📂Dynamical Systems

Symmetry of Dynamical Systems

Definition ^{1 2}

Let $I$ be the identity matrix. For a square matrix $R \ne I$, if the vector field $\dot{v} = f(v)$ satisfies the following, then this dynamical system is said to have a symmetry. $$ \begin{align*} R f(v) =& f (R v) \\ R^{2} =& I \end{align*} $$ In this case, $R : v \mapsto Rv$ is called a symmetry transformation.

Explanation

From the definition alone, a concrete form of $R$ may not readily come to mind, but simply put, it suffices that the original equation is preserved even when the order of the variables is swapped or their signs are reversed. As the most familiar example, consider the Lorenz attractor.

$$ \begin{align*} \dot{x} =& - \sigma x + \sigma y \\ \dot{y} =& - xz + \rho x - y \\ \dot{z} =& xy - \beta z \end{align*} $$

Here, since the diagonal matrix $\diag \left( -1, -1, 1 \right)$ exists, if we define $R : (x, y, z) \mapsto (-x, -y, z)$, then $R f(v)$ and $f(R v)$ are as follows. $$ \begin{align*} R \begin{bmatrix} - \sigma x + \sigma y \\ - xz + \rho x - y \\ xy - \beta z \end{bmatrix} =& f \begin{pmatrix} -x \\ -y \\ z \end{pmatrix} \\ \implies \begin{bmatrix} \sigma x - \sigma y \\ xz - \rho x + y \\ xy - \beta z \end{bmatrix} =& \begin{bmatrix} \sigma x - \sigma y \\ xz - \rho x + y \\ xy - \beta z \end{bmatrix} \end{align*} $$

Simply put, $R$ has $1$ or $-1$ as its diagonal entries, or occasionally a permutation may be multiplied in. The reason the definition excludes the case $R = I$ is, of course, that allowing $R = I$ would mean every system could be said to have a symmetry.

A symmetry that is independent of the order of the variables is called a cyclic symmetry, and examples of such systems include the Lorenz 96 system and the Thomas attractor.