Deterministic and random synthesis of discrete chaos Miguel Romera a, * , Michael Small b , Marius-F. Danca c a Instituto de Fı ´sica Aplicada, Consejo Superior de Investigaciones Cientı ´ficas, Serrano 144, 28006 Madrid, Spain b Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong c Department of Mathematics, ‘‘Tehnofrig’’ Technical College, 3400 Cluj-Napoca, Romania Abstract In this paper, two anticontrol algorithms for synthesis of discrete chaos are introduced. In these algorithms, the control parameter of a discrete dynamical system is switched, either randomly or in a deterministic way, between two or more val- ues corresponding to periodic motions, the result being chaotic behavior. These algorithms require no knowledge of spe- cific mathematical properties of the underlying map modeling the system. The existence of chaos is demonstrated using various tools including graphical iteration, histogram, Lyapunov exponent and surrogate tests. In this paper, these very simple and implementable chaotifiers are applied to the logistic map. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Logistic map; Discrete chaos; Anticontrol; Superstable orbits; Surrogate tests 1. Introduction The chaotification of a dynamical system (variously also described as chaos synthesis, chaos generation, or anticontrol of chaos) represents a significant non-trivial problem and has seen a very rapid development (see for example [1,2] for discrete dynamical systems, [3,4] for continuous dynamical systems, and [5] for discon- tinuous dynamical systems). Chaos synthesis seems to be a continuously expanding area, due to the great potential of chaos in non-tra- ditional applications. The presence of chaos has been observed to be beneficial and even desirable in many domains, including mechanical, electronic, optical, and biological systems. In particular, the clinical study of human heart-beat analysis and regulation is one specific example of great relevance (e.g. [6] and references therein, [7]). The most widely known method to make an arbitrarily given system chaotic or to enhance the existing chaos of a chaotic deterministic system is the use of small controls via time-delay feedback. Briefly, anticontrol in the case of a general one-dimensional discrete dynamical system x nþ1 ¼ f ðx n Þ, x 0 2 M  R; f : M ! R, 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.142 * Corresponding author. E-mail address: miguel@iec.csic.es (M. Romera). Applied Mathematics and Computation 192 (2007) 283–297 www.elsevier.com/locate/amc