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Piecewise linear differential systems with only centers can create limit cycles?
https://ddd.uab.cat/record/199361
In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle. Llibre, JaumeMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993612018Piecewise linear differential systems without equilibria produce limit cycles?
https://ddd.uab.cat/record/182496
In this article we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual. Llibre, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1824962017Periodic orbits of continuous and discontinuous piecewise linear differential systems via first integrals
https://ddd.uab.cat/record/182491
Llibre, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1824912017Birth of limit cycles for a classe of continuous and discontinuous differential systems in (d 2)-dimension
https://ddd.uab.cat/record/169449
The orbits of the reversible differential system ˙x = −y, ˙y = x, ˙z = 0, with x, y ∈ R and z ∈ R d, are periodic with the exception of the equilibrium points (0, 0, z1, . . . , zd). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system ˙x = −y, ˙y = x, ˙z = 0, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case this maximum number is nd(n − 1)/2, and in the second is nd(n − 1). Llibre, JaumeMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694492016Periodic solutions of Lienard differential equations via averaging theory of order two
https://ddd.uab.cat/record/169442
For ε = 0 sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x + f(x)x + n2x + g(x) = ε2p1(t) + ε3p2(t), where n is a positive integer, f : R → R is a C3 function, g : R → R is a C4 function, and pi : R → R for i = 1, 2 are continuous 2π–periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained. Llibre, JaumeMon, 23 Jan 2017 15:21:45 GMThttps://ddd.uab.cat/record/1694422015On the periodic solutions of perturbed 4D non-resonant systems
https://ddd.uab.cat/record/169432
We provide sufficient conditions for the existence of periodic solutions of a 4D non-resonant system perturbed by smooth or non-smooth functions. We apply these results to study the small amplitude periodic solutions of the non-linear planar double pendulum perturbed by smooth or non-smooth function. Llibre, JaumeMon, 23 Jan 2017 15:21:45 GMThttps://ddd.uab.cat/record/1694322015Higher order averaging theory for finding periodic solutions via Brouwer degree
https://ddd.uab.cat/record/150723
In this paper we deal with nonlinear differential systems of the form x′(t) = Xki=0εiFi(t, x) + εk+1R(t, x, ε), where Fi : R × D → Rn for i = 0, 1, · · · , k, and R : R × D × (−ε0, ε0) → Rn are continuous functions, T–periodic in the first variable, being D an open subset of Rn, and ε a small parameter. For such differential systems, which do not need to be of class C1, under convenient assumptions we extend the averaging theory for computing their periodic solutions to k–th order in ε. Some applications are also performed. Llibre, JaumeFri, 06 May 2016 08:59:51 GMThttps://ddd.uab.cat/record/1507232014Corrigendum: Higher order averaging theory for finding periodic solutions via Brouwer degree (2014 Nonlinearity 27 563)
https://ddd.uab.cat/record/150680
Llibre, JaumeFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506802014Piecewise linear differential systems with two real saddles
https://ddd.uab.cat/record/150653
In this paper we study piecewise linear differential systems formed by two regions separated by a straight line so that each system has a real saddle point in its region of definition. If both saddles are conveniently situated, they produce a transition flow from a segment of the splitting line to another segment of the same line, and this produces a generalized singular point on the line. This point is a focus or a center and there can be found limit cycles around it. We are going to show that the maximum number of limit cycles that can bifurcate from this focus is two. One of them appears through a Hopf bifurcation and the second when the focus becomes a node by means of the sliding. Artés, Joan CarlesFri, 06 May 2016 08:54:16 GMThttps://ddd.uab.cat/record/1506532013Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems witha a straight line of separation
https://ddd.uab.cat/record/150641
In this paper we study the maximum number of limit cycles for planar discontinuous piecewise linear differential systems defined in two half–planes separated by a straight line. Here we only consider non-sliding limit cycles. For that systems, the interior of any limit cycle only contains a unique singular 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 analyze for each of these types of discontinuous differential systems the maximum number of known limit cycles. Llibre, JaumeFri, 06 May 2016 08:54:16 GMThttps://ddd.uab.cat/record/1506412013On the periodic orbits of the fourth-order differential equation u' ' ' ' qu' '−u= F (u,u',u' ',u' ' ')
https://ddd.uab.cat/record/150556
We provide sufficient conditions for the existence of periodic solutions of the fourth-order differential equation u′′′′ + qu′′ − u = εF(u, u′, u′′, u′′′), where q and ε are real parameters, ε is small and F is a nonlinear function. Llibre, JaumeFri, 06 May 2016 08:49:23 GMThttps://ddd.uab.cat/record/1505562012Synchronization and non-smooth dynamical systems
https://ddd.uab.cat/record/150525
In this article we establish an interaction between nonsmooth systems, geometric singular perturbation theory and synchronization phenomena. We find conditions for a non-smooth vector fields be locally synchronized. Moreover its regularization provide a singular perturbation problem with attracting critical manifold. We also state a result about the synchronization which occurs in the regularization of the fold-fold case. We restrict ourselves to the 3-dimensional systems (` = 3) and consider the case known as a T-singularity. Llibre, JaumeFri, 06 May 2016 08:49:22 GMThttps://ddd.uab.cat/record/1505252012A universal constant for semistable limit cycles
