Computing and Controlling Basins of Attraction in Multistability Scenarios

Abstract

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

Coexisting Attractor in a Gyrostat Chaotic System via Basin of Attraction and Synchronization of Two Nonidentical Mechanical Systems

“…The green colored trajectory shows the slave system and the blue color illustrates the error between master and slave systems. As the estimated parameters start approaching their original values, using the parameter updated law (17), the error terms e i = y i − x i , i = 1, 2, 3, tend to zero which indicate the slave system in green color will converge towards the master system as time advances.…”

“…In 2016, a period-two solution was taken into account for basins of attraction, where the boundaries of two basins were asymptotically stable manifolds [15] and similar work can be seen in [16]. There are also several methods provided to compute basins in discrete and fractional dynamical systems [17][18][19]. Since the last decade, researchers have focused on the computation of basins for chaotic systems.…”

Análisis de la Multiplicidad de los Estados Estacionarios de Dos Sistemas Químicos Industriales usando la Metodología de Mapeo de Celdas

“…Sobre esta metodología y sus principales características, Merillas Santos (2006) propuso, a grandes rasgos, un algoritmo para su implementación. Por otra parte, Taborda y Angulo (2015) propusieron un método de control inductivo de punto fijo para determinar y controlar dichas regiones de atracción. Más recientemente, Erazo et.…”

Controlling multistability in a vibro-impact capsule system

“…Sevilla-Escoboza et al [20] proposed a robust control method that allows a periodic or a chaotic multistable system to be transformed to a monostable system at an orbit with dominant frequency of any of the coexisting attractors. Another control method was developed in [21], which was based on the computation of basins of attraction and allows the switching from multistable to monostable dynamical scenarios. An intermittent control strategy was developed in [2] for switching between coexisting attractors, which can be applied to non-autonomous dynamical systems.…”