The nature of chaos in two systems of ordinary nonlinear differential equations

When a suitable method is applied for the numerical integration of systems of ordinary differential equations with one (Rössler) or two (Lorenz) nonlinear terms, the integral curve is always smooth, without any trace of chaotic motion, and the numerical results are supported by a simple analysis of the equations and by Poincaré’s qualitative theory of the ordinary nonlinear differential equations. It is also demonstrated that a degenerate system with two nonlinear terms can be integrated analytically. The system is reduced to the differential equation for a harmonic oscillator (an electric LC circuit), where the only difference lies in the fact that the independent variable is not the time variable but an exponential function of time. For a certain set of constants the integral of the Lorenz system is asymptotically, very fast, approaching an integral of the degenerate system. The results of numerical integration by two methods discussed in the paper are compared with the analytical solution. They show that the methods are stable and accurate.