Homoclinic orbits of slowly periodically forced and weakly damped beams resting on weakly elastic bearings. (Q1413040)

The present paper is concerned with the equation \[ \begin{aligned} & u_{tt}+u_{xxxx}+\varepsilon \delta u_t +\varepsilon \mu h(x,\sqrt{\varepsilon } t)=0, \\ & u_{xx}(0,\cdot)=u_{xx}(\pi/4, \cdot)=0, \\ & u_{xxx}(0,\cdot)=-\varepsilon f(u(0,\cdot )),\\ &u_{xxx}(\pi/4, \cdot)=\varepsilon f(u(\pi/4, \cdot )), \end{aligned} \] where \(\varepsilon >0\) and \(\mu\) are small parameters, \(\delta \geq 0\) is a constant, \(f\in C^2(\mathbb R, \mathbb R)\), \(h\in C^1([0,\pi/4] \times \mathbb R, \mathbb R)\) and \(h\) is \(1\)-periodic in \(t\). Such equation describes vibrations of a beam resting on two identical bearings with purely elastic responses which are determined by \(f\). The length of the beam is \(\pi/4\). The authors show the existence of bounded and periodic solutions for such equation under some conditions. The homoclinicity of the bounded solutions to periodic ones are also derived.