Now showing items 1-20 of 57

• #### A general mechanism of diffusion in hamiltonian systems: qualitative results ﻿

(2019-01-01)
Article
Open Access
We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a ...
• #### A geometric approach to phase response curves and its numerical computation through the parameterization method ﻿

(2019-01-01)
Article
Open Access
The phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new ...
• #### A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T<sup>2</sup> ﻿

(1999)
Article
Open Access
We give a proof based in geometric perturbation theory of a result proved by J.N. Mather using variational methods. Namely, the existence of orbits with unbounded energy in perturbations of a generic geodesic flow in T^2 ...
• #### A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. ﻿

(2003)
Article
Open Access
We introduce a geometric mechanism for di?usion in a priori unstable nearly integrable dynamical systems. It is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and ...
• #### A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Announcement of results ﻿

(2003)
Article
Open Access
We present a geometric mechanism for diﬀusion in Hamiltonian systems. We also present tools that allow to verify it in a concrete model. In particular, we verify it in a system which presents the large gap problem.
• #### A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems ﻿

(2017-09-01)
Article
Open Access
Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should the dynamics be given ...
• #### Adiabatic invariant of the harmonic oscillator, complex matching and resurgence ﻿

(1997)
Article
Open Access
The linear oscillator equation with a frequency depending slowly on time is used to test a method to compute exponentially small quantities. This work present the matching method in the complex plane as a tool to obtain ...
• #### An analytical approach to codimension-2 sliding bifurcations in the dry friction oscillator ﻿

(2010)
Article
Restricted access - publisher's policy
In this paper, we analytically consider sliding bifurcations of periodic orbits in the dry-friction oscillator. The system depends on two parameters: F, which corresponds to the intensity of the friction, and ω, the ...
• #### Asymptotic behaviour of the domain of analyticity of invariant curves of the standard map ﻿

(1999)
Article
Open Access
In this paper we consider the standard map, and we study the invariant curve obtained by analytical continuation, with respect to the perturbative parameter E, of the invariant circle of rotation number the golden mean ...
• #### Asymptotic size of Herman rings of the complex standard family by quantitative quasiconformal surgery ﻿

(2003)
Article
Open Access
In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\ep}(u)=u{\rm e}^{{\rm i}\alpha} {\rm e}^{\frac{\ep}{2}(u-\frac{1}{u})}$, with \$\alpha=\alpha( ...
• #### Breakdown of a 2D heteroclinic connection in the hopf-zero singularity (I) ﻿

(2018-10-01)
Article
Open Access
In this paper we study a beyond all orders phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated ...
• #### Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II): The generic case ﻿

(2018-04-17)
Article
Open Access
In this paper, we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singularity. The method consists in obtaining ...
• #### Breakdown of heteroclinic orbits for analytic unfoldings of the hopf-zero singularity. The singular case ﻿

(2011)
External research report
Open Access
In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity. This heteroclinic orbit appears at any order if one performs the normal form around ...
• #### Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity ﻿

(2004)
Article
Open Access
In this paper we study the exponentially small splitting of a heteroclinic orbit in some unfoldings of the central singularity also called Hopf-zero singularity. The fields under consideration are of the form: dx dτ = ...
• #### Cálculos numéricos de la Hopf zero ﻿

(2009)
Conference lecture
Open Access
Consideramos la siguiente familia de campos X en R3 :  dx dt = − xz − y( + c z) + p+1f( x, y, z, ) dy dt = − yz + x( + c z) + p+1g( x, y, z, ) dz dt = (−1 + b(x2 + y2) + z2) + p+1h( x, y, z, ...
• #### Chaos in the hysteretic grazing-sliding codimension-one saddle-node bifurcation of piecewise dynamical systems ﻿

(UP4, Institute of Sciences, S.L., 2017)
Conference lecture
Open Access
We present two ways of regularizing a parameter family of piecewise smooth dynamical systems undergoing a grazing- sliding bifurcation. We use the Sotomayor-Teixeira regularization and prove that the bifurcation is a ...
• #### Computation of invariant curves in the analysis of periodically forced neural oscillators ﻿

(Springer, 2018)
Part of book or chapter of book
Open Access
Background oscillations, reflecting the excitability of neurons, are ubiq- uitous in the brain. Some studies have conjectured that when spikes sent by one population reach the other population in the peaks of excitability, ...
• #### Critical velocity in kink-defect interaction models: rigorous results ﻿

(2020-01-01)
Article
Open Access
In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More ...
• #### Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation ﻿

(2011-07)
External research report
Open Access
In this paper we study the splitting of separatrices phenomenon which arises when one considers a Hamiltonian System of one degree of freedom with a fast periodic or quasiperiodic and meromorphic in the state variables ...
• #### Exponentially Small Heteroclinic Breakdown in the Generic Hopf-Zero Singularity ﻿

(2013)
Article
Restricted access - publisher's policy
In this paper we prove the breakdown of a heteroclinic connection in the analytic versal unfoldings of the generic Hopf-zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order ...