The paper deals with the boundary value problem $$u^{\text{'}\text{'}}+ku=g\left(u\right)+e\left(t\right),u\left(0\right)=u\left(2\pi \right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(2\pi \right),$$ where $k\in ℝ$, $gI\mapsto ℝ$ is continuous, $e\in 𝕃J$ and ${lim}_{x\to 0+}{\int }_x^{1}g\left(s\right)\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in L^{\infty }(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary...

This paper presents vector versions of some existence results recently published for certain fourth order differential systems based on generalisations drawn from possibilities arising from the underlying auxiliary equation. The results obtained also extend some known works involving third order differential systems to the corresponding fourth order.

Henri Poincaré avait déjà remarqué que les variétés stable et instable du pendule perturbé, défini par l’hamiltonien$$H(q,p,t)=p^2/2+(-1+\cos q)(1-\mu \sin (t/ϵ\left)\right),$$ne coïncident pas lorsque que le paramètre $\mu$ n’est pas nul, mais qu’on peut leur associer un même développement formel divergent en puissance de $ϵ$. Cette divergence est ici analysée au moyen de la récente théorie de la résurgence, et du calcul étranger qui permet de trouver un équivalent asymptotique de l’écart des deux variétés pour $ϵ$ tendant vers zéro - du moins cela est-il montré...

Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions.

Nous considérons un germe $\omega$ de 1-forme analytique dans ${ℝ}^2,0$ dont le 1-jet est $ydy$. Nous montrons que si l’équation $\omega =0$ définit un centre (i.e toutes les courbes solutions sont des cycles) il existe une involution analytique de ${ℝ}^2,0$ préservant le portrait de phase du système. Géométriquement ceci signifie que les centres analytiques nilpotents sont obtenus par image réciproque par des applications pli. Un théorème de conjugaison équivariante permet d’obtenir une classification complète de ces centres.

We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and the existence of a symmetric homoclinic orbit is rigorously verified as an example.