Also we devised some method (as in the plotting of the dynamical evolution and use of normally hyperbolic fixed points).

Near hyperbolic fixed point, a nonlinear dynamical system could be linearized and stability of the fixed point is found by Hartman-Grobman theorem.

Given the nonlinear DE (29) in [R.sup.n], where f is derivable with flow [[phi].sub.t], [x.sub.c] is a hyperbolic fixed point, then there exists a neighbourhood N of [x.sub.c], on which is homeomorphic to the flow of linearization of the DE near [x.sub.c].

There is a separate class of important nonhyperbolic fixed points known as normally hyperbolic fixed points, which are rarely considered in literature (see [77]).

Table 2 shows the eigenvalues corresponding to the fixed points given in Table 1 and existence for hyperbolic, nonhyperbolic, or normally hyperbolic fixed points with the nature of stability (if any).

Among them, [P.sub.7] and [P.sub.8] are stable set of normally hyperbolic fixed points, which resembles "cosmological constant," so it explains the current phase of acceleration of universe.

The perturbed hyperbolic fixed point and the associated stable and unstable manifolds were obtained numerically.

The initial points are arranged on the circle with radius 0.05, which is centered at the hyperbolic fixed point at z = 0 (Fig.

To visualize the perturbed unstable and stable manifolds, one should find the exact location of the perturbed hyperbolic fixed points and the associated directions of eigenvectors.

The initial 3000 points at z = 0 are arranged along the eigenvector 1 direction at the exponential scale within the distance [10.sup.-3] from the perturbed hyperbolic fixed point (Table 1).

To preserve the flow rate through the region according to the volume conservation property of the two-dimensional map (Eq 16), the enclosed regions must seriously deform near the perturbed hyperbolic fixed point.