* it is distinguished by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e., any homeomorphism
h : ([T.sup.1], [D.sub.o] + [D.sup.1]) [right arrow] ([T.sup.2], [D.sub.o] + [D.sup.2]) necessarily satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and
The mapping [[chi].sub.[DELTA]]: [M.sub.[DELTA]] [right arrow] [X.sub.[DELTA]] is a homeomorphism
which is equivariant with respect to the actions of D3 and [D.sup.3.sub.C], respectively.
The nonlinear derivation operator is represented by an increasing homeomorphism
[phi]: E [right arrow] E satisfying [phi]([theta]) = [theta] and such that [phi] is expansive, i.e.,
It is equivalent because the chaotic attractor differential in the two spaces is homeomorphism
. The reconstructed system that includes the evolution information of all state variables is capable of calculating the future state of the system based on its current state, which provides a basic mechanism for chaotic time series prediction.
Let (X, f) and (Y, g) be compact spaces; we say f and g are topologically conjugate if there is a homeomorphism
h : X [right arrow] Y, such that h [omicron] f = g [omicron] h.
Some sufficient conditions for assuring the existence, uniqueness, and exponential stability of the equilibrium point of the system are derived using the vector Lyapunov function method, homeomorphism
mapping lemma, and the matrix theory.
Let f: (X, [tau]) [right arrow] (Y, [tau]) be a [theta]C homeomorphism
and f (G) [contains not equal to] G'.
If there exists another regular curve [??]:[??] [right arrow] [G.sub.3] and a homeomorphism
[sigma]: I [right arrow] [??] such that:
Two dynamical systems [([F.sub.t]).sub.t[greater than or equal to]0] and [([G.sub.t]).sub.t[greater than or equal to]0], where [F.sub.t] : X [right arrow] X and [G.sub.t] : Y [right arrow] Y; are topologically equivalent if there exists homeomorphism
h : X [right arrow] Y, such that the following diagram is commutative:
By assumptions on T, h is a homeomorphism
of Y onto X.
Also let the sets of coordinates be transformed so that the map becomes a homeomorphism
of a class [C.sub.k].
Then there exists a [GAMMA] equivariant homeomorphism
germ ([R.sup.2n] x R, (0,0)) [right arrow] ([R.sup.2n] x R, (0,0)) which maps the nonlinear normal modes of H to those of [H.sub.2] + [H.sub.k], preserving their symmetry groups.