Nonlinear Stiffness, Lyapunov Exponents, and Attractor Dimension

I propose that stiffness may be defined and quantified for nonlinear systems using Lyapunov exponents, and demonstrate the relationship that exists between stiffness and the fractal dimension of a strange attractor: that stiff chaos is thin chaos. What constitutes a stiff dynamical system? Stiffness is closely related to numerical methods [1]: the signature of stiffness in a problem is that, upon integration with a general numerical scheme — a method not specially designed for stiff problems — the routine takes extremely small integration steps [2], which makes the process computationally expensive. One view is that stiffness is inextricably linked with the numerical integration scheme used, so that there would be no such thing as an intrinsically stiff dynamical system, and the best we could hope for is an operational definition such as that above [3]. Moreover, it has been proposed that chaotic problems cannot be stiff [4]. I argue below that this is not the case, and provide a definition and a quantitative measure of stiffness for nonlinear dynamical systems. I demonstrate how stiffness affects the geometry of the strange attractor of a chaotic system: that stiff chaos is thinner — has smaller fractional part of the fractal dimension — than nonstiff chaos. When integrating a stiff problem with a variable-step explicit numerical integration scheme, the initial step length chosen causes the method to be at or near numerical instability, which leads to a large local truncation error estimate. This causes the numerical routine to reduce the step length substantially, until the principal local truncation error is brought back within its prescribed bound. The routine then integrates the problem successfully, but uses a far greater number of steps than seems reasonable, given the smoothness of the solution. Because of this, computation time and round-off error are a problem when using conventional numerical integration techniques on stiff problems, and special methods have been developed for them. Traditionally in numerical analysis, a linear stiff system of size n is defined Article published in Physics Letters A 264 (1999) 298–304