8 research outputs found
A birational mapping with a strange attractor: Post critical set and covariant curves
We consider some two-dimensional birational transformations. One of them is a
birational deformation of the H\'enon map. For some of these birational
mappings, the post critical set (i.e. the iterates of the critical set) is
infinite and we show that this gives straightforwardly the algebraic covariant
curves of the transformation when they exist. These covariant curves are used
to build the preserved meromorphic two-form. One may have also an infinite post
critical set yielding a covariant curve which is not algebraic (transcendent).
For two of the birational mappings considered, the post critical set is not
infinite and we claim that there is no algebraic covariant curve and no
preserved meromorphic two-form. For these two mappings with non infinite post
critical sets, attracting sets occur and we show that they pass the usual tests
(Lyapunov exponents and the fractal dimension) for being strange attractors.
The strange attractor of one of these two mappings is unbounded.Comment: 26 pages, 11 figure
Nonlinear optical response of a two-dimensional quantum-dot supercrystal: Emerging multistability, periodic and aperiodic self-oscillations, chaos, and transient chaos
Examples of Lorenz-like Attractors in Hénon-like Maps
We display a gallery of Lorenz-like attractors that emerge in a class of
three-dimensional maps. We review the theory of Lorenz-like attractors for diffeomorphisms
(as opposed to flows), define various types of such attractors, and find sufficient
conditions for three-dimensional Henon-like maps to possess pseudohyperbolic Lorenz-like
attractors. The numerically obtained scenarios of the creation and destruction of these
attractors are also presented