This work is devoted to study the existence of periodic solutions for a family of discontinuous differential systems $Z(x,y;ε)$ with many zones. We show that for $ε$ sufficiently small the averaged functions at any order control the existence of crossing limit cycles for systems in this family. We also provide some examples dealing with nonlinear centers.
These last years an increasing interest appeared for studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study their limit cycles. In this paper we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line, and formed by combinations of linear centers (consequently isochronous) and cubic isochronous centers with homogeneous nonlinearities. For these …
In this paper, we study the number of limit cycles that can bifurcating from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove that if the central subsystem, i.e. the system defined between the two parallel lines, has a real center and the others subsystems have centers or saddles, then we have at least three limit cycles that appear after perturbations of periodic annulus. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system and present a norm…
In this paper, we study the existence of limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential systems with two or three zones separated by straight lines and such that the linear systems that define the piecewise one have isolated singular points, i.e. centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones, then it has no limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, then it ha…
考虑R^2中的半平面N={y>0}和S={y<0} $ having as common boundary the straight line D={y=0}$。在N和S中分别定义多项式向量场X和Y，导致不连续的分段多项式向量场Z=（X，Y）。本文研究了Z在N和S之间的解的稳定性和过渡分析，这是由Filippov（1988）和Kozlova（1984）开始的，并由Sotomayor-Teixeira（1995）用正则化方法重新表述。该方法是分析一个单参数连续向量场族Z_ε，它由X和Y的平均值定义。当参数变为零时，该族接近Z。将Sotomayor-Teixeira和Sotomayor-Machado（2002）关于（X，Y）上正则向量场在平面紧连通区域上结构稳定的条件推广到R^2上不连续的分段多项式向量场.满足上述稳定性条件的向量场的相关泛生性结果也推广到了目前的情况。本文提出了通过紧化研究无穷远处不连续分段向量场的方法。
Dynamics of 1D discontinuous maps with multiple partitions and linear functions having the same fixed point. An application to financial market modeling
For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such `subsumed' homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy $0 < |λ| < 1 < σ< \frac{1}{|λ|}$, in a two-parameter unfolding there exists an infinite sequ…
Stability of non-sticking periodic oscillations obtained via the averaging method in discontinuous systems. I. Smooth outside of the discontinuity surfaces systems
From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps
The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $\mathcal{Z}\subset\mathbb{R}^n$ of periodic solutions satisfying $\dim(\mathcal{Z})<n.$ Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential syste…
In this paper some piecewise smooth perturbations of a three-dimensional differential system are considered. The existence of invariant manifolds filled by periodic orbits is obtained after suitable small perturbations of the original differential system. These manifolds emerge from a continuum of cylinders of $\mathbb{R}^3$ which does exist for the piecewise smooth differential systems after a rotation of some planar algebraic polynomial curves. The main tool used in order to obtain the results is the averaging theory for piecewise smooth differential systems.
We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being $3$ such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of $7$ limit cycles of small amplitude.
In this paper, we first show that any nonlinear monotonic increasing contracting maps with one discontinuous point on a unit interval which has an unique periodic point with period $n$ conjugates to a piecewise linear contracting map which has periodic point with same period. Second, we consider one parameter family of monotonic increasing contracting maps, and show that the family has the periodic structure called Arnold tongue for the parameter which is associated with the Farey series. This implies that there exist a parameter set with a positive Lebesgue measure such that the map has a per…
Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform …
