A new type of relaxation oscillation in a model with rate-and-state friction

Abstract

In this paper, we prove the existence of a new type of relaxation oscillation occurring in a one-block Burridge–Knopoff model with Ruina rate-and-state friction law. In the relevant parameter regime, the system is a slow-fast ordinary differential equation with two slow variables and one fast. The oscillation is special for several reasons: firstly, its singular limit is unbounded, the amplitude of the cycle growing like as . As this estimate reflects, the unboundedness of the cycle—for this non-polynomial system—cannot be captured by a simple -dependent scaling of the variables, in contrast to e.g. Gucwa and Szmolyan (2009 Discrete Continuous Dyn. Syst. S 2 783–806). We therefore obtain its limit on the Poincaré sphere. Here we find that the singular limit consists of a slow part on an attracting critical manifold, and a fast part on the equator (i.e. at ) of the Poincaré sphere, which includes motion along a center manifold. The reduced flow on this center manifold runs out along the manifold’s boundary, in a special way, leading to a complex return to the slow manifold. We prove the existence of the limit cycle by showing that a return map is a contraction. The main technical difficulty lies in the fact that the critical manifold loses hyperbolicity at an exponential rate at infinity. We therefore use the method in Kristiansen (2017 Nonlinearity 30 2138–84), applying the standard blowup technique in an extended phase space. In this way, we identify a singular cycle, consisting of 12 pieces, all with desirable hyperbolicity properties, that enables the perturbation into an actual limit cycle for . The result proves a conjecture in Bossolini et al (2017 Nonlinearity 30 2805–34). The Bossolini et al (2017 Nonlinearity 30 2805–34) also includes a preliminary analysis based on the approach in Kristiansen (2017 Nonlinearity 30 2138–84) but several details were missing. We provide all the details in the present manuscript and lay out the geometry of the problem, detailing all of the many blowup steps.