Existence of Invariant Cones in General 3-Dim Homogeneous Piecewise Linear Differential Systems with Two Zones

2017 ◽  
Vol 27 (12) ◽  
pp. 1750189 ◽  
Author(s):  
Song-Mei Huan
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Special Cases ◽  
Line Separation ◽  
Linear Differential ◽  
Invariant Cones

Existence and number of invariant cones in general 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane are investigated. Implicit parametric expressions of two proper half slope maps whose intersections determine the existence and number of invariant cones are obtained. Based on these expressions, some sufficient conditions for the existence of at most three invariant cones are provided, and it is proved that the maximum number of invariant cones for some special cases is equal to 1 plus the maximum number of limit cycles in planar piecewise linear systems with a straight line separation. Moreover, it is illustrated by a numerical example with four invariant cones that the maximum number of invariant cones is not less than four. Specially, the main results provide a method to completely solve the existence and number of invariant cones in any specific 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane by using numerical 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 LOWER BOUNDS FOR THE MAXIMUM NUMBER OF LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH A STRAIGHT LINE OF SEPARATION

2013 ◽  
Vol 23 (04) ◽  
pp. 1350066 ◽  
Author(s):  
J. LLIBRE ◽  
M. A. TEIXEIRA ◽  
J. TORREGROSA
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Unique Equilibrium ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding segment. Moreover, the linear differential systems that we consider in every half-plane can have either a focus (F), or a node (N), or a saddle (S), these equilibrium points can be real or virtual. Then, we can consider six kinds of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS, SS. We provide for each of these types of discontinuous differential systems examples with two limit cycles.

The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Marina Esteban ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Hilbert Problem ◽  
Piecewise Linear ◽  
Upper Bounds ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential ◽  
16Th Hilbert Problem ◽  
To Receive

The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin in 1920’s, and nowadays such systems still continue to receive the attention of many researchers mainly due to their applications. We study the discontinuous piecewise differential systems formed by two linear centers separated by a nonregular straight line. We provide upper bounds for the maximum number of limit cycles that these discontinuous piecewise differential systems can exhibit and we show that these upper bounds are reached. Hence, we solve the extended 16th Hilbert problem for this class of piecewise differential systems.

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.

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.

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