Constructing Continuous-Time Chaos-Generating Templates Using Polynomial Approximation

Abstract

Aiming at developing a methodology for constructing continuous-time chaotic dynamical systems as flexible pattern generators, this paper discusses a strategy for binding desired unstable periodic orbits into a chaotic attractor. The strategy is comprised of the following two stages: constructing an interim “chaos-generating template”, and deforming the template according to the specifically desired orbits. INTRODUCTION Chaos is a fascinating phenomenon arising from nonlinearities in dynamical systems, and investigations on its applications to intelligent and flexible systems have appeared in many fields. An important aspect of chaos is that chaotic attractors embed or “host” an infinite number of unstable periodic orbits (UPO’s) bifurcated from prechaotic states (Ott, 2002). Among them, some distinctive orbits can be used for characterization or control purposes. For example, a variety of chaos control methods (Ott et al., 1990; Pyragas, 1992; Zhang et al., 2009) can stabilize UPO’s embedded in chaotic attractors, enlarging the operation range and/or enhancing the functionality of the system. In more recent years, the synthesis of chaos from various approaches (Chen and Ueta, 2002; Zelinka et al., 2008; Munoz-Pacheco and Tlelo-Cuautle, 2009) has been an active direction of research along the line of exploiting chaos. Primary concerns of these efforts include statistical and topological characteristics (e.g., invariant measure, Lyapunov spectrum, novel scrolling behaviors) that would be important in designing chaos-based information processing and communication applications. While the present study share some common motivation with the above-mentioned studies, we have put more focus on the geometrical shape and dynamical properties of UPO’s themselves from the viewpoint of the adaptive generation of periodic behaviors. Here our intention lies in extending the functionality of (stable) periodic pattern generators based on function approximation of vector fields, e.g., polynomial approximation (Okada and Nakamura, 2002) and neural network learning (Kuroe and Miura, 2006). In this paper, we consider continuous-time chaotic attractors as a container of UPO’s (patterns) where they can be stabilized, entrained, or targeted by external inputs into the dynamical system. In what follows, we propose a design strategy for binding desired UPO’s into a chaotic attractor governed by a polynomial vector field. The strategy is comprised of the following two stages: constructing an interim “chaos-generating template”, and deforming the template according to the specifically desired orbits. POLYNOMIAL VECTOR FIELDS We consider polynomial dynamical systems of the form ẋ = f(x)(x ∈ R ) where the vector field f(x) is represented as f(x) = [f1(x) · · · fN (x)] T = Φ(a(p1p2···pN ))θ(x), (1) θ(x) = [x1 · · · x ` N x `−1 1 x2 · · · 1] T . (2) Here, ` denotes the maximum degree of the polynomials, and the matrix Φ is comprised of the coefficients a(p1p2···pN ) of the polynomials. For example, with 3rdorder polynomials, the first element of a two-dimensional vector field is represented as f1(x) = a1(30)x 3 1 + a1(21)x 2 1x2 + a1(12)x1x 2 2 + a1(03)x 3 2 + a1(20)x 2 1 + a1(11)x1x2 + a1(02)x 2 2 + a1(10)x1 + a1(01)x2 + a1(00).(3) In our design process of the dynamical system, the coefficients are obtained by least-square fitting. To this end, we set up target vectors f(ηi) (i = 1, 2, · · · , L) at design points ηi on and in the vicinity of the target orbits, and construct matrices F = [f(η1) f(η2) · · · f(ηL)], (4) Θ = [θ(η1) θ(η2) · · · θ(ηL)]. (5) With these matrices, the least-square solution for Φ is given by Φ(a(p1 p2 ··· pN )) = FΘ # (6) where Θ is the Moore-Penrose pseudo inverse of Θ. Proceedings 24th European Conference on Modelling and Simulation ©ECMS Andrzej Bargiela, Sayed Azam Ali David Crowley, Eugene J.H. Kerckhoffs (Editors) ISBN: 978-0-9564944-0-5 / ISBN: 978-0-9564944-1-2 (CD)