In this paper a billiard problem in nonlinear and nonequilibrium systems is
investigated. This is an interesting problem where a traveling pulse solution
behaves as if it is a billiard ball at a glance in some kind of
reaction-diffusion system in a rectangular domain. We would like to elucidate
the characteristic properties of the solution of this system. For the purpose,
as the first step, we try to make a reduced model of discrete dynamical system
having the important properties which the original system must have. In this
paper we present a discrete toy model, which is reduced intuitively as one of
the candidates by use of numerical experiments and careful observation of the
solutions. Moreover, we discuss about the similar and important points between
the solution in the original ordinary differential equation (which describes the
pulse behavior) and the one in the toy model by computing numerically the
characteristic quantities in view of the dynamical system, for example, global
and local Lyapunov exponents and Lyapunov dimensions. As a result, we elucidate
that the system possesses an intermittent-type chaotic attractor.