View source for Equivalence of dynamical systems

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Two autonomous systems of ordinary differential equations (cf. [[Autonomous system|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)|Flow (continuous-time dynamical system)]]), are topologically equivalent [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]] if there exists a [[Homeomorphism|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Geometrical methods in the theory of ordinary differential equations" , ''Grundlehren math. Wiss.'' , '''250''' , Springer  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Guckenheimer,   Ph. Holmes,   "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Yu.A. Kuznetsov,   "Elements of applied bifurcation theory" , Springer  (1995)</TD></TR></table>