Definition of Class Equivalence and Conjugacy Between Dynamical Systems 📂Dynamical Systems

Definition of Class Equivalence and Conjugacy Between Dynamical Systems

Definition¹

Let two dynamical systems be represented by the following vector fields for $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $g : \mathbb{R}^{n} \to \mathbb{R}^{n}$. $$ \begin{align*} \dot{x} =& f(x) \\ \dot{y} =& g(y) \end{align*} $$ $f$ and $g$ belong to the class $C^{r}$, meaning they are differentiable $r$ times and their derivatives are continuous, and from here on we take $k \le r$. For the flows $\phi$ and $\psi$ obtained from $f$ and $g$ respectively, if there exists a diffeomorphism $h \in C^{k}$ such that the following holds for some parametrization $\alpha$, then $f$ and $g$ are said to be $C^{k}$-equivalent. $$ h \left( \phi (t , x) \right) = \psi \left( \alpha (t, x), h(x) \right) $$ If there exists $h \in C^{k}$ that also preserves the time $t$ and satisfies the following, then $f$ and $g$ are said to be $C^{k}$-conjugate. $$ h \left( \phi (t , x) \right) = \psi \left( t, h(x) \right) $$

Theorem

The following theorems are known regarding the fixed points and periodic solutions of a system.

If $f$ and $g$ are $C^{k}$-conjugate, then the following hold.

The fixed points of $f$ correspond to the fixed points of $g$.
The periodic solutions of $f$ correspond to the periodic solutions of $g$.

If $f$ and $g$ are $C^{k}$-equivalent, then the following hold.

The fixed points of $f$ correspond to the fixed points of $g$.
The periodic solutions of $f$ correspond to the periodic solutions of $g$. However, the periods may differ.

Explanation

The definitions themselves can be said to be a repetition of what we have already seen in conjugacy defined on maps and topological equivalence between dynamical systems.

Basically, class conjugacy is similar to the usual topological conjugacy, but it differs in that topological conjugacy requires only a homeomorphism, whereas class conjugacy also takes differentiability into account. The difference between equivalence and conjugacy may be regarded as whether time is preserved or not, that is, in plain terms, the difference in the speed carried by the vector field. Equivalence only requires that the appearance be similar, while conjugacy requires that not only the appearance but also the behavior, including periodicity, be similar.