$ on the periodic orbit as a phase shift.

### Example 6: Coupled Oscillators

Suppose that the dynamical system $\frac{d X}{dt} = F(X), \qquad X \in \mathbb R^M$ has a stable periodic orbit $$\zeta(t)$$ with minimal period $$T\ .$$ We call such a dynamical system an oscillator. For example, an oscillator could be a periodically firing neuron or a pacemaker cell in the heart. Now, suppose a set of $$N$$ identical oscillators are weakly coupled to each other, with identical all-to-all coupling, $\frac{d X_k}{dt} = F(X_k) + \epsilon \sum_j p(X_k,X_j), \qquad k=1,\cdots,N,$ where $$X_k$$ describes the state of the $$k^{\rm th}$$ oscillator. We define the phase $$\theta_k$$ of the $$k$$th oscillator according to isochrons, so that in the absence of coupling $$\frac{d\theta_k}{dt} = 2 \pi/T \equiv \omega$$ for all $$k\ ;$$ this gives $\frac{d \theta_k}{dt} = \frac{\partial \theta_k}{\partial X_k} \cdot \frac{d X_k}{dt} = \frac{\partial \theta_k}{\partial X_k} \cdot \left( F(X_k) + \epsilon \sum_j p(X_k,X_j) \right)= \omega + \epsilon \frac{\partial \theta_k}{\partial X_k} \cdot \sum_j p(X_k,X_j).$ To lowest order in $$\epsilon$$ (that is, for weak coupling), $\frac{d \theta_k}{dt} = \omega + \epsilon Z(\theta_k) \cdot \sum_j p(\zeta(\theta_k),\zeta(\theta_j)),$ where $Z(\theta_k) = \left. \frac{\partial \theta_k}{\partial X_k} \right|_{\zeta(\theta_k)}$ is called the phase response curve. It follows that the phases $$\phi_k\equiv\theta_k - \omega t$$ satisfy $\frac{d \phi_k}{dt} = \epsilon Z(\zeta(\phi_k + \omega t)) \cdot \sum_j p(\zeta(\phi_k + \omega t), \zeta(\phi_j+\omega t)),$ and hence that for $$O(\epsilon^{-1})$$ times $$\phi_k$$ obeys the averaged equation (Guckenheimer and Holmes (1983)) $\frac{d \phi_k}{dt} = \frac{\epsilon}{T} \int_0^T Z(\phi_k+\omega t) \cdot \sum_j p(\zeta(\phi_k + \omega t),\zeta(\phi_j + \omega t)) dt$ $= \frac{\epsilon}{T} \int_0^T Z(\phi_k+\omega t) \cdot \sum_j p(\zeta(\phi_k + \omega t),\zeta(\phi_j-\phi_k+\phi_k + \omega t)) dt.$ Thus $\frac{d \theta_k}{dt} = \omega + \frac{\epsilon}{2 \pi} \sum_j \int_0^{2 \pi} Z(s) \cdot p(\zeta(s),\zeta(\theta_j-\theta_k+s)) ds.$

This equation can be rewritten in the form $\frac{d \theta_k}{dt} = \omega + \sum_{j=1}^N g(\theta_j - \theta_k),$ where $$\theta_k \in [0,2 \pi)\ ,$$ $$k = 1, \cdots, N\ ,$$ and corresponds to the phase reduction of a set of identical coupled oscillators with weak, identical all-to-all coupling to other oscillators (see phase model, also Ashwin and Swift 1992, and Brown, Holmes, and Moehlis 2003).

