A Bendixon–Dulac theorem for some piecewise systems

Abstract

The Bendixson-Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems. We apply it to three different Liénard piecewise differential systemsẍ + f ± (x)ẋ + x = 0. The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one.

On the Poincaré–Bendixson Formula for Planar Piecewise Smooth Vector Fields

“…On one hand, it is obvious that the existence and uniqueness theorem is not true in the piecewise smooth context [12]. While on the other hand, under suitable assumptions, Poincaré-Bendixson theorem [6], Bendixson-Dulac theorem [8], Peixoto theorem [26] and Poincaré recurrence theorem [10] have been generalized to piecewise smooth differential systems. This paper focuses on the Poincaré index formula and its generalization to piecewise smooth differential systems.…”

“…We define the index of an isolated equilibrium point q ∈ Σ of a piecewise smooth vector field (4) using its regularization vector field (7) as follows (8) i…”