What is a symmetric ODE / what does it mean when an ODE is symmetric?

[I_laff]

How can an ODE be symmetric? How would you plot an ODE to show off this property? (i.e. what would be the axes?)

 

[pasmith]

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ and $A : \mathbb{R}^n \to \mathbb{R}^n$ be an invertible linear map. Then $A$ is a symmetry of the ODE $$\dot x = f(x)$$ if and only if $y(t) = Ax(t)$ is also a solution of the ODE, ie. $\dot y = f(y)$. This requires that $f = A^{-1} \circ f \circ A$.

If you plotted all of the solution curves in phase space, then the resulting diagram would have $A$ as a symmetry.

Example 1: $$\begin{pmatrix} \dot x \\ \dot y \end{pmatrix} = \begin{pmatrix} y \\ -x \end{pmatrix}$$ is symmetric with respect to $$\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$ for any $\theta$.

Example 2: $$\dot x_1 = x_1 + x_2x_3, \qquad \dot x_2 = x_2 + x_1x_3, \qquad \dot x_3 = x_3 + x_1x_2$$ is symmetric with respect to any permutation of $(x_1, x_2, x_3)$.