Undergrad 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 <br /> \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 <br /> \begin{pmatrix} \cos \theta &amp; -\sin \theta \\ \sin \theta &amp; \cos \theta \end{pmatrix} for any \theta.

Example 2: <br /> \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).