Alessandro Colombo Mike Jeffrey J. Tomàs Lázaro - Extended Abstracts Spring 2016

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Springer International Publishing AG 2017

Alessandro Colombo , Mike Jeffrey , J. Toms Lzaro and Josep M. Olm (eds.) Extended Abstracts Spring 2016 Trends in Mathematics 10.1007/978-3-319-55642-0_1

On Degenerate Cycles in Planar Filippov Systems

Kamila da S. Andrade 1

(1)

Department of Mathematics, UFG, IME, Goinia, GO, 74690-900, Brazil

(2)

Department of Mathematics, Unicamp, IMECC, Campinas, SP, 13083-970, Brazil

Kamila da S. Andrade (Corresponding author)

Email:

R. M. Martins

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Marco Antonio Teixeira

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Abstract

The main objective of this paper is to study bifurcations of a vector field in a neighborhood of a cycle having a homoclinic-like connection at a saddle-regular point. In order to perform such a study it is necessary to analyze how the cycle can be broken, in this way the approach is to look separately at local bifurcations and at the structure of the first return map defined near the cycle.

Preliminary Concepts

Consider the set of all -vector fields defined in endowed with the -topology. For define

where is a smooth function such that is a codimension 1 embedded submanifold, called the switching manifold. Such a vector field Z , as defined above, is called a piecewise smooth vector field and it is usually denoted by . We also denote and .

Let be the set of all with , then can be identified with and thus it is naturally endowed with the product topology. In the switching manifold three regions are distinguished (see Fig. ): the crossing region ,

the sliding region ,

and the escaping region ,

where is the Lie derivative of h in relation to the vector field X , analogously is the Lie derivative of h in relation to Y .

Fig. 1

Illustrations of the regions in

The concept of solutions and trajectories follows Filippovs convention: it implies that multiple solutions through points in are being considered; see [].

Statement of the Problem and Results

The main objective is to study bifurcations of a homoclinic-like connection through a saddle-regular point, then consider a piecewise smooth vector field . In order to conduct an analysis as simple as possible some generic conditions are stated:

1. (i)
   BS (1): has a hyperbolic saddle , the unstable and stable manifold of , and , respectively, are transversal to at ;
2. (ii)
   BS (2): is transversal to , , and at ;
3. (iii)
   BS (3): the sliding vector field near the saddle point has as a non-degenerate singularity;
4. (iv)
   BSC (1): the unstable manifold of which lies in

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