Since this system is defined on the $$N$$-torus manifold $$T^N\ ,$$ some care must be used in determining its equivariance properties. It is convenient to embed this system in $$\mathbb C^N$$ by letting $$z_k = e^{i \theta_k}\ .$$ Then $\frac{dz_k}{dt} = i z_k \left[\omega + \sum_{j=1}^N g(\theta_j - \theta_k) \right] \equiv f_k(z),$ where $$z = (z_1,z_2,\cdots,z_N)\ ,$$ and for ease of notation we write $$\theta_j$$ for $${\rm arg}(z_j)\ ,$$ etc. We first show that this system is equivariant with respect to the simple transposition $\sigma: (z_1,\cdots,z_k,\cdots,z_l,\cdots,z_N) \rightarrow (z_1,\cdots,z_l,\cdots,z_k,\cdots z_N).$ Specifically $(f_k(\sigma \cdot z),f_l(\sigma \cdot z)) = \left( i z_l \left[\omega + \sum_{j=1}^N g(\theta_j - \theta_l) \right] ,i z_k \left[ \omega + \sum_{j=1}^N g(\theta_j - \theta_k) \right] \right) = \sigma \cdot (f_k(z),f_l(z))$ for any $$k$$ and $$l\ .$$ We next show that the system is equivariant with respect to the group $$T^1 = \{\gamma_\alpha | \alpha \in [0,2 \pi) \}$$ where $$\gamma_\alpha: \theta_k \rightarrow \theta_k + \alpha$$ for all $$k = 1,\cdots,N\ .$$ This induces the equivalent action $$\gamma_\alpha: z_k \rightarrow e^{i \alpha} z_k\ .$$ Therefore $f_k(\gamma_\alpha \cdot z) = i z_k e^{i \alpha} \left[ \omega + \sum_{j=1}^N g(\theta_j + \alpha - (\theta_k + \alpha)) \right] = i z_k e^{i \alpha} \left[ \omega + \sum_{j=1}^N g(\theta_j - \theta_k) \right] = \gamma_\alpha \cdot f_k(z).$ Note that this symmetry ultimately comes from averaging the equations, as described above.

Thus, our system is equivariant with respect to both the permutation group $$S_N$$ and the circle group $$T^1\ .$$ Putting these together properly, our system is equivariant with respect to the direct product $$S_N \times T^1\ .$$

One solution to these equations is $$\theta_k = (\omega + N g(0)) t$$ for all $$k = 1,\cdots, N\ .$$ This is an example of a phase-locked solution; see Ashwin and Swift (1992), and Brown, Holmes, and Moehlis (2003). The elements of $$S_N$$ act trivially on this solution. The elements of the circle group $$T^1$$ shift the phase to give the solutions $$\theta_k = (\omega + N g(0)) t + \alpha$$ for all $$k = 1,\cdots, N$$ and any $$\alpha\ .$$

## Symmetry Properties of Solutions of Equivariant Dynamical Systems

It is possible to classify solutions of equivariant dynamical systems based on their symmetry properties. Specifically, the symmetry of a solution $$x_0(t)\in M$$ is characterized by the isotropy subgroup $\Sigma_{x_0(t)} = \{\gamma \in \Gamma: \gamma \cdot x_0(t) = x_0(t) \},$ that is, the set (in fact, subgroup) of all group elements which leave the solution $$x_0(t)$$ unchanged. The isotropy subgroups of $$x_0(t)$$ and $$\gamma \cdot x_0(t)$$ are related by the conjugacy $$\Sigma_{\gamma \cdot x_0(t)} = \gamma \cdot \Sigma_{x_0(t)} \cdot \gamma^{-1}\ ,$$ because if $$\sigma \cdot x_0(t) = x_0(t)\ ,$$ then $$(\gamma \cdot \sigma \cdot \gamma^{-1}) \cdot (\gamma \cdot x_0(t)) = \gamma \cdot x_0(t)\ .$$ Isotropy subgroups are often usefully organized according to the isotropy lattice; see Golubitsky, Stewart, and Schaeffer (1988).

Associated with an isotropy subgroup is the fixed point subspace ${\rm Fix}[\Sigma_{x_0(t)}] = \{x(t) \in M: \sigma \cdot x(t) = x(t) \; {\rm for} \; {\rm all} \; \sigma \in \Sigma_{x_0(t)} \},$ that is the set of points in phase space fixed by all elements of $$\Sigma_{x_0(t)}\ .$$ Fixed-point subspaces are invariant under the flow generated by $$f\ ,$$ that is, a trajectory starting in a fixed point subspace will stay in it for all time. This follows from the fact that $$\sigma \cdot f(x) = f(\sigma \cdot x) = f(x)\ .$$

A periodic solution $$x_0(t)$$ with (rescaled) period $$2 \pi$$ can also have a spatiotemporal symmetry characterized by the isotropy subgroup $\Sigma_{x_0(t)} = \{(\gamma,\theta) \in \Gamma \times S^1 : \gamma \cdot x_0(t + \theta) = x_0(t) \}.$

More generally, a solution $$x(t)$$ to a $$\Gamma$$-equivariant dynamical system has a symmetry $$\gamma\in \Gamma$$ if $$\gamma\overline{\{x(t)\}}= \overline{\{x(t)\}}\ .$$

