Pergamon Chaos, Solitions & Fractals, Vol.9, No. 1/2,pp. 83-93, 1998 ~) 1998Elsevier Science Ltd. All rightsreserved Printedin Great Britain 0960-0779/98 $19.00+0.00 PII: S0960-0779(97)00051-9 Modelling Chaos and Hyperchaos with 3-D Maps KRZYSZTOF STEFANSKIt Institute of Astronomy, Nicholas Copernicus University, 87-100 Torufi, Poland Abstract Two generalizations on R 3 of the H6non nonconservative map are defined. Although, in many respects, they are closely related to each other, one of them can produce strange attractors with only one positive Lyapunov exponent, while the other has strange attractors with two positive Lyapunov exponents, thus showing a "hyperchaotic" behaviour. The results of numerical tests to be discussed, show, that apart from obvious differences in the number of positive Lyapunov exponents and the shape of the chaotic atractor, the hyperchaos may not be much different from regular chaos. ~) 1998 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION The term "hyperchaos" was first introduced by R6131er [1], who named chaotic phenomena with more than one positive Lyapunov exponent that way. The term itself, though not very common, is nevertheless occasionally used (see e.g.[2]). It may be interesting to find out, testing simple models, whether the hyperchaos is essentially different from the regular chaos in any respect apart from the obvious one--the number of positive Lyapunov exponents. The problem may be of some interest due to the fact that there are many systems whose chaotic behaviour is characterized by multidimensional chaotic attractors (cf. e.g.[3-5]). Here we present results of numerical tests for two maps designed in such a way that one can produce a chaotic attractor with topological dimension 1 and the other a chaotic attractor with topological dimension 2, which corresponds to 1 or 2 positive Lyapunov exponents respectively. We focus our attention on scenarios that lead to chaos in both maps, and on the autocorrelation functions that are a primary tool for classification of behaviours in systems whose explicit evolution laws are unknown, and a time series of one variable is the only measurable quantity [6]. We compare these functions for both maps in the case of noisily periodic and fully chaotic evolution. The paper consists of 3 sections apart from the Introduction. In Section 2 the maps are defined, and their most elementary features are discussed. Section 3 gives an overview of numerical tests, and in Section 4 the results are briefly summarized and concluded. 2. HI'NON-LIKE MAPS IN R 3 2.1 Construction of the maps Let us construct two dissipative, invertible maps Fi:R3---~ R 3 for i = 1,2, (1) which would be two generalizations of the H6non map [7]--one with expansion in 1 and tPresent address: Institute of Mathematics, Pedagogical University, PI. Hoyssenhoffa, 85-072 Bydgoszoz, Poland. 83