Chaos in numerical analysis of ordinary differential equations

The discretisation of the ordinary nonlinear differential equation $\text{d}\text{y}\text{d}\text{t}\text{= y(1?y)}$ by the entral difference scheme is studied for fixed mesh size. In the usual numerical computation, this method produces some “ghost solution” for the long range calculation. Regarding this discretisation as a dynamical system in R², these pathological behaviors are shown to be a kind of “chaos” in the dynamical system for any mesh size. Moreover, some combination of the central difference scheme and the Euler's scheme is studied for the above equation. It gives some motivation for Hénon's model. The usual discretisation of a second order differential equation are studied also. It gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by Hénon-Heiles.