scholarly journals Crossing Periodic Orbits via First Integrals

2020 ◽  
Vol 30 (11) ◽  
pp. 2050163
Author(s):  
Jaume Llibre ◽  
Durval José Tonon ◽  
Mariana Queiroz Velter
Keyword(s):  
Periodic Orbits ◽  
First Integrals ◽  
The Other ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Families Of Periodic Orbits ◽  
Linear Differential ◽  
Nonlinear Differential Systems

We characterize the families of periodic orbits of two discontinuous piecewise differential systems in [Formula: see text] separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems.

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.

HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ3

2007 ◽  
Vol 17 (04) ◽  
pp. 1171-1184 ◽  
Author(s):  
JAUME LLIBRE ◽  
ENRIQUE PONCE ◽  
ANTONIO E. TERUEL
Keyword(s):  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Homoclinic Orbits ◽  
Parametric Family ◽  
Homoclinic Loop ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Analytical Proof ◽  
Linear Differential

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ = 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit.

scholarly journals Fabricación de (sin)sentido

2017 ◽  
Author(s):  
Juan F. Navarro
Keyword(s):  
Periodic Orbits ◽  
Applied Mathematics ◽  
Differential Systems ◽  
International Conference ◽  
Linear Differential Systems ◽  
Non Linear ◽  
Linear Differential ◽  
Symbolic Algorithm

FABRICATION OF (NON)SENSE es un proyecto artístico que trata de cuestionar los sistemas discursivos sobre los que se construyen la Ciencia y el Arte, mediante la fabricación de un resultado científico `falso’ con apariencia de veracidad y su posterior difusión en el ámbito científico internacional: ha sido presentado en la 2nd International Conference on Mathematics and Computers in Sciences and Industry, celebrada en Malta en agosto de 2015, y publicado en el Journal of Advances in Applied Mathematics (Vol. 1, No. 3, pp. 160-174) en 2016. La aportación, titulada A Symbolic Algorithm for the Computation of Periodic Orbits in Non-Linear Differential Systems, presenta un método para el cálculo de órbitas periódicas en sistemas diferenciales no lineales, empleando para ello una adaptación del método de Poincaré-Lindstedt. Como proyecto artístico, ha sido seleccionado en la convocatoria `Arte en la Casa Bardín’ del Instituto Juan Gil-Albert.Esta comunicación al III Congreso Internacional de Investigación en Artes Visuales analiza, en primer lugar, las partes en las que se ha estructurado el proyecto. En segundo lugar, establece los diferentes niveles de lectura de la pieza, como cuestionamiento de los sistemas de la Ciencia y el Arte y, finalmente, como una reducción al absurdo. Por último, articula una serie de conexiones con otras piezas, estudios y movimientos adscritos al ámbito del arte contemporáneo.http://dx.doi.org/10.4995/ANIAV.2017.4970

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