# Equivalence of dynamical systems

Two autonomous systems of ordinary differential equations (cf. Autonomous system)

$$\tag{a1 } {\dot{x} } = f ( x ) , \quad x \in \mathbf R ^ {n} ,$$

and

$$\tag{a2 } {\dot{y} } = g ( y ) , \quad y \in \mathbf R ^ {n}$$

(and their associated flows, cf. Flow (continuous-time dynamical system)), are topologically equivalent [a1], [a2], [a3] if there exists a homeomorphism $h : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} }$, $y = h ( x )$, which maps orbits of (a1) into orbits of (a2) preserving the direction of time. The systems (a1) and (a2) are locally topologically equivalent near the origin if $h$ is defined in a small neighbourhood of $x = 0$ and $h ( 0 ) = 0$.

If the systems depend on parameters, the definition of topological equivalence is modified as follows. Two families of ordinary differential equations,

$$\tag{a3 } {\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {m} ,$$

and

$$\tag{a4 } {\dot{y} } = g ( y, \beta ) , \quad y \in \mathbf R ^ {n} , \beta \in \mathbf R ^ {m} ,$$

are called topologically equivalent if:

i) there is a homeomorphism $p : {\mathbf R ^ {m} } \rightarrow {\mathbf R ^ {m} }$, $\beta = p ( \alpha )$;

ii) there is a family of parameter-dependent homeomorphisms ${h _ \alpha } : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} }$, $y = h _ \alpha ( x )$, mapping orbits of (a3) at parameter values $\alpha$ into orbits of (a4) at parameter values $\beta = p ( \alpha )$.

The systems (a3) and (a4) are locally topologically equivalent near the origin, if the mapping $( x, \alpha ) \mapsto ( h _ \alpha ( x ) ,p ( \alpha ) )$ is defined in a small neighbourhood of $( x, \alpha ) = ( 0,0 )$ in $\mathbf R ^ {n} \times \mathbf R ^ {m}$ and $h _ {0} ( 0 ) = 0$, $p ( 0 ) = 0$.

The above definitions are applicable verbatim to discrete-time dynamical systems defined by iterations of diffeomorphisms.

How to Cite This Entry:
Equivalence of dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_of_dynamical_systems&oldid=46842
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article