Estimation of periodic-like motions of chaotic evolutions using detected unstable periodic patterns Daolin Xu a, * , Zhigang Li a , Steven R. Bishop b , Ugo Galvanetto c a Department of Engineering Mechanics, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore b Centre for Nonlinear Dynamics and its Applications, University College London, Gower Street, London WC1 6BT, UK c Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK Received 9 January 2001; received in revised form 15 May 2001 Abstract We introduce an approach to detect cyclical patterns embedded within chaotic data and make use of the detected patterns to estimate periodic-like motions in a chaotic process. A chaotic attractor contains many unstable periodic orbitsUPOs).TheUPOsarehiddencyclicalpatternsthatdominatethedynamicalevolutionofthesystem.Knowledge of UPOs can be used for estimating the trends of chaotic evolutions. A numerical experiment is conducted to illustrate an application on the business cycle detection. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Pattern; Chaos; Estimation; Unstable periodic orbits 1. Introduction Estimation of chaotic behaviors is one of the challenging topics in the prediction of dynamical systems. A chaotic process is deterministic rather than random, such that the variation of the cha- otic system may be predictable in the short term based on the deterministic property. Deterministic models could be constructed based on state space reconstruction from time series with delay coor- dinates Takens, 1981). In deterministic approach, local linear approximations have been widely adopted for the estimation of evolutions based on optimal metrics Garcia et al., 1996) and ordinary leastsquaresKugiumtzisetal.,1998)asexamples. The deterministic approach is a localized linear method. It omits useful information of the inher- ent cyclical patterns that are embedded within chaotic data. In this paper, we introduce a pattern-based ap- proach to estimate the evolution of chaotic mo- tions by making use of unstable periodic orbits UPOs) in chaos. The method is developed based on the following nature of chaos. An initially chaotic motion may suddenly be tamed by a UPO, andappeartobeaperiodic-likemodesimilartothe UPO pattern. Such a periodic-like motion may persistforawhileandwilleventuallyrevertbackto theirregularbehaviorduetotheunstablenatureof the UPO. Occurrence of many dierent types of periodic-like motions in a chaotic evolutionary Pattern Recognition Letters 23 2002) 245±252 www.elsevier.com/locate/patrec * Corresponding author. Tel.: +65-7906940; fax: +65- 7911859. E-mail address: mdlxu@ntu.edu.sg D. Xu). 0167-8655/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0167-865501)00100-3