Equivalence of dynamical systems

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

(a1)

and

(a2)

(and their associated flows, cf. Flow (continuous-time dynamical system)), are topologically equivalent [a1], [a2], [a3] if there exists a homeomorphism , , 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 is defined in a small neighbourhood of and .

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

(a3)

and

(a4)

are called topologically equivalent if:

i) there is a homeomorphism , ;

ii) there is a family of parameter-dependent homeomorphisms , , mapping orbits of (a3) at parameter values into orbits of (a4) at parameter values .

The systems (a3) and (a4) are locally topologically equivalent near the origin, if the mapping is defined in a small neighbourhood of in and , .

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)