Difference between revisions of "Equivalence of dynamical systems"
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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]]) | ||
| − | + | $$ \tag{a1 } | |
| + | {\dot{x} } = f ( x ) , \quad x \in \mathbf R ^ {n} , | ||
| + | $$ | ||
and | 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]] | + | (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, | 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 | and | ||
| − | + | $$ \tag{a4 } | |
| + | {\dot{y} } = g ( y, \beta ) , \quad y \in \mathbf R ^ {n} , \beta \in \mathbf R ^ {m} , | ||
| + | $$ | ||
are called topologically equivalent if: | are called topologically equivalent if: | ||
| − | i) there is a homeomorphism | + | 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 | + | 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 | + | 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) |
Equivalence of dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_of_dynamical_systems&oldid=46842