On the relationship between instability and Lyapunov times for the 3-body problem

In this study we consider the relationship between the survival time and the Lyapunov time for 3-body systems. It is shown that the Sitnikov problem exhibits a two-part power law relationship as demonstrated previously for the general 3-body problem. Using an approximate Poincare map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the 3-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general 3-body problem.Comment: 9 pages, 19 figures, accepted for publication in MNRA

Shadowing unstable orbits of the Sitnikov elliptic 3-body problem

Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative of true dynamics. Using the Sitnikov Problem as an example, it is demonstrated that unstable orbits of the 3-body problem can be shadowed for long periods of time. In addition, it is shown that the stretching of phase space near escape and capture regions is a cause for the failure of the shadowing refinement procedure.Comment: 9 pages, 13 figures, accepted in MNRA

We Do What We Are: Representation of the Self-Concept and Identity-Based Choice

The current research proposes a novel approach to identity-based choice that focuses on consumers’ representations of the self-concept, as captured by the perceived cause-effect relationships among features of an individual consumer’s self-concept. More specifically, the studies reported here test the proposal that the causal centrality of an identity—the number of other features of a consumer’s self-concept that the consumer believes influenced or were influenced by the identity—underlies identity importance and is a determinant of identity-based consumer behaviors. Across seven studies, using both measured and manipulated causal centrality, the current research provides evidence for the role of causal centrality in identity-based choice. Among consumers who share an identity (belong to the same social category), those who believe that the identity is more causally central perceive the identity as more important and are more likely to engage in behaviors consistent with the norms of the social category