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)