Zhou, "Center stable manifold
for planar fractional damped equations," Applied Mathematics and Computation, vol.
However, we can find the dimension of stable manifold
(if exists) with the help of centre manifold theorem.
(b) If [[alpha].sub.2] < [delta] then [[lambda].sub.1] > 0 and [[lambda].sub.2] < 0; then [U.sup.*.sub.0] is an unstable saddle point, the unstable manifold of which is [E.sup.u] = <(1; 0)> and the stable manifold
is [E.sup.s] = <(0;1)>.
For standard definitions of attracting fixed point, saddle point, stable manifold
, and related notions see .
(b) if [a.sub.0] + [a.sub.1] = 2, then the zero equilibrium of (131) is globally asymptotically stable within its basin of attraction which is the region below the global stable manifold
[W.sup.s](E), E(1, 1) in the Northeast ordering.
By (H2), there exists an n-dimensional stable manifold
near the point ([phi](0), 0) on the phase plane ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), which is in some region [G.sub.1] of vector function [[PI].sub.0]y.
We will combine the nonlinear equations with information about the local dynamics to trace out the global stable manifold
of the low-inflation steady state.
which indicates that the stable manifold
of P is four-dimensional.
Thus, if the initial size of the two populations is on the basin of attraction of [E.sub.1] or [E.sub.2] then one population will survive and the other population will become extinct; and if the initial population is on the stable manifold
of [E.sub.3] then both populations will co-exist at the saddle [E.sub.3].
Then F has the 1-dimensional local stable manifold
[W.sup.s.sub.loc]([P.sub.+]) at the point [P.sub.+].
On the other hand, the past history of the deformation is determined by the (perturbed) stable manifold
. Both manifolds constitute the homoclinic tangle, a signature of chaos, and the configuration of the manifolds determines the material transports in the chaotic region, which will be discussed throughout this work.
Given z [elements of] [lambda] we define the [epsilon]-local stable manifold
through z by: