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Two autonomous systems of ordinary differential equations (cf. [[Autonomous system|Autonomous system]])
 
Two autonomous systems of ordinary differential equations (cf. [[Autonomous system|Autonomous system]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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$$ \tag{a1 }
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{\dot{x} } = f ( x ) , \quad x \in \mathbf R  ^ {n} ,
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$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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$$ \tag{a2 }
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{\dot{y} } = g ( y ) , \quad y \in \mathbf R  ^ {n}
 +
$$
  
(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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100904.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100905.png" /> is defined in a small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100907.png" />.
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(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} } $,  
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$  y = h ( x ) $,  
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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,
 
If the systems depend on parameters, the definition of topological equivalence is modified as follows. Two families of ordinary differential equations,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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$$ \tag{a3 }
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{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {n} ,  \alpha \in \mathbf R  ^ {m} ,
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e1100909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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$$ \tag{a4 }
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{\dot{y} } = g ( y, \beta ) , \quad y \in \mathbf R  ^ {n} ,  \beta \in \mathbf R  ^ {m} ,
 +
$$
  
 
are called topologically equivalent if:
 
are called topologically equivalent if:
  
i) there is a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009011.png" />;
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i) there is a homeomorphism $  p : {\mathbf R  ^ {m} } \rightarrow {\mathbf R  ^ {m} } $,  
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$  \beta = p ( \alpha ) $;
  
ii) there is a family of parameter-dependent homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009013.png" />, mapping orbits of (a3) at parameter values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009014.png" /> into orbits of (a4) at parameter values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009015.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009016.png" /> is defined in a small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110090/e11009020.png" />.
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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.
 
The above definitions are applicable verbatim to discrete-time dynamical systems defined by iterations of diffeomorphisms.

Latest revision as of 19:37, 5 June 2020


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)
How to Cite This Entry:
Equivalence of dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_of_dynamical_systems&oldid=16565
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article