Homoclinic trajectory

The critical nucleus, which can be found explicitly, is described by a homoclinic trajectory of the Euler-Lagrange equation ew = g w) (see, for instance Bates and Fife, 1993). The fact that this perturbation plays a role of a threshold is clear from Fig.9 which demonstrates extreme sensitivity of the problem to slight variations around the critical nucleus representing particular initial data (see Ngan and Truskinovsky (1996b) for details). 

If the x-motion is whole, not P+ -stable [x ca(x)] and a(x) (°) co(x) 4>> then the x-trajectory will be called a loop. An example of a loop on the plane is a loop of the sepatrix that is a trajectory going from a singular point and back to it. Another example is a homoclinic trajectory for which the same saddle limit cycle is both a a-limit and an co-limit set. 

Fig. 7.4 On the left [right], supercritical [subcritical] saddle-centre bifurcation, Eq. (7.65) [Eq. (7.66)], for /t = —1,0, +1, from top to bottom. Regular orbits are blue. The homoclinic trajectories through saddle points and the trajectories joining at the cusp, i.e., at the degenerate critical point corresponding to the bifurcation, are red...

The stochastic aspect of a complex bifurcation arising in a two variables chemical system is studied. The dynamics reduces, in a suitable region of the phase space, to a normal form for which both roots of the characteristic equation vanish simultaneously. In conditions close to this degenerate situation, the normal form can be viewed as a perturbation of an exactly soluble hamiltonian system, of hamiltonian h, which exhibits a homoclinic trajectory, h = 0. BAESENS and IMICOLIS [l ] have shown that the phase portrait of the dissipative sytem displays two steady states that coalesce, a focus F and a saddle S. [ Moreover, as one moves in the parameter space, a limit cycle surrounding F, bifurcates from a homoclinic trajectory and then disappears by Hopf bifurcation. ... 

The two kinds of expansions of U around F and S are suitable to describe respectively the Hopf bifurcation and the bifurcation of the limit cycle from the homoclinic trajectory. The results are qualitatively in agreement with those obtained by BAESENS and NICOLIS but our approach does not lead to the exact condition of existence of, the homo-clinic trajectory. 

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

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

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. 

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. 

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. 

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. 

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. 

Fig 12.1.1. Bifurcation sequence of a saddle-node equilibrium with a homoclinic trajectory (a) before, (b) at, and (c) after the bifurcation. 

Gonchenko, S. V. and Shilnikov, L. P. [1990] Invariants of fi-conjugacy of diffeomorphisms with a structurally unstable homoclinic trajectory, Ukrainian Math. J. 42(2), 134-140. 

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