Leontovich theorem

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. 

The original proof of the Andronov-Leontovich theorem assumes that the system is defined on the plane. We choose here a somewhat different approach which can be easily adopted to the case of systems defined on non-orientable two-dimensional surfaces as well. 

Remark 4. The above proof can be easily adopted to the case of a separatrix loop on a general two-dimensional surface, regardless whether it is orientable or non-orientable. In both cases the map will have the form (13.2.9). Note, however, that if a small neighborhood of the separatrix loop is homeomorphic to an annulus, then A>0 and if a neighborhood of f is a Mobius band, then A < 0 (the latter corresponds, obviously, to the non-orientable case). In the case > 0, the Andronov-Leontovich theorem holds without changes. 

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

Remark. Theorem 11.1 was proven by Andronov and Leontovich via constructing and studying the mapping without using explicitly the theory of normal forms. Figures 11.5.1 and 11.5.2 are copied from the book Theory of Oscillations by Andronov, Vitt and Khaikin where they illustrated the phenomena of the soft and the rigid generation of self-oscillations (see Chap. 14). 

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). 

The question of the bifurcations of a separatrix loop to a saddle with zero saddle value (Tq was first considered by E. Leontovich. She had proven the following theorem ... 

In her proof of the above theorem, Leontovich had assumed C -smoothness for the system, where r > 4n + 6. First of all, she proved that when the first saddle value is close to zero, a system near the saddle can be transformed into... 

On the next step of the proof Leontovich had evaluated the local map. She considered, in fact, the map from the cross-section Si y = d to 5o x = d, i.e. the inverse of the local map To in our notations. Note that by assumption of the theorem only the last saddle value an is bounded away from zero, whereas ai,.., an-i are small. Therefore, by rescaling time variable the system may... 

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