Algebraic determination of limit cycles in a family of three-dimensional piecewise linear differential systems

Abstract

We study a one-parameter family of symmetric piecewise linear differential systems in R 3 which is relevant in control theory. The family, which has some intersection points with the adimensional family of Chua’s circuits, exhibits more than one attractor even when the two matrices defining its dynamics in each zone are stable, in an apparent contradiction to the three-dimensional Kalman’s conjecture. For these systems we characterize algebraically their symmetric periodic orbits and obtain a partial view of the one-parameter unfolding of its triple-zero degeneracy. Having at our disposal exact information about periodic orbits of a family of nonlinear systems, which is rather unusual, the analysis allows us to assess the accuracy of the corresponding harmonic balance predictions. Also, it is shown that certain conditions in Kalman’s conjecture can be violated without losing the global asymptotic stability of the origin.