### Example 1 Continued

The isotropy subgroup of the equilibrium at $$x_0 = 0$$ is the full group $$Z_2 = \langle \gamma \rangle.$$ The isotropy subgroup for each of the equilibria $$x = 1$$ and $$x=-1$$ is the identity element $$e.$$ The fixed point subspace of $$Z_2$$ is the set of all points for which $$\gamma \cdot x = x,$$ that is, for which $$-x = x;$$ thus $${\rm Fix} [Z_2] = \{ 0 \}.$$

### Example 4 Continued

We have the following isotropy subgroups and fixed point subspaces:

Isotropy Subgroup Fixed Point Subspace Dimension of Fixed Point Subspace
$$D_4$$ $$\{0,0\}$$ 0
$$Z_2(\gamma_1)$$ $$\{(a,0) | a \in \mathbb R \}$$ 1
$$Z_2(\gamma_2 \cdot \gamma_1)$$ $$\{(a,a) | a \in \mathbb R \}$$ 1
$$e$$ $$\{(a,b) | (a,b) \in \mathbb R^2 \}$$ 2

Here $$Z_2(\gamma_1) = \langle \gamma_1 \rangle\ ,$$ $$Z_2(\gamma_2 \cdot \gamma_1) = \langle \gamma_2 \cdot \gamma_1 \rangle\ ,$$ and $$e$$ is the identity element.

Other fixed point subspaces can be found by acting on these fixed point subspaces with group elements. In particular, $$\gamma_2 \cdot (a,0) = (0,a)\ ,$$ so $$\{ (0,a) | a \in \mathbb R \}$$ is a fixed point subspace. Such points have isotropy subgroup $$\gamma_2 \cdot Z_2(\gamma_1) \cdot \gamma_2^{-1} = \langle \gamma_2 \cdot \gamma_1 \cdot \gamma_2^3 \rangle = \langle \gamma_1 \cdot \gamma_2^2 \rangle \equiv Z_2(\gamma_1 \cdot \gamma_2^2)\ .$$ Similarly, $$\gamma_1 \cdot (a,a) = (a,-a)\ ,$$ so $$\{ (a,-a) | a \in \mathbb R \}$$ is a fixed point subspace. Such points have isotropy subgroup $$\gamma_1 \cdot Z_2(\gamma_2 \cdot \gamma_1) \cdot \gamma_1^{-1} = \langle \gamma_1 \cdot \gamma_2 \rangle \equiv Z_2(\gamma_1 \cdot \gamma_2 )\ .$$

### Example 6 Continued

Ashwin and Swift (1992) show that the isotropy subgroups of solutions for this example take the form

where $$N=m(k_1 + \cdots + k_{l_B})\ ,$$ and denotes the semi-direct product. The fixed-point subspace $${\rm Fix}[\Sigma_{\mathbf{k},m}]$$ may be thought of as being partitioned into $$m$$ blocks each containing $$k = (k_1 + \cdots + k_{l_B})$$ oscillators. The solution is invariant under time shifts of the period divided by $$m\ ,$$ coupled with a cyclic permutation of the blocks, giving the $$Z_m$$ symmetry. Each block is partitioned into clusters of $$k_i$$ oscillators, and the solution is invariant under permutations of oscillators within these clusters, giving the $$S_{k_1} \times \cdots \times S_{k_{l_B}}$$ symmetry. These permutations all commute, hence the direct products, while the $$Z_m$$ symmetry does not commute with the permutations, hence the semi-direct product. Examples of such solutions, labeled by their isotropy subgroups, are shown in the figures. Here each dot corresponds to a cluster of oscillators, with identically colored dots corresponding to clusters with the same number of oscillators. See Ashwin and Swift (1992), and Brown, Holmes, and Moehlis (2003) for more discussion.

## Construction of Equivariant Dynamical Systems

Here we describe a systematic way of constructing the most general dynamical system that is equivariant with respect to a given symmetry $$\Gamma\ ;$$ see Golubitsky, Stewart, and Schaeffer (1988). An invariant function $$g(x)$$ satisfies $g(x) = g(\gamma \cdot x)$ for all $$\gamma \in \Gamma\ .$$ For a given representation of the group $$\Gamma\ ,$$ there exists a finite set of invariant polynomials $$H(\Gamma) \equiv \{u_1(x),u_2(x),\cdots,u_k(x)\},$$ called a Hilbert basis, such that any invariant polynomial may be expressed as a polynomial function of elements of $$H(\Gamma)\ .$$ Similarly, there is a finite set of equivariant vector field generators $$\{f_1(x),f_2(x),\cdots,f_n(x) \},$$ such that $f(x) = \sum_{j=1}^n g_j(x) f_j(x),$ where the $$g_j$$ are invariant functions, i.e., $g_j(x) \equiv g_j(u_1(x),u_2(x),\cdots,u_k(x)), \qquad j = 1,\cdots,n.$

