Shilnikov condition

This theorem is a part of a more general assertion [including also the case of a multi-dimensional unstable manifold as well as saddle-foci of types (2,1) and (2, 2)] on complex dynamics near the homoclinic loop of a saddle-focus [136]. Condition p < 1 also known as the Shilnikov condition is very important here, because the structure of the phase space near the homoclinic loop is drastically changed in comparison to the case p > 1 covered by Theorem 13.6. The main bifurcations in the boundary case p = 1, when a small perturbation trigging the system into a homoclinic explosion from simple dynamics (p > 1) to complex dynamics (p < 1) were first considered in [29]. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

According to the Shilnikov s theorem, the reactor presents a chaotic behavior. In order to test the presence of a strange attractor, it is necessary to raise the value of xe ax to introduce a perturbation in the vector field around the homoclinic orbit. Taking xemax = 5, the results of the simulation are shown in Figure 18, where the sensitive dependence on initial conditions has been corroborated. 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

We should, however, stress that such a reduction to the two-dimensional case is not always possible. In particular, it cannot be performed when the equilibrium state is a saddle-focus. Moreover, under certain conditions, we run into an important new phenomenon when infinitely many saddle periodic orbits coexist in a neighborhood of a homoclinic loop to a saddle-focus. Hence, the problem of finding the stability boundaries of periodic orbits in multidimensional systems requires a complete and incisive analysis of all cases of homoclinic loops of codimension one, both with simple and complex dynamics. This problem was solved by L. Shilnikov in the sixties. 

Theorem 13.9. (Shilnikov [134]) Let a saddle O with saddle value cr < 0 have a homoclinic loop F which satisfies the non-degeneracy conditions (1) and (2). Let U he a small neighborhood ofT. If the homoclinic loop splits inward on the invariant manifold Ad, then a single periodic orbit L with an n-dimensional unstable manifold will be bom. Furthermore the only orbits which stay in U for all times are the saddle O, the cycle L and a single heteroclinic orhi which is ot-limit to O and u)-limit to L. 

Theorem 13.10. (Shilnikov [136]) Let a saddle-focus O have a homoclinic loop r which satisfies the non-degeneracy conditions (1) and (2). Then in an arbitrarily small neighborhood of P, there exist infinitely many saddle periodic orbits. 

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