View source for Equivariant dynamical systems

===Example 6: Coupled Oscillators===
Suppose that the dynamical system
:<math>
\frac{d X}{dt} = F(X), \qquad  X \in \mathbb R^M
</math>
has a stable [[periodic orbit]] <math>\zeta(t)</math> with minimal period <math>T\ .</math>  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 <math>N</math> identical oscillators are weakly coupled to each other, with 
identical all-to-all coupling,
:<math>
\frac{d X_k}{dt} = F(X_k) + \epsilon \sum_j p(X_k,X_j), \qquad k=1,\cdots,N,
</math>
where <math>X_k</math> describes the state of the <math>k^{\rm th}</math> oscillator.
We define the phase <math>\theta_k</math> of the <math>k</math>th oscillator according to [[isochrons]], so that in the absence of coupling <math>\frac{d\theta_k}{dt} = 2 \pi/T \equiv \omega</math> for all <math>k\ ;</math> this gives
:<math>
\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).
</math>
To lowest order in <math>\epsilon</math> (that is, for weak coupling),
:<math>
\frac{d \theta_k}{dt} = \omega + \epsilon Z(\theta_k) \cdot \sum_j p(\zeta(\theta_k),\zeta(\theta_j)),
</math>
where
:<math>
Z(\theta_k) = \left. \frac{\partial \theta_k}{\partial X_k} \right|_{\zeta(\theta_k)}
</math>
is called the [[phase response curve]].
It follows that the phases <math>\phi_k\equiv\theta_k - \omega t</math> satisfy
:<math>
\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)),
</math>
and hence that for <math>O(\epsilon^{-1})</math> times <math>\phi_k</math> obeys the averaged equation (Guckenheimer and Holmes (1983)) 
:<math> 
\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 
</math>
:<math>
= \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.
</math>
Thus
:<math>
\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.
</math>

This equation can be rewritten in the form
:<math>
\frac{d \theta_k}{dt} = \omega + \sum_{j=1}^N g(\theta_j - \theta_k),
</math>
where <math>\theta_k \in [0,2 \pi)\ ,</math> <math>k = 1, \cdots, N\ ,</math> 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 <math>N</math>-torus manifold <math>T^N\ ,</math> some care must be used in determining its equivariance properties.  It is convenient to embed this system in <math>\mathbb C^N</math> by letting <math>z_k = e^{i \theta_k}\ .</math>  Then
:<math>
\frac{dz_k}{dt} = i z_k \left[\omega + \sum_{j=1}^N g(\theta_j - \theta_k) \right] \equiv f_k(z),
</math>
where <math>z = (z_1,z_2,\cdots,z_N)\ ,</math> and for ease of notation we write <math>\theta_j</math> for <math>{\rm arg}(z_j)\ ,</math> etc.  We first show that this system is equivariant with respect to the simple transposition
:<math>
\sigma: (z_1,\cdots,z_k,\cdots,z_l,\cdots,z_N) \rightarrow (z_1,\cdots,z_l,\cdots,z_k,\cdots z_N).
</math>
Specifically
:<math>
(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))
</math>
for any <math>k</math> and <math>l\ .</math>  We next show that the system is equivariant with respect to the group <math>T^1 = \{\gamma_\alpha | \alpha \in [0,2 \pi) \}</math> where <math>\gamma_\alpha: \theta_k \rightarrow \theta_k + \alpha</math> for all 
<math>k = 1,\cdots,N\ .</math>  This induces the equivalent action <math>\gamma_\alpha: z_k \rightarrow e^{i \alpha} z_k\ .</math>  Therefore
:<math>
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).
</math>
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 <math>S_N</math> and the circle group <math>T^1\ .</math>  Putting these together properly, our system is equivariant with respect to the '''direct product''' <math>S_N \times T^1\ .</math>

One solution to these equations is <math>\theta_k = (\omega + N g(0)) t</math> for all <math>k = 1,\cdots, N\ .</math>  This is an example of a '''phase-locked solution'''; see Ashwin and Swift (1992), and Brown, Holmes, and Moehlis (2003).  The elements of <math>S_N</math> act trivially on this solution.  The elements of the circle group <math>T^1</math> shift the phase to give the solutions <math>\theta_k = (\omega + N g(0)) t + \alpha</math> for all <math>k = 1,\cdots, N</math> and any <math>\alpha\ .</math>