# Entrainment and migration controls of two-dimensional maps

- Authors
- Journal
- Physica D Nonlinear Phenomena 0167-2789
- Publisher
- Elsevier
- Publication Date
- Volume
- 54
- Issue
- 3
- Identifiers
- DOI: 10.1016/0167-2789(92)90038-o
- Disciplines

## Abstract

Abstract A general approach to several types of controls of complex dynamic systems, formulated by Jackson, is applied to two classic two-dimensional maps x k+1=E(x k)(xϵ R 2) ; the Hénon and the Ikeda maps. The convergent regions of their phase spaces, C k , are shown to respectively possess translational and rotatational symmetries. Several topologically distinct entrainment goals (e-goals), { g k }⊂C k , are illustrated for the Hénon map. For both maps, the basins of entrainment, BE( g 0)={ x 0| lim k→∞ | x k − g 0|=0}, of the controlled dynamics for fixed point goals in each convergent region, g 0 ϵC k , are illustrated. The BE( g 0) are invariant, relative to g 0, when g 0 is varied by the above symmetries. It is proved that, for the Ikeda map, BE(g 0=0)= R 2 . For experimental purposes, spheres of entrainment are introduced SE( r 0)={ x 0|| x 0 − g 0| < r( g 0)}, where r( g 0) = max r { r|| x 0 − g 0| < r → lim k→∞ | x k − g 0| = 0}. The Ikeda map has two attractors in a bounded region so that migration controls can be illustrated. This is done, using several migration goals (m-goals) which transfer this system from one attractor to the basin of attraction of the other attractor (A 1 → BA 2, or A 2 → BA 1). These require controls only for finite times. The influence of noise on the entrainment process is analyzed and compared with numerical examples.

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