Wada property in systems with delay

Abstract

Delay differential equations take into account the transmission time of the information. These delayed signals may turn a predictable system into chaotic, with the usual fractalization of the phase space. In this work, we study the connection between delay and unpredictability, in particular we focus on the Wada property in systems with delay. This topological property gives rise to dramatical changes in the final state for small changes in the history functions.

Ascertaining when a basin is Wada: the merging method

“…Therefore, in this context, Wada boundaries are usually referred to as those that separate three or more basins at a time, but the basins need not to be connected. Since the earliest references to the Wada property in dynamical systems, many authors claim that the boundaries have the Wada property for disconnected basins [7][8][9][10][11][12]. In this work, we adopt this latter definition: Wada boundaries are those that separate three or more basins, no matter whether the basins are connected or not.…”

“…Despite our primary intuition, Wada basins are a common feature appearing in many dynamical systems. Since its first report, Wada basins have been found in open Hamiltonian systems [10], ecological models [11], delayed differential equations [12], hydrodynamical systems [13], and many engineering problems [14][15][16]. This is possible because Wada bound-aries are related to iterative processes and fractal structures, which are a common feature in the basins of nonlinear dynamical systems [17].…”

Wada structures in a binary black hole system

“…Later, a numerical method based on successive refinements of a grid was introduced. This approach allows one to test the Wada property in a variety of situations up to a given resolution [41,42]. Recently, a third numerical method has been proposed [24].…”

Computing Complex Horseshoes by Means of Piecewise Maps

“…After examining two well-known examples, and as a completely new feature, we show that a three-fold horseshoe can give rise to basin boundaries exhibiting the so-called Wada property [Yoneyama , 1917]. Given the fact that homoclinic tangles give rise to infinitely many successive foldings, this fact suggests why the Wada property appears so frequently in nonlinear multistable dynamical systems and dispersive systems with several escapes [Aguirre et al, 2009;Vandermeer , 2004;Portela et al, 2007;Toroczkai et al, 1997;Daza et al, 2017;Seoane & Sanjuán , 2012]. Finally, we show how one-dimensional unimodal or multimodal maps can be in general computed from horseshoes with one or more foldings, which can simplify enormously the manipulation of a dynamical system.…”