Dynamic Stability of High Dimensional Dynamical Systems

We investigate the dynamical stability conjectures of Palis and Smale, and Pugh and Shub from the standpoint of numerical observation and put forth a stability conjecture of our own. We find that as the dimension of a dissipative dynamical system is increased, the number of positive Lyapunov exponents increases monotonically, the number of observable periodic windows decreases at least below numerical precision, and we observe a subset of parameter space such that topological change is very common with small parameter perturbation. However, this seemingly inevitable topological variation is never catastrophic (the dynamic type is preserved) if the dimension of the system is high enough.