HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ3

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit.