Discontinuous generalized synchronization of chaos

Abstract

International audienceWe study synchronization functions in basic examples of discontinuous forced systems with contractive response and chaotic driving. The forcing is given by baker-type maps and the response is assumed to depend monotonically on the drive. The resulting synchronization functions have dense sets of discontinuities and their graphs appear to be extremely choppy. We show that these functions have bounded variation when the contraction is strong, and conversely, that their total variation is infinite when the contraction becomes weak. In the first case, we also analyze in detail smoothness properties of the corresponding continuous component

The variation of invariant graphs in forced systems

“…where T : [0, 1] → [0, 1] (and, again, the mapping f 2 will play no role here). For a systematic way of defining f such that f −1 is as here, see [10]. Assuming in addition that the filter satisfies…”

“…The sync function has been shown to be of bounded variation under the condition e htop(f −1 ) γ < 1 where h top (f −1 ) is the topological entropy of f −1 . On the other hand, the total variation of φ becomes infinite when γ is sufficiently large [10]. A sync function of bounded variation makes it easier to show that the factor statistics inherits absolute continuity of the forcing statistics [11].…”

Statistical properties of invariant graphs in piecewise affine discontinuous forced systems

“…Moreover, when h is of bounded variation, a decomposition into the difference h c − h d of increasing functions, where in addition h c is continuous and h d is a step function, is provided. (This is the only result of [5] that is being used here, see lemma 3. )…”

“…In [5], the first author analyses properties of the function h. In particular, it is shown that the total variation is finite iff λ < 1/2. Moreover, when h is of bounded variation, a decomposition into the difference h c − h d of increasing functions, where in addition h c is continuous and h d is a step function, is provided.…”

“…Proof. We already know from [5] that h d is a strictly increasing step function with a dense set of discontinuities and h c is strictly increasing and continuous. There remains only to prove that h c is absolutely continuous.…”