Morse-Smale class

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]. 

Note that a transverse homoclinic orbit is, obviously, preserved under small smooth perturbations of the system. Therefore, Theorem 7.11 implies that when a transverse homoclinic exists, any close system is away from the Morse-Smale class. This gives us a robust and simple indicator for detecting the complex dynamics. By now, the presence of transverse homoclinics is regarded as a universal criterion of chaos. 

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. 

Consider next a Banach space B of dynamical systems X of the Morse- Smale class in a compact region G. Let dB denote the boundary of B, Any system Xq G dB is structurally unstable. We will assume then that a system Xo G dB is a boundary system of the Morse-Smale class, if in any of its neighborhoods there are systems with infinitely many periodic orbits (basically, this means the presence of transverse homoclinic trajectories). The other systems on dB correspond to internal bifurcations within the Morse-Smale class. 

This idea was proposed by Andronov and Leontovich in their first work [9] which deals with primary bifurcations of limit cycles on the plane. Further developments of the theory of bifurcations, internal to the Morse-Smale class, has also confirmed the sufficiency of using finite-parameter families for a rather large number of problems. 

Note that only cases (1) and (2) correspond to structurally stable systems the other cases are non-rough. In essence, a bifurcation of a homoclinic-8 with a negative saddle value is an internal bifurcation in the Morse-Smale class. 

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. 

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. 

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