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.

References

[a1]	V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a2]	J. Guckenheimer, Ph. Holmes, "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer (1983)
[a3]	Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995)