Data-driven discovery of quasiperiodically driven dynamics

Quasiperiodically driven dynamical systems are nonlinear systems which are driven by some periodic source with multiple base-frequencies. Such systems abound in nature, and are present in data collected from sources such as astronomy and traffic data. We present a completely data-driven procedure which reconstructs the dynamics into two components - the driving quasiperiodic source with generating frequencies; and the driven nonlinear dynamics. Unlike conventional methods based on DMD or neural networks, we use kernel based methods. We utilize our structural assumption on the dynamics to conceptually separate the dynamics into a quasiperiodic and nonlinear part. We then use a kernel based Harmonic analysis and kernel based interpolation technique in a combined manner to discover these two parts. Our technique is shown to provide accurate reconstructions and frequency identification for datasets collected from three real world systems

Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise-linear they are relatively amenable to an exact analysis and we are able to explicitly identify parameter values for heterodimensional cycles and blenders. For a one-parameter subfamily we identify bifurcations involved in a transition through unstable dimension variability. This is facilitated by being able to compute periodic solutions quickly and accurately, and the piecewise-linear form should provide a useful test-bed for further study

Attractor-repeller collision and the heterodimensional dynamics

We study the heterodimensional dynamics in a simple map on a three-dimensional torus. This map consists of a two-dimensional driving Anosov map and a one-dimensional driven M\"obius map, and demonstrates the collision of a chaotic attractor with a chaotic repeller if parameters are varied. We explore this collision by following tangent bifurcations of the periodic orbits, and establish a regime where periodic orbits with different numbers of unstable directions coexist in a chaotic set. For this situation, we construct a heterodimensional cycle connecting these periodic orbits. Furthermore, we discuss properties of the rotation number and of the nontrivial Lyapunov exponent at the collision and in the heterodimensional regime