Estimates on the number of limit cycles of a generalized Abel equation
Naeem M. H. Alkoumi Pedro J. Torres
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 25-34 doi: 10.3934/dcds.2011.31.25
We prove new results about the number of isolated periodic solutions of a first order differential equation with a polynomial nonlinearity. Such results are applied to bound the number of limit cycles of a family of planar polynomial vector fields which generalize the so-called rigid systems.
keywords: Polynomial nonlinearity limit cycle polynomial vector field. periodic solution
Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass
Jifeng Chu Zaitao Liang Pedro J. Torres Zhe Zhou
Discrete & Continuous Dynamical Systems - B 2017, 22(7): 2669-2685 doi: 10.3934/dcdsb.2017130

We study the existence and stability of periodic solutions of a differential equation that models the planar oscillations of a satellite in an elliptic orbit around its center of mass. The proof is based on a suitable version of Poincaré-Birkhoff theorem and the third order approximation method.

keywords: Satellite equation twist periodic solutions unstable periodic solutions Poincaré-Birkhoff theorem third order approximation
Non-collision periodic solutions of forced dynamical systems with weak singularities
Pedro J. Torres
Discrete & Continuous Dynamical Systems - A 2004, 11(2&3): 693-698 doi: 10.3934/dcds.2004.11.693
We prove the existence of periodic solutions in a second order differential system with a singular potential of attractive or repulsive type and forced periodically. The proof is based on a Krasnoselskii fixed point theorem for absolutely continuous operators on a Banach space, and this makes possible to avoid any kind of "strong force" condition.
keywords: weak singularity. Periodic solution singular potential
Vortex interaction dynamics in trapped Bose-Einstein condensates
Pedro J. Torres R. Carretero-González S. Middelkamp P. Schmelcher Dimitri J. Frantzeskakis P.G. Kevrekidis
Communications on Pure & Applied Analysis 2011, 10(6): 1589-1615 doi: 10.3934/cpaa.2011.10.1589
Motivated by recent experiments studying the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates (BECs), we illustrate that such systems can be accurately described by ordinary differential equations (ODEs) incorporating the precession and interaction dynamics of vortices in harmonic traps. This dynamics is tackled in detail at the ODE level, both for the simpler case of equal charge vortices, and for the more complicated (yet also experimentally relevant) case of opposite charge vortices. In the former case, we identify the dynamics as being chiefly quasi-periodic (although potentially periodic), while in the latter, irregular dynamics may ensue when suitable external drive of the BEC cloud is also considered. Our analytical findings are corroborated by numerical computations of the reduced ODE system.
keywords: vortices vortex dipoles. Bose-Einstein condensates
Solvability for some boundary value problems with $\phi$-Laplacian operators
J. Ángel Cid Pedro J. Torres
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 727-732 doi: 10.3934/dcds.2009.23.727
We study the existence of solution for the one-dimensional $\phi$-laplacian equation $(\phi(u'))'=\lambda f(t,u,u')$ with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate $\lambda_0$ is given such that the problem possesses a solution for any $|\lambda|<\lambda_0$.
keywords: mixed boundary value problem Schauder fixed point theorem Dirichlet boundary value problem $\phi$-Laplacian
Periodic solutions of twist type of an earth satellite equation
Daniel Núñez Pedro J. Torres
Discrete & Continuous Dynamical Systems - A 2001, 7(2): 303-306 doi: 10.3934/dcds.2001.7.303
We study Lyapunov stability for a given equation modelling the motion of an earth satellite. The proof combines bilateral bounds of the solution with the theory of twist solutions.
keywords: earth satellite equation. Odd periodic solution twist Lyapunov stability
On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian
J. Ángel Cid Pedro J. Torres
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 141-152 doi: 10.3934/dcds.2013.33.141
In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$
keywords: pendulum equation Periodic solution stability. $\phi$-Laplacian degree theory
Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities
Juan Belmonte-Beitia Víctor M. Pérez-García Vadym Vekslerchik Pedro J. Torres
Discrete & Continuous Dynamical Systems - B 2008, 9(2): 221-233 doi: 10.3934/dcdsb.2008.9.221
Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to study different examples and use the qualitative theory of dynamical systems to obtain some properties of these solutions.
keywords: Nonlinear Schrodinger equations dynamical systems. Solitons Lie symmetries
Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem
Jifeng Chu Pedro J. Torres Feng Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1921-1932 doi: 10.3934/dcds.2015.35.1921
For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.
keywords: Gylden-Meshcherskii-type problem Radial stability twist. periodic solutions
Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations
Pedro J. Torres Zhibo Cheng Jingli Ren
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2155-2168 doi: 10.3934/dcds.2013.33.2155
We analyze the non-degeneracy of the linear $2n$-order differential equation $u^{(2n)}+\sum\limits_{m=1}^{2n-1}a_{m}u^{(m)}=q(t)u$ with potential $q(t)\in L^p(\mathbb{R}/T\mathbb{Z})$, by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.
keywords: superlinear uniqueness $2n$-order differential equation. Non-degeneracy semilinear

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