Seminar in PSA 107

ABSTRACT

Title:  Chaos or numerical errors?

By:    Lun-Shin Yao, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, Arizona

Discrete numerical methods with finite time steps are probably the most practical technique, if not the only one, to solve non-linear differential equations. This is particularly true for chaos. Using the Lorenz equations as an example, we clearly demonstrate that a trajectory mistakenly penetrates the virtual separatrix due to finite time steps and truncation errors. This leads to the explosive growth of numerical errors; hence, these computed "solutions" are unshadowable and are not valid solutions. This is generic for chaotic systems and occurs repeatedly since the l-lemma guarantees a trajectory travels arbitrarily close to the inset of a saddle point over and over again. Hence, numerical solutions of differential equations sensitive to the integration-time step with finite statistical properties are divergent, bounded numerical errors. At present, all numerical methods introduce truncation errors; consequently, they are incapable of solving chaotic differential equations. The convergence of these numerical chaotic solutions has never been proven. Realistically, it cannot be proven; nevertheless, it is common practice, perhaps out of convenience and for lack of an alternative, to blindly accept them as adequate solutions to these problems!