On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line (original) (raw)
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On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems
Applied Mathematics and Computation, 2015
In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. We provide sufficient conditions to ensure the existence of a limit cycle bifurcating from the infinity. The main tools used are the Bendixson transformation and the averaging theory.
Three Limit Cycles in Discontinuous Piecewise Linear Differential Systems with Two Zones
2015
In this paper we study a planar piecewise linear differential system formed by two regions separated by a straight line so that one system has a real unstable focus and the other a virtual stable focus which coincides with the real one. This system was introduced by S.-M. Huan and X.-S. Yang in [8] who numerically showed that it can exhibit 3 limit cycles surrounding the real focus. This is the first example that a non–smooth piecewise linear differential system with two zones can have 3 limit cycles surrounding a unique equilibrium. We provide a rigorous proof of this numerical result.
Existence of piecewise linear differential systems with exactly n limit cycles for all
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In this paper, we prove that the piecewise linear di erential systeṁ x = −y − (x);ẏ = x with = 0 and an odd piecewise linear periodic function of period 4, has exactly n limit cycles in the strip |x| 6 2(n + 1). Consequently, there are piecewise linear di erential systems having inÿnitely many limit cycles in the real plane. We also provide examples of piecewise linear di erential systems having exactly n limit cycles for all n ∈ N. ?
Dynamical Systems
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 classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles.
On the limit cycles of a class of piecewise linear differential systems in with two zones
Mathematics and Computers in Simulation, 2011
We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems with two zones. Our main result shows that three is an upper bound for the number of limit cycles that bifurcate from a center, up to first order expansion of the displacement function. Moreover, this upper bound is reached. The main technique used is the averaging method.
International Journal of Bifurcation and Chaos, 2022
First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise diffe...
Two Limit Cycles in Liénard Piecewise Linear Differential Systems
Journal of Nonlinear Science, 2018
Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise-linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry.
On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones
International Journal of Bifurcation and Chaos, 2015
Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.
On the number of limit cycles in piecewise planar quadratic differential systems
arXiv (Cornell University), 2023
We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply our technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S 1 , S 2 , S 3 , and S 4 , as well as to non-smooth perturbations of non-smooth centers given by putting different S i 's in each zone. To show the coverage of our approach, we apply its first order, which recovers the averaging method of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply its second order to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.