Chaotic dynamics in an elastic medium with surface disorder

We investigate the dynamics of an elastic medium described by a two-dimensional network of nodes of equal mass connected by springs whose force constants are equal inside the network and chosen at random at its surface. The system can be considered a billiard in the sense that the network is ordered all throughout its bulk. Being an eigenvalue problem its complexity is manifested in a frequency statistics which, in most of the spectrum, can be described by the Wigner-Dyson distribution. At low frequencies the dispersion relation is linear in the wave number and the network shows regular behavior (frequency statistics according to Poisson distribution). We study the dynamical behavior of this model by investigating how the system escapes from a normal mode of the ordered network, and calculate the Lyapunov exponent $\ensuremath{\lambda}$ in different frequency regions.