TITLE:
Three-dimensional H\'enon--like maps and wild Lorenz--like attractors

AUTHORS:
S.V. Gonchenko ^(1), I.I. Ovsyannikov^(2), C. Sim\'o^(3) and D. Turaev^(4)

(1) Inst. of Appl. Math. and Cyb., Nizhny Novgorod State Univ.,
    10 Uljanova st., Nizhny Novgorod, 603005, Russia

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

(3) Departament Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona,
    Gran Via, 585, 08007 Barcelona, Spain

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

E-mail: gosv100@uic.nnov.ru, ovii@mail.ru, carles@maia.ub.es,
        turaev@math.bgu.ac.il

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
H\'enon 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 attractors domains in the parameter space
of the 3D H\'enon map. In all cases the maximal Lyapunov exponent,
Lambda_1, is positive. Concerning next Lyapunov exponent, Lambda_2,
there are open domains where it is definitely positive, other where
it is definitely negative and, finally, domains where it cannot be
distinguished numerically from zero (i.e.,|Lambda_2|<rho, where rho is
some tolerance ranging between 10^{-5} and 10^{-6}). Furthermore, several
other kinds of interesting attractors have been found in this family of
3D H\'enon maps.