We study the bifurcation of limit cycles from the periodic orbits of a two-dimensional (resp. four-dimensional) linear center in ℝn perturbed inside a class of discontinuous piecewise linear differential systems. Our main result shows that at most 1 (resp. 3) limit cycle can bifurcate up to first-order expansion of the displacement function with respect to the small parameter. This upper bound is reached. For proving these results, we use the averaging theory in a form where the differentiability of the system is not needed.
Abstract
For the two-dimensional linear differential system
with Lebesgue integrable coefficients 𝑝𝑖𝑘 : [𝑎, 𝑏] → ℝ (𝑖 = 1, 2), a Beurling–Borg type theorem is proved on an upper estimate of the number of zeros of an arbitrary non-trivial solution.
We study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.
LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS
