Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations
Persistence behavior and bifurcations of Lyapunov manifolds are studied for classes of a parametrically excited wave equations. The semilinear IVP \[ \frac{d w}{ dt } + A w =\varepsilon f(w,t,\varepsilon), \, w(0)=w_0 \] is considered, where \(A\) generates a uniformly bounded \(C_0\)-group \(G(t)\) on the Banach space \(X.\) The second object of this paper is the following wave equation: \[ u_{tt} - c^2 u_{xx} + \varepsilon \beta u_t +(\omega_0^2 + \varepsilon \gamma \cos t) u = \varepsilon \alpha u^3,\quad t\geq 0,\;0<x<\pi \] \[ u_x(0,t) = u_x(\pi,t)=0,\quad \beta>0 \] The fast dynamics behavior is segregated for a finite, resonant part of the spectrum and slow dynamics is found elsewhere. The manifold where the fast dynamics takes place is almost-invariant. It is to be noted that a number of the phenomena have been found in this paper, i.e. periodic and quasi-periodic solutions, are stable and open for experimental investigation.
scientific article; zbMATH DE number 5641443