Complex Behaviour of Systems Due to Semi-Stable Attractors: Attractors That Have Been Destabilized but which Still Temporarily Dominate the Dynamics of a System

Abstract

Semi-stable attractors are provisionally defined as invariant sets around which volumes are contracting. Three examples are referred to: one is extensively documented by Grebogi, et al. (1986), a second is generated by an age-structured model, and the last is seen in the logistic model. The first example is described by Grebogi as a chaotic transient. However, this transient lasts in excess of 80,000 iterations, which, for real ecological systems in which each iteration represents a year, is longer than many systems can survive. The second looks very much like a saddle point, and indeed, saddle points may be semi-stable attractors, but only if the absolute value of the determinant of the Jacobian at the saddle point is less than 1. In the last example, the trajectories don’t stay near the semi-stable attractor very long, but they generate interesting dynamics when ‘r’ is permitted to vary, and illustrate the relation between semi-stable attractors and the stable attractors they’re derived from.