International Journal of Bifurcation and Chaos, Vol. 23, No. 1 (2013) 1350003 (11 pages) c  World Scientific Publishing Company DOI: 10.1142/S021812741350003X A TWO-PARAMETER METHOD FOR CHAOS CONTROL AND TARGETING IN ONE-DIMENSIONAL MAPS DANIEL FRANCO Departamento de Matem´atica Aplicada, Universidad Nacional de Educaci´on a Distancia, Apartado de Correos 60149, 28080 Madrid, Spain dfranco@ind.uned.es EDUARDO LIZ ∗ Departamento de Matem´atica Aplicada II, Universidad de Vigo, E. I. Telecomunicaci´ on, Campus Marcosende, 36310 Vigo, Spain eliz@dma.uvigo.es Received September 23, 2011; Revised November 24, 2011 We investigate a method of chaos control in which intervention is proportional to the difference between the current state and a fixed value. We prove that this method allows to stabilize the most usual one-dimensional maps used in discrete-time models of population dynamics about a globally stable positive equilibrium. From the point of view of targeting, this technique is very flexible, and we show how to choose the control parameter values to lead the system towards the desired target. Another important feature of this control scheme in the ecological context is that it can be designed to prevent the risk of extinction in models with the so-called Allee effect. We provide a useful geometrical interpretation, and give some examples to illustrate our theoretical results. Keywords : One-dimensional maps; chaos control; global stability; population dynamics; Allee effect. 1. Introduction Control strategies aiming to stabilize chaotic sys- tems about fixed points or periodic orbits should exhibit a number of good features. To list some of them, the method should be easy to implement, the control action should not be too strong, and the controlled system should drive most solutions to the stabilized equilibrium or periodic orbit. Such properties have been discussed for several methods of control; for example, for the proportional feed- back method (PF) [G¨ u´ emez & Mat´ ıas, 1993; Liz, 2010a; Braverman & Liz, 2012; Carmona & Franco, 2011], and for the prediction-based control (PBC) [de Sousa Vieira & Lichtenberg, 1996; Ushio & Yamamoto, 1999; Polyak, 2005; Liz & Franco, 2010]. Depending on the related problem, some other aspects are of special interest; for example, if the method is applied in the context of popula- tion dynamics, one can seek to stabilize the system about a high population level (e.g. in exploited pop- ulations), or to a low population level (e.g. in the control of plagues). Another important aspect in the * Author for correspondence 1350003-1 Int. J. Bifurcation Chaos 2013.23. Downloaded from www.worldscientific.com by 211.144.81.67 on 06/19/14. For personal use only.