Mathematical Problems in Engineering

Volume 2015, Article ID 313154, 13 pages

http://dx.doi.org/10.1155/2015/313154

## Computing and Controlling Basins of Attraction in Multistability Scenarios

^{1}Facultad de Ingeniería, Programa de Ingeniería Electrónica, Magma Ingeniería, Universidad del Magdalena, Apartado Postal 2121630, Santa Marta, Colombia^{2}Departamento de Ingeniería Eléctrica, Electrónica y Computación, Percepción y Control Inteligente, Facultad de Ingeniería y Arquitectura, Universidad Nacional de Colombia (Sede Manizales), Campus La Nubia, Bloque Q, Manizales 170003, Colombia

Received 17 October 2015; Accepted 16 November 2015

Academic Editor: Rongwei Guo

Copyright © 2015 John Alexander Taborda and Fabiola Angulo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Complex bifurcation scenarios have been observed in nonlinear dynamic systems from virtually all areas of science, including a broad range of natural sciences, mechanical and electrical engineering, and economics and other areas of the social sciences [1–3]. Theoretical and applied researches have explained various bifurcation scenarios [4–6], and analytical, numerical, and experimental works have contributed to unraveling the complexity inherent to chaotic motion [3].

Coupled chaotic maps are a set of special discrete-time dynamical systems that can describe chemical, epidemiological, physiological, biological, or engineering systems [7]. Interesting nonlinear phenomena have been reported in them. For example, in [8, 9], coupled logistic maps were analyzed and new scenarios for transition to chaos were found via the creation and destruction of multilayered tori. In those papers, novel routes to chaos were described, and authors found that, depending on the coupling constant value, the system approaches different periodic attractors.

Control methods of coexisting attractors in multistability scenarios have been widely studied in the last decades. In [10], periodic signals were replaced by chaotic ones in order to eliminate multiple domains of attraction. In [11, 12], the influence of noise on preference and dominance of attractors in multistable systems was studied. In [13, 14], multistability was controlled using small periodic modulation of a system parameter. In [15], the basins of attraction (BA) were controlled using harmonic and stochastic modulation. In [16], an impulsive force was used in order to perturb one attractor of the system and to change its response such that the system response evolved to another attractor. In [17], control of multistability scenarios based on the selection of a particular attractor by periodic external modulation was presented. A complete report of control of multistability can be found in [18].

In this paper, we prove a different technique to control domains of attraction in multistability scenarios which is called Fixed-Point Inducting Controller (FPIC). The FPIC is a feed-forward controller that forces the system solution to evolve to an existing desired attractor which cannot be always reached for the uncontrolled case because of initial conditions. This technique was initially proposed in [19] and successfully applied in [20, 21]. In particular, in [21], experiments showed the good performance of the controlled system, where the FPIC was used as a second control loop. However, in all previous works published until now, FPIC has been only used to stabilize unstable periodic orbits.

The main contributions of this paper lie in numerical analysis and control design areas. The development of a novel methodology to compute and analyze the basins of attraction reinforced the numerical analysis. The methodology consists in decomposing the map of the system in submaps, where corresponds to the order of the periodic orbit to be controlled. To have the BA, the long-time response terms of this submap are depicted according to key colours. Apart from help to the control design, this way to compute the basin of attraction can be seen as the basins of attraction of period-1 orbits which are computed every sixth or fifth iteration for linear and nonlinear coupling, respectively, and the control goal can be thought of as the control of period-1 orbit. On the other hand, the methodology to design the controller based on FPIC technique belongs to control theory. When the proposed controller is used, the system is forced to evolve to a known periodic orbit that exists in the uncontrolled map. Moreover, by using the information obtained from bifurcation diagrams and basins of attraction, it is possible to compute the minimum control effort required to stabilize the target orbit in the defined region. In particular, the coexistence of periodic solutions in coupled logistic maps [8, 9, 22] is controlled by widening the basin of attraction of a period- orbit that coexists with another one, and the minimum control effort is computed aided by the unstable period-2 orbit for the linear coupling and by unstable period-1 orbits for the nonlinear coupling case.

The paper is organized as follows. In Section 2, the coupled logistic maps are presented. The coexistence of period-6 and period-5 orbits is identified. In Section 3, the methodology to compute the basins of attraction is explained, and the procedure is applied to the linear and nonlinear coupling maps. In Section 4, the methodology to compute the controller is presented and applied to the considered systems and the minimum control effort required to guarantee monostabilization of the periodic orbit is computed. In Section 5, a brief discussion of the results is presented, and finally, in Section 6, the conclusions are given.

#### 2. Coupled Logistic Maps

Coupled chaotic maps provide a source of bifurcation scenarios with nonlocal phenomena and coexistence of attractors. To design the proposed controller, two maps with different coupling mechanisms are chosen. For linear coupling map, there are two period-6 orbits that coexist in a range of the parameter set. Similarly, for nonlinear coupling case, two period-5 orbits coexist.

##### 2.1. Coupled Logistic Maps: Linear Coupling

This system is described bywhere is the coupling constant and , , is the parameter associated with the nonlinear part of (1). Since the system is symmetrical, there is an invariant line and the restriction of the 2D map to reduces it to a 1D map (the logistic map). The dynamics on this invariant set help us to study the dynamics and bifurcations of the 2D system. Moreover, symmetrical trajectories are generated by symmetrical initial conditions leading to symmetrical basins of attraction [22]. Novel routes to chaos through torus-breakdown mechanism of this system were reported in [9] and different dynamics were characterized in the parameter region given by .

Figure 1 shows the dynamic behavior of (1) when is varied and two symmetrical initial conditions are considered. Note that a pair of coexisting period-6 orbits are identified in these diagrams for , which makes the main difference between Figures 1(a) and 1(b). The dynamical behavior in the rest of the interval is very similar and other differences cannot be easily seen. Table 1 shows the specific values of the two coexisting period-6 orbits. It can be observed that the solutions are mutually symmetrical in the sense that .