https://ddd.uab.cat/record/150475
We consider one–parameter families of 2–dimensional vector fields Xµ having in a convenient region R a semistable limit cycle of multiplicity 2m when µ = 0, no limit cycles if µ / 0, and two limit cycles one stable and the other unstable if µ ' 0. We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integers n sufficiently large there is a power law in the parameter µ of the form µn ≈ Cnα < 0 with C, α ∈ R, such that the orbit of Xµn through a point of p ∈ R reaches the position of the semistable limit cycle of X0 after given n turns. The exponent α of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p ∈ R and of the family Xµ. In fact α = −2m/(2m − 1). Moreover the constant C is independent of the initial point p ∈ R, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle Γ. Artés, Joan CarlesFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504752011On the birth of minimal sets for perturbed reversible vector fields
https://ddd.uab.cat/record/150437
The results in this paper fit into a program to study the existence of periodic orbits, invariant cylinders and tori filled with periodic orbits in perturbed reversible systems. Here we focus on bifurcations of one-parameter families of periodic orbits for reversible vector fields in R4. The main used tools are normal forms theory, Lyapunov-Schmidt method and averaging theory. Llibre, JaumeFri, 06 May 2016 08:30:49 GMThttps://ddd.uab.cat/record/1504372011On the periodic solutions of a perturbed double pendulum
https://ddd.uab.cat/record/150406
We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion ¨θ1 = −2aθ1 + aθ2 + εF1(t, θ1, ˙θ1, θ2, ˙θ2), ¨θ2 = 2aθ1 − 2aθ2 + εF2(t, θ1, ˙θ1, θ2, ˙θ2), where a and ε are real parameters. The two masses of the unperturbed double pendulum are equal, and its two stems have the same length l. In fact a = g/l where g is the acceleration of the gravity. Here the parameter ε is small and the smooth functions F1 and F2 define the perturbation which are periodic functions in t and in resonance p:q with some of the periodic solutions of the unperturbed double pendulum, being p and q positive integers relatively prime. Llibre, JaumeFri, 06 May 2016 08:30:47 GMThttps://ddd.uab.cat/record/1504062011Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations
https://ddd.uab.cat/record/145367
We provide lower bounds for the maximum number of limit cycles for the m–piecewise discontinuous polynomial differential equations x = y + sgn(gm (x, y))F (x), y = −x, where the zero set of the function sgn(gm (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane in sectors of angle 2π/m, and sgn(z) denotes the sign function. Llibre, JaumeTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453672015On the birth of limit cycles for non-smooth dynamical systems
https://ddd.uab.cat/record/145361
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 result are presented to ensure the existence of limit cycles of such systems. These result may represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail. Llibre, JaumeTue, 12 Jan 2016 16:38:11 GMThttps://ddd.uab.cat/record/1453612015Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds
https://ddd.uab.cat/record/145354
We consider a differential equation ˙p = X(p), p ϵ R3 with discontinuous right-hand side and discontinuities occurring on an algebraic variety ∑. We discuss the dynamics of the sliding mode which occurs when for any initial condition near p ϵ ∑ the corresponding solution trajectories are attracted to ∑. First we suppose that ∑ = H-1(0) where H is a polynomial function and 0 ϵ R is a regular value. In this case ∑ is locally di↵eomorphic to the set F = {(x, y, z) ϵ R3; z = 0} (Filippov). Second we suppose that ∑ is the inverse image of a non–regular value. We focus our attention to the equations defined around singularities as described in [8]. More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of R3 in terms of implicit functions and immersions are broken in a stable manner. In this case ∑ is locally diffeomorphic to one of the following sets D = {(x, y, z) ϵ R3; xy = 0} (double crossing); T = {(x, y, z) ϵ R3; xyz = 0} (triple crossing); C = {(x, y, z) ϵ R3; z2-x2-y2 = 0}(cone) or W = {(x, y, z) ϵ R3; zx2-y2 = 0} (Whitney's umbrella). Llibre, JaumeTue, 12 Jan 2016 16:38:11 GMThttps://ddd.uab.cat/record/1453542015Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones
https://ddd.uab.cat/record/145336
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 its is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential systems 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. Llibre, JaumeTue, 12 Jan 2016 16:38:10 GMThttps://ddd.uab.cat/record/1453362015Maximum number of limit cycles for certain piecewise linear dynamical systems
https://ddd.uab.cat/record/145304
This paper deals with the question of the determinacy of the maximum number of limit cycles of some classes of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line \Sigma. We restrict ourselves to the non-sliding limit cycles case, i. e. , limit cycles that do not contain any sliding segment. Among all cases treated here, it is proved that the maximum number of limit cycles is at most 2 if one of the two linear differential systems of the discontinuous piecewise linear differential system has a focus in \Sigma, a center, or a weak saddle. We use the theory of Chebyshev systems for establishing sharp upper bounds for the number of limit cycles. Some normal forms are also provided for these systems. Llibre, JaumeTue, 12 Jan 2016 16:38:08 GMThttps://ddd.uab.cat/record/1453042015Generic bifurcation of reversible vector fields on a 2-dimensional manifold
https://ddd.uab.cat/record/12931
In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed. Teixeira, Marco AntonioWed, 22 Nov 2006 17:12:02 GMThttps://ddd.uab.cat/record/129311997