Toward quantitative characterization of simulated earthquake- cycle complexities

Earthquake cycle simulations based on the rate-and-state friction formulation are evolutions of nonlinear dynamical systems (NDS). The term “cycle” implies an overall stable structure that is a phase-space attractor naturally traced out by trajectories of NDS as it evolves. Quantitatively characterizing these attractors should be a basis for measuring complexities of the simulated earthquake cycles, i.e. to determine if and how regular or chaotic they are. I first revisit the textbook-standard quasi-dynamic spring-slider system from an NDS perspective, explicitly showing the attractors, their relationship with the parameters of the NDS, and how they can be characterized taken advantage of their low-dimensionality while aiming to extend the analysis to high-dimensionality. I evaluate two approaches, computing the Lyapunov exponents (LEs) and measuring correlation dimensions, with the simple spring-slider and earthquake-cycle examples whose phase-space attractors can be visually verified. I conclude LEs are too inconvenient and computationally expensive to use whereas measuring correlation dimensions is an easy and effective approach even with highly non-uniform time sampling present in all simulations. For earthquake-cycle simulations, an attractor reconstruction is performed based on Taken’s theorem to corroborate my correlation-dimension results. The current method is limited in its ability to detect chaos in a dichotomous manner, which illuminates the direction for future study. An improving ability to quantitatively characterize earthquake-cycle simulations as an overall stable structure offers new opportunities to understand exotic seismic observations such as slow-slip events and enables more informative comparison with real data, particularly from paleoseismology, which could have far-reaching implications in earthquake forecasting.