Computing fractal dimension in supertransient systems directly, fast and reliable

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices and they are a common phenomena in dynamical systems. Superlong chaotic transients are caused by the presence of chaotic saddles whose stable sets have fractal dimensions that are close to phase-space dimension. For many physical systems chaotic saddles have a big impact on laboratory measurements, and it is important to compute the dimension of such stable sets including fractal basin boundaries through a direct method. In this work, we present a new method to compute the dimension of stable sets of chaotic saddles directly, fast, and reliable. Chaotic transients are common phenomena in dynamical systems. They were observed in the Lorenz system [1, 2], in fluid experiments [3, 4, 5], in many low-dimensional dynamical systems [6, 7, 8, 9], in spatiotemporal chaotic dynamical systems [10, 11], simulations of the plane Couette flow (a shear flow between two parallel walls) [12, 13], and in coupled map lattices [14]. Generally, these transients are caused by the presence of chaotic saddles [15, 23, 17, 18] (i.e., strange repellers). When there is a chaotic saddle in the phase space, trajectories originating from random initial conditions in a neighborhood of the stable set of the chaotic saddle (such as in the neighborhood of a fractal basin boundary (BB)), usually wander in the vicinity of the chaotic saddle for a finite amount of time before escaping the neighborhood of the chaotic saddle and settling into a final attractor. Chaotic saddles can strongly effect observed experiments and a “turbulent” state in experiments could be supported by a chaotic saddle rather than an attractor. If there are two or more attractors in the system, then chaotic saddles may be buried in the boundaries of basins of attraction leading to fractal BBs. Supertransients are superlong (chaotic) transients and occur commonly in spatiotemporal chaotic dynamical systems [10, 14, 11]. In such a case, trajectories starting from random initial conditions in a neighborhood of the stable set of a chaotic saddle wander chaotically for a long time before settling into a final attractor. It was observed in numerical experiments that spatially extended systems exhibit supertransients so that the observation of the system’s asymptotic attractor is practically impossible [10, 14]. Reference [11] investigates the geometric properties of the chaotic saddles which are responsible for the supertransients in spatiotemporal chaotic systems, and reports that the stable set of the chaotic saddle possesses a fractal dimension which is close to the phase-space dimension. There are a few tools for computing the dimension of (one-dimensionally unstable) stable sets of chaotic saddles [9, 19, 15, 23, 17, 20, 11]. These methods involve the computation of Lyapunov exponents. A frequently used method is the combination of the Sprinkle method [9] and the method for computing the maximal Lyapunov exponent of a typical Saddle Straddle Trajectory [15, 23, 17]. However, computing the Lyapunov exponents of a Saddle Straddle Trajectory in a supertransient system is extremely time consuming. The purpose of this Letter is to present a simple, general method (involving no Lyapunov exponents) for computing the dimension of the stable set of a chaotic saddle that is responsible for the supertransients and the computation is direct, fast and reliable. We refer to this algorithm as the “Straddle Grid Dimension Algorithm” (SGDA); it can be applied to flows (differential equations) or maps. Our main result is that the SGDA can be applied to supertransient systems having one-dimensionally unstable stable sets such as basin boundaries in d-dimensional space (d ≥ 1). To illustrate the effectiveness of our method for computing dimension see Fig. 2. As indicated by our numerics, the SGDA is superior to applying the box-counting algorithm (BCA) by utilizing straddle pairs for the end points of the boxes.