Riddled basins in complex physical and biological systems

Abstract

Complex systems have typically more than one attractor, either periodic or chaotic, and their basin structure ultimately determines the final-state predictability. When certain symmetries exist in the phase space, their basins of attraction may be riddled, which means that they are so densely intertwined that it may be virtually impossible to determine the final state, given a finite uncertainty in the determination of the initial conditions. Riddling occurs in a variety of complex systems of physical and biological interest. We review the mathematical conditions for riddling to occur, and present two illustrative examples of this phenomenon: coupled Lorenz-like piecewise-linear maps and a deterministic model for competitive indeterminacy in populations of flour beetles.

Paleoanthropology of Consciousness, Culture and Oral Language

“…9 A type of basin topology that may occur in systems that, through a symmetry or some other constraint, have a smooth invariant manifold, i.e. there exists a smooth surface or hypersurface in the phase space, such that any initial condition in the surface generates an orbit that remains in the surface [18] [19]. 10 In physics resonance or coupling-phase is a condition under which an oscillating system responds to an alternative driving force with the maximum amplitude.…”

“…Utilitarian finds from the second half of the Lower Paleolithic are artifacts that [52]; stone tools (choppers, bifaces, scrapers, blades, handaxes and spikes) [53] belonging to the so called core-tool tradition, developed and diversified in terms of their intended use [54], but also because of the location of their makers perspective, a fact of great importance in the process of psycho-relational individuation that has transformed the inner life and the relational approach of human communities, because it indicates that the human position, in the binocular perception of reality, is taken as the psycho-spatial coordinate that gives a sense of depth to the reality itself: the human actor's position, as psycho-spatial coordinate for the depth, is the dependent variable which orients the two-dimensional extension of the line and the one-dimensional point (dot) generating the tridimensionality 18 . Non-utilitarian finds from this period are stone carvings, such as cupules [56] [57] (a shallow, non-functional cup-like depression, cut into the surface of a rock as an engraved dot) and petroglyphs found in Bhimbetka and Daraki-Chattan Caves, India (dated between 700 to 290 -200 tya), lines and dots as engravings which add the size of the depth to the two-dimensional perspective, creating the suggestion of three-dimensionality; artifacts with zoomorphic and anthropomorphic forms realized according to a two-dimensional spatial perspective; specimens of anthropomorphic statuettes or figurines classified as Venus ( 18 With respect to the 3-dimensional sense Wynn observes [55]: Perhaps the most critical new spatial concept is the understanding and coordination of multiple points of view. The intentionally straight edges and parallels on some of the Isimila bifaces require attention to a stable point of view, which is a projective notion.…”

On predicting climate under climate change

“…For some intransitive nonlinear systems, a very small change in the initial state of the system does not just limit predictability due to chaos but can also change the attractor (and therefore climate) to which the system evolves (McDonald et al 1985). If dynamical system features of this nature, such as riddled basins of attraction (Viana et al 2009), were found in climate models then the model distributions from an IC ensemble could be dependent on the finest details of the chosen ICs (Lorenz 1968(Lorenz , 1976. The above interpretation of climate would then be ill-defined.…”

The occurrence of riddled basins and blowout bifurcations in a parametric nonlinear system

“…The first one is the parallel Lyapunov exponent which describes the evolution on the invariant subspaces and must be positive for emergence of riddled basins. The second is the normal Lyapunov exponent that characterizes evolution transverse to the subspaces [Ashwin et al(1996), Cazelles(2001), Viana et al(2009)].…”