### Example 7

Suppose we have a dynamical system equivariant with respect to $$D_4 = \langle \gamma_1,\gamma_2 \rangle\ ,$$ with $$\gamma_1$$ and $$\gamma_2$$ as defined in Example 4. Letting $$z = x_1 + i x_2 \in \mathbb C\ ,$$ we see that the equivalent actions on $$z$$ are $\gamma_1: z \rightarrow \bar{z},$ $\gamma_2: z \rightarrow i z.$ Golubitsky, Stewart, and Schaeffer (1988) show that the Hilbert basis for this system is $$u_1(z) = |z|^2$$ and $$u_2(z) = z^4 + \bar{z}^4\ ,$$ and that the equivariant vector field generators are $$f_1(z) = z\ ,$$ $$f_2(z) = \bar{z}^3\ .$$ Thus, any vector field equivariant with respect to this representation of $$D_4$$ can be written in the form $\frac{dz}{dt} = g_1(|z|^2,z^4 + \bar{z}^4) z + g_2(|z|^2,z^4 + \bar{z}^4) \bar{z}^3.$

## Normal Form Symmetry

The process of putting a dynamical system on the center manifold of an equilibrium into normal form can introduce additional symmetries into the system; see Elphick et al (1988), Golubitsky, Stewart, and Schaeffer (1988), Crawford and Knobloch (1991). Specifically, a normal form can be chosen which is equivariant with respect to the one-parameter group $\Gamma \equiv \{\exp(s A^\dagger) : s \in \mathbb R \},$ where $$A^\dagger$$ is the adjoint (transpose and complex conjugate) of the the Jacobian matrix $$A$$ evaluated at the equilibrium. By assumption all of the eigenvalues of $$A$$ lie on the imaginary axis in the complex eigenvalue plane.

### Example 8

Suppose we have a dynamical system for which the center manifold is two-dimensional, and the evolution on the center manifold has the linearization $\left( \begin{array}{c} \dot{u}_1 \\ \dot{u}_2 \end{array} \right) = \left( \begin{array}{cc} 0 & -\omega_0 \\ \omega_0 & 0 \end{array} \right) \left( \begin{array}{c} u_1 \\ u_2 \end{array} \right).$ Letting $$z = u_1 + i u_2\ ,$$ $\left( \begin{array}{c} \dot{z} \\ \dot{\bar{z}} \end{array} \right) = \left( \begin{array}{cc} i \omega_0 & 0 \\ 0 & -i \omega_0 \end{array} \right) \left( \begin{array}{c} z \\ \bar{z} \end{array} \right),$ where $$\bar{z}$$ is the complex conjugate of $$z\ .$$ Then the normal form for the $$(\dot{z},\dot{\bar{z}})$$ equations is equivariant with respect to $\exp(s A^\dagger) = \left( \begin{array}{cc} e^{i \phi} & 0 \\ 0 & e^{-i \phi} \end{array} \right), \qquad \phi \equiv -s \omega_0,$ or equivalently, the normal form for the $$\dot{z}$$ equation is equivariant with respect to the group $S^1: z \rightarrow e^{i \phi} z, \qquad \phi \in [0,2 \pi).$ It is then readily shown that on the center manifold $\frac{d z}{d t} = g_1(|z|^2) z.$ This $$S^1$$ symmetry arises from coordinate transformations, and does not represent a true symmetry of the system. It is, however, a symmetry of periodic solutions. In the present case the normal form symmetry corresponds to a phase-shift symmetry, and since the solution to the normal form equation is to leading order $$z \sim e^{i \omega_0 t}\ ,$$ it may be interpreted as a time translation symmetry. The coordinate transformations extend the symmetry of the vector field evaluated on the periodic orbit to a neighborhood of the periodic orbit.

## Heteroclinic Cycles

A heteroclinic cycle is a collection of solution trajectories that connect invariant solutions such as equilibria. Heteroclinic cycles occur robustly in many equivariant dynamical systems but do not generally occur in systems without symmetries. Cycles are commonly constructed using connections in fixed point subspaces for equivariant dynamical systems.