Douglas D. Novaes

Higher order averaging theory for finding periodic solutions via Brouwer degree

In this paper we deal with nonlinear differential systems of the form where for i = 0, 1, ..., k, and are continuous functions, and T-periodic in the first variable, D being an open subset of , and e a small parameter. For such differential systems, which do not need to be of class , under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in e. Some applications are also performed.

On the birth of limit cycles for non-smooth dynamical systems

Abstract The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non-smooth dynamical systems theory. An application is presented in careful detail.

Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

Regularization of hidden dynamics in piecewise smooth flows

Abstract This paper studies the equivalence between differentiable and non-differentiable dynamics in R n . Filippovs theory of discontinuous differential equations allows us to find flow solutions of dynamical systems whose vector fields undergo switches at thresholds in phase space. The canonical convex combination at the discontinuity is only the linear part of a nonlinear combination that more fully explores Filippovs most general problem: the differential inclusion. Here we show how recent work relating discontinuous systems to singular limits of continuous (or regularized ) systems extends to nonlinear combinations. We show that if sliding occurs in a discontinuous systems, there exists a differentiable slow–fast system with equivalent slow invariant dynamics. We also show the corresponding result for the pinching method, a converse to regularization which approximates a smooth system by a discontinuous one.

A Simple Solution to the Braga–Mello Conjecture

Recently Braga and Mello conjectured that for a given n ∈ ℕ there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper, we prove a result from which the conjecture is an immediate consequence. Several explicit examples are given where location and stability of limit cycles are provided.

On the periodic solutions of a perturbed double pendulum

We provide sufficient conditions for the existence of peri- odic solutions of the planar perturbed double pendulum with small oscillations having equations of motion ¨

On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

In this paper we consider the linear differential center ( x ? , y ? ) = ( - y , x ) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0 . Using the Bendixson transformation we provide sufficient conditions to ensure the existence of a crossing limit cycle coming purely from the infinity. We also study the displacement function for a class of discontinuous piecewise smooth differential system.

On the periodic solutions of perturbed 4D non-resonant systems

We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non–linear planar double pendulum perturbed by smooth or non–smooth functions.

Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones

Abstract This work is devoted to study the existence of periodic solutions for a class of e -family of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers.

An Equivalent Formulation of the Averaged Functions via Bell Polynomials

We use Bell polynomials to provide an alternative formula for the averaged functions. This new formula can make the computational implementation of the averaged functions easier.