Chaotically driven sigmoidal maps

Abstract

Abstract. We consider skew product dynamical systems f : Θ × R → Θ × R, f (θ, y) = (T θ, f θ (y)) with a (generalized) baker transformation T at the base and uniformly bounded increasing C 3 fibre maps f θ with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for f we prove that the presence of these fibres restricts considerably the possible structures of invariant measuresboth topologically and measure theoretically, and that this finally allows to provide a "thermodynamic formula" for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.

Invariant graphs for chaotically driven maps

“…Although we think that also the finer quantitative results from [18] on the structure of P and φ * have close analogues in the present setting, we do not see how to transfer the corresponding proofs in a routine way. In general it is also not possible to apply the results from [18] directly to the partially conjugated system, because in that system fibres over base points θ ∈ P may be degenerated to single points.…”

“…The observation that the pinching set P is a union of global unstable fibres of the base map, when the base map S is an Anosov diffeomorphism of T 2 (theorem 2.18), allows to prove, along the lines of the proof of theorem 2.1 in [18]:…”

“…Our major goal here is to describe the structure of invariant graphs and study the properties of the pinching set, the set of points where the values of all of the invariant graphs coincide. In [18], the authors studied skew product systems driven by a generalized Baker map S : T 2 → T 2 with the restrictive assumption that f θ depend on θ = (ξ, x) only through the stable coordinate x of θ. Our aim is to relax this assumption and construct a fibre-wise conjugation between the original system and a new system for which the fibre maps depend only on the stable coordinate of the derive.…”

“…Invariant graphs have a wide variety of applications in many branches of nonlinear dynamics, some knowledge of such applications is also available, through works of several authors (e.g. [6,7,9,10,11,15,18,23,34] etc. ).…”

“…Namely, they assumed that the two outermost invariant graphs φ − and φ + are constant, say -1 and 1, and both of them are attracting simultaneously. More recently, G. Keller and A. Otani studied in [18] the hyperbolic case where S : T 2 → T 2 is a generalized Baker map with the restrictive assumption that f θ depend on θ = (ξ, x) only through the stable coordinate x, so that f θ can be written as f x , and x → f x is continuous on T 1 , where T 1 is the one-dimensional torus. Also f x > 0 and Sf x < 0.…”