Existence of a homoclinic orbit in a generalized Liénard type system

The object of this paper is to study the existence and nonexistence of an important orbit in a generalized Liénard type system. This trajectory is doubly asymptotic to an equilibrium solution, i.e., an orbit which lies in the intersection of the stable and unstable manifolds of a critical point. Such an orbit is called a homoclinic orbit

A hyperbolic Lindstedt-poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators

A hyperbolic Lindstedt-Poincaré method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Liénard oscillator is studied in detail, and the present method's predictions are compared with those of Runge- Kutta method to illustrate its accuracy. © 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-verlag GmbH.postprin

Slow divergence integrals in generalized Liénard equations near centers

Using techniques from singular perturbations we show that for any $n\ge 6$ and $m\ge 2$ there are Liénard equations $\{\dot{x}=y-F(x),\ \dot{y}=G(x)\}$, with $F$ a polynomial of degree $n$ and $G$ a polynomial of degree $m$, having at least $2[\frac{n-2}{2}]+[\frac{m}{2}]$ hyperbolic limit cycles, where $[\cdot]$ denotes "the greatest integer equal or below"

Complex oscillations with multiple timescales - Application to neuronal dynamics

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations in three-dimensional systems. Two families of such complex oscillations are considered: mixed-mode oscillations (MMOs) in systems with two slow variables, and bursting oscillations in systems with two fast variables. In the last chapter, we present recent results on systems with two slow and two fast variables, where both MMO-type dynamics and bursting-type dynamics can arise and where complex oscillations are also organised by canard solutions. The main application area that we consider here is that of neuroscience, more precisely low-dimensional point models of neurons displaying both sub-threshold and spiking behaviour. We focus on analysing how canard objects allow to control the oscillatory patterns observed in these neuron models, in particular the crossings of excitability thresholds

Isochronous and Unexpected Behavior for Complex-Valued Nonlinear Oscillators with Parametric Excitation ​

Usually oscillators with periodic excitations show a periodic motion with frequency equal to the forcing one. A complex-valued nonlinear oscillator under parametric excitation is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four differential equations for two nonlinearly coupled oscillators are derived. Approximate solutions are obtained and their stability is discussed. We found that the resulting motion is periodic with a frequency equal to the forcing one, if appropriate inequalities are satisfiedand then for a large parameter range. The system is isochronous because periodic solutions are possible in a well defined phase region and not only for certain discrete values. Moreover we demonstrate that if we insert a gyroscopic term the motion can be always periodic for a well defined parameter range but with a frequency different from the forcing frequency.Analytic approximate solutions are checked by numerical integration.

On solutions of neumann boundary value problem for the liénard type equation

We provide conditions on the functions f(x) and g(x), which ensure the existence of solutions to the Neumann boundary value problem for the equation x'' + f(x)x'2+g(x)=0. First Published Online: 14 Oct 201

A Bendixson-Dulac theorem for some piecewise systems

The Bendixson-Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems. We apply it to three different Liénard piecewise differential systems ¨ x+f±(x)˙ x+x = 0. The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one

Qualitative Analysis of Solutions to the Semiclassical Einstein Equation in homogeneous and isotropic Spacetimes

In der vorliegenden Arbeit werden Methoden aus der Theorie der dynamischen Systeme verwendet, um das qualitative Verhalten von Lösungen der semiklassischen Einsteingleichung für Friedmann-Lamaître-Robertson-Walker Raumzeiten zu untersuchen. Es werden ausschließlich masselose und konform gekoppelte Quantenfelder betrachtet. Bei der Renormierung des Energie-Impuls-Tensors solcher Quantenfelder treten Ambiguitäten auf, die sich als freie Parameter in der semiklassischen Einsteingleichung manifestieren. Mit Hilfe der Theorie der dynamischen Systeme ist es möglich, Lösungen nach ihren qualitativen Verhalten zu klassifizieren und dadurch Argumente für oder gegen bestimmte Werte der Renormierungskonstanten herauszuarbeiten. Befindet sich das Quantenfeld im konformen Vakuumzustand, erhält man ein zweidimensionales dynamisches System. Für dieses dynamische System werden die strukturell stabilen Fälle und Bifurkationsdiagramme herausgearbeitet, sowie das globale Stabilitätsverhalten der Minkowski und De-Sitter Gleichgewichtspunkte. Mittels dieser Analyse wird das qualitative Verhalten der semiklassischenLösungen mit dem qualitativen Verhalten der Lösungen des Lambda-CDM Modells der Kosmologie verglichen. Es zeigt sich, dass das semiklassische Modell in der Lage ist das qualitative Verhalten von Lösungen des klassischen Lambda-CDM Modells wiederzugeben. Weiterhin wird gezeigt, das im Vakuumfall Lösungen existieren, welche sich, im Gegensatz zu Lösungen des klassischen Lambda-CDM Modells, im Allgemeinen nicht eindeutig durch ihre Anfangsdaten bestimmen lassen. Um dieses atypische Verhalten aufzulösen müssen die Trajektorien dieser Lösungen in einem dreidimensionalen Phasenraum betrachtet werden.Das entsprechende dreidimensionale dynamische System beschreibt das dynamische Verhalten der Lösungen für beliebige Quantenzustände. Für allgemeine Quantenzustände wird die lokale (Lyapunov-) Stabilität der Gleichgewichtspunkte untersucht und für eine spezielle Wahl der Renormierungskonstanten und des Quantenzustandes neue Lösungen gefunden und mit Lösungen des klassischen Lambda-CDM Modells verglichen. Auch hier besteht eine qualitative Äquivalenz

