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THREE-DIMENSIONAL HÉNON-LIKE MAPS AND WILD LORENZ-LIKE ATTRACTORS

S. V. GONCHENKO

Inst. Appl. Math. and Cyb., Nizhny Novgorod State Univ., 10 Ulyanova st., Nizhny Novgorod, 603005, Russia

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I. I. OVSYANNIKOV

Radio and Physical Dept., Nizhny Novgorod State Univ., 23 Gagarina av., Nizhny Novgorod, 603000, Russia

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C. SIMÓ

Departament Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain

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, and

D. TURAEV

Department of Math., Ben Gurion University, Beer Sheva 84105, Israel

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https://doi.org/10.1142/S0218127405014180Cited by:99 (Source: Crossref)

Abstract

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ₁, is positive. Concerning the next Lyapunov exponent, Λ₂, there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ₂| < ρ, where ρ is some tolerance ranging between 10^-5 and 10^-6). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.

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