Asymptotically orbitally stable

Theorem 4.2. Let n — of the Floquet multipliers of (4.6) lie inside the unit circle in the complex plane. Then 7 is an asymptotically orbitally stable trajectory of (4.5). 

Corollary 5.2. If (3.4) holds then T is asymptotically orbitally) stable. 

Proof. The quantity under the integral sign in the definition of A in (5.2) is the trace of the Jacobian matrix for the system (5.1) evaluated along the periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only two multipliers and one of them is 1, is the remaining one. The periodic orbit is asymptotically orbitally stable because, in view of Lemma 5.1, A<0. ... 

When the inequality in (3.4) is reversed, there will be a periodic orbit (by an application of the Poincare-Bendixson theorem). By our assumption of hyperbolicity, this orbit must be asymptotically orbitally stable since it is so from the inside. These comments establish the next result. 

Theorem 3.2. Suppose that the parameters ai, mi, and T[ are chosen so that (3.2) has a (linearly) asymptotically orbitally stable periodic solution (S(t), x(t)) with period T>0. Fix ai and ti > tj. Then there exist a critical value ml and a branch of periodic orbits of (3.1), with positive Xi component, bifurcating from the hypothesized orbit for m2 near ml-... 

Definition 14.2. A point eo on the stability boundary of a periodic trajectory Le is said to be safe if L q is asymptotically orbitally stable. 

If C is orbitally stable and, in addition, the distance between B and C tends to zero as t - oo, this form of stability is called asymptotic orbital stability. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain < in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit. ... 

Proof. Let 7 = (a (/), (/), 0) be the orbitally asymptotically stable periodic orbit of period T given by Theorem 5.4. (We have already noted that if there are several orbits then one must be asymptotically stable, by our assumption of hyperbolicity.) Let the Floquet multipliers of 7, viewed as a solution of (3.1), be 1 and p, where 0periodic orbit, define p( 3) by... 

The differential equations were solved for a variety of values of less than a. The program was run for considerable time and the last 100 points saved. If the limiting periodic orbit were asymptotically stable, these points would be near the periodic orbit - equal as well as the eye can determine. These periodic orbits, corresponding to different parameters and hence to different systems of differential equations, were then plotted on a single three-dimensional graph (Figure 8.1). This illustrates the stability. 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

Then x is globally asymptotically stable for all initial conditions, x(r) —> x as r oo. In particular the system has no closed orbits. (For a proof, see Jordan and Smith 1987.)... 

In Figure 3.10(b), the point (qi = q°, Pi = 0) is a saddle point of the potential. While those orbits that asymptotically approach the saddle point constitute the stable manifold, those that asymptotically leave the saddle form the unstable manifold. These two manifolds do not in general coincide with each other. For systems of one degree of freedom under a periodic external force and those of two degrees of freedom, these two manifolds have intersections. When the two manifolds have the same saddle point in common as shown Figure 3.10(h), their intersections are called homoclinic. When they do not, their intersections are called heteroclinic. 

This phenomenon is often called "hard self-excitation" because there exists a self-excited (i.e. orbitally asymptotically stable) limit cycle, but to reach the self-excited oscillation requires a "hard" (i.e. finite) perturbation from the steady state. (In contrast, a "soft self-excitation" is illustrated in Fig. I.l.) There is some experimental indication of hard self-excitation in the Belousov-Zhabotinskii reaction. Notice in Fig. II. 1 that after a short induction period the oscillations appear suddenly with large amplitude. This is to be expected for hard self-excitation during the induction period the system is trapped in a locally stable steady state until the kinetic parameters change such that the steady state loses its stability and the system jumps to large amplitude stable oscillations. In the case of soft self-excitation it is expected that as the steady state loses stability, small amplitude stable oscillations first appear and then grow in size. 

Problem 7. For = 0, show that there exists a unique orbitally asymptotically stable limit cycle. See Problem 1.5. 

In the limit - 0, y(T) changes much more rapidly than x(t) Except near Q = 0 the vector field (x,y) is everywhere nearly horizontal. The two falling sections of the one-dimensional manifold Q 0 are stable, but the middle section is unstable. (We referred to this fact earlier.) For 0 < 6 < 6 and 6 < 6 < /e find that the steady state is globally asymptotically stable (as -> 0). However, under these conditions the system is excitable in the sense described in Chapter IV (pp. 76f) For 6q < 6 < 6 we find an orbitally as3nmptoti-cally stable periodic solution illustrated in Fig. 4. 

Since the contraction in the local map can be made arbitrarily strong and the derivative of the global map is bounded, the superposition T = To oTi inherits the contraction of the local map for all small p as well. It then follows from the Banach principle of contracting mappings (Sec. 3,15) that the map T has a unique stable fixed point on So- As this is a map defined along the trajectories of the system, it follows that the system has a stable periodic orbit in V which attracts all trajectories in V. The period of this orbit is the sum of two times the dwelling time t of local transition from Sq to S and the flight time from Si to Sq. The latter is always finite for all small p. It now follows from (12.1.4) that the period of the stable periodic orbit increases asymptotically of order tt/x/a This completes the proof. 

We have foimd that the region of existence of the stable periodic orbit is given by the condition e > x (/x, ), which can obviously be rewritten in the form > hkomil ) where the smooth function hhom behaves asymptotically as y/ fi /l2> The boundary of the region corresponds to the point Me on E, i.e. to a homoclinic loop of Oi. End of the proof. 

It may turn out that T is one-side stable, but not asymptotically. For example, in the case of Hamiltonian systems the region U may be filled by periodic orbits. 

Figure 4. A stable limit cycle. The broken lines represent the perturbed path of an element approaching the stable orbit (solid line) asymptotically.

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