Hyperbolic fixed points

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below... 

The system (7) with e = 0 is referred as unperturbed system. About it we shall assume that it possesses a hyperbolic fixed point xQyh connected to itself by a homoclinic orbit Xh(t) = x (t), x (t)). 

When viewed in three-dimensional space R2xS, the hyperbolic fixed point x0yh turns to hyperbolic periodic orbit of the system... 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

Proof. We easily see that ( 7r, 0) is a hyperbolic fixed point of... 

For larger T (T = 1.6), chaotic behavior arises, the hyperbolic fixed point is disrupted and the tori are perturbed (see Figure 1.20) [28]. A chaotic region appears with homoclinic tangle and formation of new hyperbolic and elliptic points. 

Figure 3. Phase portrait of a hyperbolic fixed point, for a 1-DOF linear Hamiltonian.

Figure 3. The stable and unstable manifolds of the critical points A and B on the TCM for Z = 2. (a) The collinear eZe configuration (a = ti). The critical points A and B on the TCM are hyperbolic fixed points, (b) The Wannier ridge configuration (x = ti/2). The critical points A and B on the TCM are a stable focus and an unstable focus, respectively.

In the interval q [—hyperbolic fixed points of the unperturbed Hamiltonian Hq are given by... 

In this section, we consider the breakdown of the condition of normal hyperbolicity. First, we explain a simple example where breakdown of normal hyperbolicity leads to a bifurcation in reaction processes. In the Belousov-Zhabotinsky (BZ) reaction [40], the bifurcation from the stable fixed point to the limit cycle takes place through the breakdown of normal hyperbolicity. This is the simplest case where mathematical analyses are in progress [41]. 

Moreover, the breakdown of normal hyperbolicity leads to the bifurcation from the fixed point to the limit cycle. Suppose that under a smooth variation of parameters we change the flow from the one in Fig. 30 to the one in Fig. 31. Then, in order for the fixed point PI in Fig. 30 to shift to P2 in Fig. 31, it should go through the point where normal hyperbolicity breaks down. 

The behavior of orbits of P near a fixed point x can be described in the case where x is a hyperbolic fixed point, that is, when no eigenvalue (multiplier) of the Jacobian of P at x has modulus equal to 1. In this case there exist (local) stable and unstable manifolds M (x) and M (x) (respectively) containing the point x which are tangent to the stable (resp. unstable) subspace of the Jacobian of P at x. (The stable (unstable) subspace... 

More is known about the dynamics generated by P than has been discussed in this chapter. In [DS] and [HaS] it is shown that there is a curve C joining Ei and E2 which is the graph of a strictly decreasing continuous function. This curve C forms the boundary of the unstable manifold of Eq and every fixed point, except Eq must lie on C. Therefore, every orbit of P except Eq is attracted to a fixed point on C. If each fixed point of P is assumed to be hyperbolic, then there are finitely many fixed points. Moving along the curve C, the fixed points alternate between saddle points and attractors. In particular, if the hypotheses of Corollary 5.2 hold then there are an odd number of positive fixed points on C, at least one of which is an attractor. See [S5] for more details. 

Since i is quite small, the pendulum (l, tp,) shows approximately periodic motion. /2 deviates from its initial value when (l, tp,) moves away from the hyperbolic fixed point. When (7i, tpj) comes back near the hyperbolic fixed point, h also comes near to the initial value. The difference A/2, [Eq. (12)] is the quantity we are interested in. 

Hyperbolic Fixed Points, Topological Equivalence, and Structural Stability... 

If Re(A) 0 for both eigenvalues, the fixed point is often called hyperbolic. (This is an unfortunate name—it sounds like it should mean saddle point —but it has become standard.) Hyperbolic fixed points are sturdy their stability type is unaffected by small nonlinear terms. Nonhyperbolic fixed points are the fragile ones. 

We ve already seen a simple instance of hyperbolicity in the context of vector fields on the line. In Section 2.4 we saw that the stability of a fixed point was accurately predicted by the linearization, as long as f x ) 0. This condition is the... 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

Hyperbolic fixed points also illustrate the important general notion of structural stability. A phase portrait is structurally stable if its topology cannot be changed by an arbitrarily small perturbation to the vector field. For instance, the phase portrait of a saddle point is structurally stable, but that of a center is not an arbitrarily small amount of damping converts the center to a spiral. 

In the case of real eigenvalues, Ai 2 = A, x is called a hyperbolic fixed point, corresponding to a saddle point of the streamfunction, and it is of unstable character. It is located at the intersection of two special streamlines, called separatrices, which are the stable and unstable manifolds of the hyperbolic fixed point. These lines are defined as the set of points that approach the fixed point in the limits t —> Too, respectively. The motion around a hyperbolic point can be obtained as a solution of (2.40)... 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

A. Anikin, Normal form of a quantum Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point, Regul. Chaotic Dyn. 13 (2008) 377-A02. 

Figure 10 Poincare map for two noniinearly coupled oscillators at fixed energy. The energetic periphery is labeled X. (A) The elliptic fixed point of an orbit with a),/o)2 = /i. (B, C) Elliptic fixed points of two distinct orbits with mjuiz - Vj. (D) Sep-aratrix surrounding the Wj/wj = Vi resonance zone. The separatrix pinches off at four hyperbolic fixed points, three of which are more or less clearly visible. (E) One of four elliptic fixed points of an orbit with V (the other three are not visi-...

Trajectories initiated close to an elliptic fixed point behave in a qualitatively different manner from those near a hyperbolic fixed point. For one thing, elliptic fixed points are invariably surrounded by invariant tori, with frequency ratios not far from that of the fixed point all motion on each torus stays on the same torus. Hyperbolic fixed points may or may not be surrounded by tori. However, they are always associated with a single unique set of manifolds composed of motion asymptotic to them in positive and negative time. These manifolds are called stability manifolds or separatrix manifolds, and their continued existence in the presence of a coupling term is guaranteed by the Stable Manifold theorem. The nature of the asymptotic manifolds will be seen to be of special interest and importance, and we discuss them at length in the following section. 

For now, we note that the elliptic fixed points are separated from one another by hyperbolic fixed points, which meet along a line generally known as... 

In Figure 10 the chaotic region is extremely small. However, in Figures 11 and 12 we show a second system s phase space map as a function of en-ergy.35,119 xhjs system exhibits a mode-mode resonance at low energies, with a hyperbolic fixed point located near the center of the Poincare map. Note in Figure 11 that as the energy increases, the measure of quasiperiodic phase space decreases and approaches a limit in which most of the tori are destroyed, with... 

We have discussed the typical manifestation of periodic orbits on a Poincare map as fixed points that are either elliptic or hyperbolic. Let us now consider the properties of motion nearby these fixed points in terms of their stability properties. This is accomplished by a straightforward linear stability analysis about the fixed point. We can carry out such an analysis on any fixed point, whether or not the surrounding phase space is chaotic (as long as we can find the fixed point). 

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