Stability asymptotic orbital

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. 

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. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

The limit cycle found in the previous section holds only for 103 — 031 small. Obviously, once the limit cycle exists, it can be continued, either globally or until certain bad things happen such as the period tending to infinity or the orbit collapsing to a point. It is very difficult to show analytically that these events do not occur. Moreover, the computations necessary to actually prove the asymptotic stability of the bifurcating orbit are very difficult. We discuss briefly some numerical computations, shown in Figure 8.1, which suggest answers to both these problems. 

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. 

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. 

---