How is this system a reversable system if it defies the definition?

Question

From Strogatz's Nonlinear Dynamics and Chaos:

A reversible system is any second-order system that is invariant under $t \rightarrow -t$ and $y \rightarrow -y$. For example, any system of the form

$x ' = f(x,y)$

$y' = g(x,y)$

where $f$ is odd in $y$ and $g$ is even in $y$ (i.e., $f(x,-y) = - f(x, y)$ and $g(x, -y) = g(x, y) )$ is reversible.

Strogatz claims that the system $x' = 2 \cos x - \cos y, y' = 2 \cos y - \cos x$ should be a reversible system, but it fails to satisfy the definition as $2 \cos x - \cos (-y) \neq -[2 \cos x - \cos y]$.

Am I misunderstanding something here?

Answer 1

That example (Ex. 6.6.3) illustrates a more general definition of reversibility, stated immediately above it on the same page, which involves specifying an involution $R(\mathbf{x})$.

In Ex. 6.6.3 he's using $R(x,y)=(-x,-y)$.

In the previous examples (and in the first definition that you have quoted) it was $R(x,y)=(x,-y)$.