Transverse homoclinic trajectory

The above proof can be easily translated into the language of diffeomor-phisms with a fixed point having a transverse homoclinic trajectory. It also covers the case of a periodic point with a homoclinic trajectory. In the last case one should consider the qth iteration of the original diffeomorphism, where q is the period. 

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. 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

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. 

A more vivid characteristics of systems with complex behaviors is the presence of a Poincare homoclinic trajectory, i.e. a trajectory which is biasymptotic to a saddle periodic orbit as t —> oo. The existence of a homoclinic orbit which lies at the transverse intersection of the stable and unstable... 

Theorem 7.11. Let L be a saddle periodic orbits and let P be its homoclinic trajectory along which Wf and intersect transversely. Then, any small neighborhood of L JT contains infinitely many saddle periodic orbits. 

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 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. 

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