Shilnikov theorem

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. 

C.7. 82. Apply the Shilnikov theorem and explain what kind of behavior one should anticipate in the Rossler system [172, 188]... 

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. 

Theorem 7.13. (Afraimovich and Shilnikov [2]) The intersection Wq C Wq possesses infinitely many heteroclinic trajectories if, and only if the closure Wgi Pi Wq contains a periodic orbit L of type m- -l,n — m), other than Qi and Q2>... 

The generalization of this bifurcation for second-order systems was studied by Andronov and Leontovich. Their proof uses essentially the features of systems on a plane. Our proof of Theorem 12.1 is close to that suggested by L. Shilnikov in [130] for the high-dimensional case with the difference that we have simplified calculations by reducing the system near the origin to the form (12.1.1). 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

Theorem 13.6. (Shilnikov [130]) When the saddle value a is negative at the saddle a single stable periodic orbit L is bom from the homoclinic loop for p> 0. The separatrix Fi tends to L as t +oo. For there are no peri-... 

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]. 

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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