Equivalence of dynamical systems
Two autonomous systems of ordinary differential equations (cf. Autonomous system)
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) |
Equivalence of dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_of_dynamical_systems&oldid=46842