scholarly journals Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in with Three Zones

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yanyan Cheng
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Averaging Method ◽  
Piecewise Linear ◽  
Dimensional System ◽  
Bifurcation Of Limit Cycles ◽  
Linear Differential Systems ◽  
Main Technique ◽  
Perturbed Systems ◽  
Linear Differential

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional system when the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems. Moreover, there are perturbed systems having six limit cycles. The main technique is the averaging method.

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

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

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 have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives 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.

scholarly journals LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS

2011 ◽  
Vol 21 (11) ◽  
pp. 3181-3194 ◽  
Author(s):  
PEDRO TONIOL CARDIN ◽  
TIAGO DE CARVALHO ◽  
JAUME LLIBRE
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Displacement Function ◽  
Two Dimensional ◽  
Bifurcation Of Limit Cycles ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Order Expansion ◽  
Linear Differential

We study the bifurcation of limit cycles from the periodic orbits of a two-dimensional (resp. four-dimensional) linear center in ℝn perturbed inside a class of discontinuous piecewise linear differential systems. Our main result shows that at most 1 (resp. 3) limit cycle can bifurcate up to first-order expansion of the displacement function with respect to the small parameter. This upper bound is reached. For proving these results, we use the averaging theory in a form where the differentiability of the system is not needed.

BIFURCATION OF LIMIT CYCLES FROM A FOUR-DIMENSIONAL CENTER IN CONTROL SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2653-2662 ◽  
Author(s):  
ADRIANA BUICĂ ◽  
JAUME LLIBRE
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Piecewise Linear ◽  
Displacement Function ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Linear Differential Systems ◽  
Order Expansion ◽  
Linear Differential ◽  
Natural Way

We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems, which appears in a natural way in control theory. Our main result shows that three is an upper bound for the number of limit cycles, up to first-order expansion of the displacement function with respect to the small parameter. Moreover, this upper bound is reached. For proving this result we use the averaging method in a form where the differentiability of the system is not needed.

scholarly journals ON THE NUMBER OF LIMIT CYCLES FOR DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ2n WITH TWO ZONES

2013 ◽  
Vol 23 (02) ◽  
pp. 1350024 ◽  
Author(s):  
JAUME LLIBRE ◽  
FENG RONG
Keyword(s):  
Small Parameter ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Displacement Function ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Order Expansion ◽  
Linear Differential

We study the number of limit cycles of the discontinuous piecewise linear differential systems in ℝ2n with two zones separated by a hyperplane. Our main result shows that at most (8n - 6)n-1 limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result, we use the averaging theory in a form where the differentiability of the system is not necessary.

Bifurcation of Limit Cycles from a Quintic Center via the Second Order Averaging Method

2015 ◽  
Vol 25 (03) ◽  
pp. 1550047 ◽  
Author(s):  
Linping Peng ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Method ◽  
Second Order ◽  
Averaging Theory ◽  
Bifurcation Of Limit Cycles

This paper is concerned with the bifurcation of limit cycles from a quintic system with one center. By using the averaging theory, we show that under any small quintic homogeneous perturbations, up to order 1 in ε, at most three limit cycles bifurcate from periodic orbits of the considered system, and this upper bound can be reached. Up to order 2 in ε, at most seven limit cycles emerge from periodic orbits of the unperturbed one.

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