Chaos and bifurcation in numerical computation by the Runge-Kutta method

In this paper the chaotic phenomenon and bifurcation in numerical computation using the Runge-Kutta method to discretize the nonlinear differential equation are investigated. It is shown that the bifurcation condition in the discretized equation is given by the eigenvalue of the jacobian matrix of the original differential equation. As an example, the bifurcation and chaos when a second-order nonlinear equation is discretized by the Runge-Kutta method is investigated and it is shown that the scenario from a stable fixed point to chaos when the fourth-order Runge-Kutta method is applied is quite different from those of the second-order Runge-Kutta method