Attractors and repellers. Reconstruction of attractors from an experimental time series. Quantitative characteristics of attractors

Abstract

A point set in phase space attracting all neighbouring phase trajectories from some region named the attraction region is said to be an attractor. According to this definition a stable singular point (Fig 3.1a), a stable limit cycle (Fig 3.16) and a stable torus are attractors. However, a system with n degrees of freedom system for n > 1 may have not only such simple attractors but complexly formed attractors as well. The latter attractors are often spoken of as strange attractors. Strange at tractors can be separated into two categories: stochastic and chaotic, depending on whether they are associated with stochastic or chaotic behaviour of the system. Attractors involving only a finite or an infinite number of saddle cycles and their unstable integral manifolds are referred to as stochastic. Attractors involving both saddle and stable cycles with small attraction regions are referred to as chaotic. All phase trajectories forming a stochastic attractor have to be exponentially unstable. A chaotic attractor has to hold at least one stable trajectory. In particular, chaotic attractors can consist either of one stable multi-revolution limit cycle with sufficiently nearby coils (Fig 3.2) or of a denumerable set of stable limit cycles with sufficiently small attraction regions (the number of the cycles can be infinite). Open image in new window Figure 3.1 The simple attractors: stable singular point (a) and stable limit cycle (b).