Morse-Smale systems

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

Usually, a reasonable high-order model must exhibit both types of dynamics — simple and complex. Of course, the first step in the analysis of such models is the study of the structure of the partitioning of the phase space into trajectories in those parameter regions which correspond to simple dynamics. In the next section, we will be focusing on a rather broad class of structurally stable systems with simple dynamics which are called the Morse-Smale systems. Systems with complex dynamics require special care, and will be the subject of a further monograph. 

Morse-Smale systems are introduced axiomatically. Consider a dynamical system... 

Definition 7.9. System (7.5.1) in region G is called a Morse-Smale system if it satisfies the following two axioms. ... 

In fact, it can be shown that periodic orbits and equilibrium states are the only non-wandering trajectories of Morse-Smale systems. Axiom 1 excludes the existence of unclosed self-limit (P-stable) trajectories in view of BirkhofF s Theorem 7.2. The existence of homoclinic orbits is prohibited by Theorems 7.9 and 7.11 below. Next, it is not hard to extract from Theorem 7.12 that an u)-limit (a-limit) set of any trajectory of a Morse-Smale system is an equilibrium state or a periodic orbit. 

Theorem 7.9. Morse-Smale systems have no homoclinic trajectory to an equilibrium state. 

Both, the continuous and discrete, Morse-Smale systems on compact smooth manifolds were singled out by Smale in his article Morse inequalities for a dynamical system [142]. The title itself reveals that the work deals... 

Theorem 7.10. (Palis and Smale) Morse-Smale systems are structurally stable. 

For example, the following theorem shows that a Morse-Smale system cannot have a homoclinic trajectory to a saddle periodic orbit. 

In essence, the above proof is a close repetition of that suggested by L. Shilnikov [131]. It allows one to liberate from the axiom stipulating the absence of homoclinic trajectories in Morse-Smale systems originally postulated by Smale. 

It may be proved that the relation < defines a partial order on the set of non-wandering orbits of a Morse-Smale system. An important result is ... 

Theorem 7.12. There are no cycles in Morse-Smale systems. 

First of all, observe that there cannot exist cycles like Qo < Qo because homoclinic trajectories are not admissible in Morse-Smale systems. Also, it follows from the transversality condition [see (7.5.4)] that a cycle cannot contain equilibrium states neither can it include periodic orbits of different topological types. 

The fact that there are a finite number of non-wandering trajectories in Morse-Smale systems implies that any chain has a finite length which does not exceed the total number of non-wandering trajectories. Moreover, a maximal chain can end only at a stable equilibrium state or a periodic orbit. 

It is clear that the phase diagram is an invariant of topological equivalence of Morse-Smale systems. 

The principal feature of Morse-Smale systems which distinguishes them from Andronov-Pontryagin systems is that the former may have infinitely many special heteroclinic trajectories. As an example, let us consider a two-dimensional diffeomorphism with three fixed points of the saddle type denoted by Oi, O and O2- Suppose that O Wq 0 and n Wq 0, the... 

The violation of structural stability in Morse-Smale systems is caused by the bifurcations of equilibrium states, or periodic orbits, by the appearance of homoclinic trajectories and heteroclinic cycles, and by the breakdown of transversality condition for heteroclinic connections. However, we remark that some of these situations may lead us out from the Morse-Smale class moreover, some of them, under rather simple assumptions, may inevitably cause complex dynamics, thereby indicating that the system is already away from the set of Morse-Smale systems. 

The primary scope of this book will focus on the analysis of the internal bifurcations within the class of systems with simple dynamics, such as Morse-Smale systems. Furthermore, we will restrict our study mostly to bifurcations of codimension-one. The reason for this restriction is that some bifurcations of higher codimension turn out to be boundary bifurcations in many cases, such as when the normal forms for the equilibrium states are three-dimensional. Nevertheless, we will examine some codimension-two cases which are concerned with equilibrium states and the loss of stability of periodic orbits. Meanwhile, let us start our next section with a discussion of some questions related to structurally unstable heteroclinic connections. 

Afraimovich, V. S. and Shilnikov, L. P. [1973] On critical sets of Morse-Smale systems, Trans. Moscow Math. Soc. 28, 1761-1765. 

Meyer, K. R. [1968] Energy functions for Morse-Smale systems, Anter. J, Math. 90(4), 1031-1040. 

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