We classify the hulls of different limit-periodic potentials and show that the hull of a limit-periodic potential is a procyclic group. We describe how limit-periodic potentials can be generated from a procyclic group and answer arising questions. As an expository paper, we discuss the connection between limit-periodic potentials and profinite groups as completely as possible and review some re...

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter pro...

The paper analyses the nature of limit cycles induced through the interaction of transformer tap changing and load dynamics. Linearization of a Poincaré map is used to prove local stability. An approximation is introduced which allows the map to be formulated explicitly. An estimate of the region of attraction can then be obtained.

This paper deals with the stability of an event-based proportional-integral controller. In particular, necessary and sufficient conditions on the controller parameters for the existence of equilibrium points without limit cycles are given for a first-order-plus-dead-time process. Practical issues related to the controller implementation are also addressed. The presented conditions enable a simp...

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x′ = −y + g1(x) + f1(x)y and y′ = x + g2(x) + f2(x)y, where fi(x) and gi(x) for i = 1, 2 are polynomials of variable x with any...

We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval T . Taking T as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s me...

In this paper, we address stability and robustness properties of hybrid limit cycles for a class of hybrid systems with multiple jumps in one period. The main results entail equivalent characterizations of stability of hybrid limit cycles for hybrid systems. The hybrid limit cycles may have multiple jumps in one period and the jumps are allowed to occur on sets. Conditions guaranteeing robustne...

We give a new and short proof of the characterization of monodromic nilpotent critical points.We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

We consider a generalized pendulum equation depending on the scalar parameter μ having for μ = 0 a limit cycle Γ of the second kind and of multiplicity three. We study the bifurcation behavior of Γ for −1 ≤ μ ≤ ( √ 5 + 3)/2 by means of a Dulac-Cherkas function.