Nonlinear differential equations having non-sign-definite weights

In the present PhD thesis we deal with the study of the existence, multiplicity and complex behaviors of solutions for some classes of boundary value problems associated with second order nonlinear ordinary differential equations of the form $u''+f(u)u'+g(t,u)=s,$ or $u''+g(t,u)=0,$ $t\in I$, where $I$ is a bounded interval, $f\colon\mathbb{R}\to\mathbb{R}$ is continuous, $s\in\mathbb{R}$ and $g: I\times \mathbb{R}\to\mathbb{R}$ is a perturbation term characterizing the problems. The results carried out in this dissertation are mainly based on dynamical and topological approaches. The issues we address have arisen in the field of partial differential equations. For this reason, we do not treat only the case of ordinary differential equations, but also we take advantage of some results achieved in the one dimensional setting to give applications to nonlinear boundary value problems associated with partial differential equations. In the first part of the thesis, we are interested on a problem suggested by Antonio Ambrosetti in ``Observations on global inversion theorems'' (2011). In more detail, we deal with a periodic boundary value problem associated with the first differential equation where the perturbation term is given by $g(t,u):=a(t)\phi(u)-p(t)$. We assume that $a,$ $p\in L^{\infty}(I)$ and $\phi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying $\lim_{|\xi|\to\infty}\phi(\xi)=+\infty$. In this context, if the weight term $a(t)$ is such that $a(t)\geq 0$ for a.e. $t\in I$ and $\int_{I}a(t)\,dt>0$, we generalize the result of multiplicity of solutions given by Fabry, Mawhin and Nakashama in ``A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations'' (1986). We extend this kind of improvement also to more general nonlinear terms under local coercivity conditions. In this framework, we also treat in the same spirit Neumann problems associated with second order ordinary differential equations and periodic problems associated with first order ones. Furthermore, we face the classical case of a periodic Ambrosetti-Prodi problem with a weight term $a(t)$ which is constant and positive. Here, considering in the second differential equation a nonlinearity $g(t,u):=\phi(u)-h(t)$, we provide several conditions on the nonlinearity and the perturbative term that ensure the presence of complex behaviors for the solutions of the associated $T$-periodic problem. We also compare these outcomes with the result of stability carried out by Ortega in ``Stability of a periodic problem of Ambrosetti-Prodi type'' (1990). The case with damping term is discussed as well. In the second part of this work, we solve a conjecture by Yuan Lou and Thomas Nagylaki stated in ``A semilinear parabolic system for migration and selection in population genetics'' (2002). The problem refers to the number of positive solutions for Neumann boundary value problems associated with the second differential equation when the perturbation term is given by $g(t,u):=\lambda w(t)\psi(u)$ with $\lambda>0$, $w\in L^{\infty}(I)$ a sign-changing weight term such that $\int_{I}w(t)\,dt<0$ and $\psi\colon[0,1]\to[0,\infty[$ a non-concave continuous function satisfying $\psi(0)=0=\psi(1)$ and such that the map $\xi\mapsto \psi(\xi)/\xi$ is monotone decreasing. In addition to this outcome, other new results of multiplicity of positive solutions are presented as well, for both Neumann or Dirichlet boundary value problems, by means of a particular choice of indefinite weight terms $w(t)$ and different positive nonlinear terms $\psi(u)$ defined on the interval $[0,1]$ or on the positive real semi-axis $[0,+\infty[$