Topological equivalence

Topological equivalence is a term from the theory of dynamic systems .

Two dynamic systems are clearly equivalent in this sense if there is a self-mapping of the phase space under which the orbits of one system correspond to the orbits of the second system.

definition

Two dynamical systems on a phase space are called topologically equivalent if there is a homeomorphism such that ${\ displaystyle \ Phi _ {1}, \ Phi _ {2}}$ ${\ displaystyle X}$ ${\ displaystyle h \ colon X \ to X}$

{\ displaystyle h (\ Phi _ {1} (t, x)) = \ Phi _ {2} (t, h (x))}

applies to all . ${\ displaystyle (t, x)}$

We then say that the flow in the river conjugated. ${\ displaystyle h}$ ${\ displaystyle \ Phi _ {1}}$ ${\ displaystyle \ Phi _ {2}}$

One speaks of the topological equivalence of two ordinary differential equations (or two vector fields ) if the associated flows are topologically equivalent.

Examples

The flows of the differential equations and are topologically equivalent. Homeomorphism conjugates the flow of into the flow of . ${\ displaystyle {\ dot {x}} = x}$ ${\ displaystyle {\ dot {x}} = 2x}$ ${\ displaystyle h (x) = x ^ {2}}$ ${\ displaystyle \ Phi _ {1} (t, x) = e ^ {t} x_ {0}}$ ${\ displaystyle {\ dot {x}} = x}$ ${\ displaystyle \ Phi _ {2} (t, x) = e ^ {2t} x_ {0}}$ ${\ displaystyle {\ dot {x}} = x ^ {2}}$

The (undamped) harmonic oscillator is not topologically equivalent to the damped harmonic oscillator : while with the harmonic oscillator all paths are periodic, with the damped harmonic oscillator all paths strive to the point of equilibrium.

The Hartman-Grobman are (under certain conditions) the topological equivalence of the ordinary differential equation and its linearization. Let x '= Ax be the linearization of x' = v (x), so it holds with and . If all eigenvalues of the operator A lie in the left half-plane (i.e. have negative real parts), then the differential equation and its linearization are topologically equivalent. ${\ displaystyle v = v_ {1} + v_ {2}}$ ${\ displaystyle v_ {1} (x) = Ax}$ ${\ displaystyle v_ {2} (x) = O (\ Vert x \ Vert ^ {2})}$

literature

VI Arnold : Geometrical methods in the theory of ordinary differential equations , Grundlehren der Mathematischen Wissenschaften, Volume 250, Springer-Verlag, New York, 1983, ISBN 0-3879-0681-9