This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and helpful in investigating the communication of chaos. Work partially supported by the NSC under Grant No. 89-2115-M-008-029 and the National Center for Theoretical Sciences Mathematics Divisio...

We consider the dynamical system A, T , whereA is a class of differentiable functions defined on some interval and T :A → A is the operator Tφ : f ◦φ, where f is a differentiablem-modal map. Using an algorithm, we obtained some numerical and symbolic results related to the frequencies of occurrence of critical values of the iterated functions when the kneading sequences of f are aperiodic. More...

Abstract. Dynamical zeta functions provide a powerful method to analyze low dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand even simple one dimensional maps can show an intricate structure of the grammar rules that may lead to a non smooth dependence of global observable on parameters changes. A paradigmatic example is the fractal diffusio...

We investigate the topological zeta function for unimodal maps in general and dynamical zeta functions for the tent map in particular. For the generic situation, when the kneading sequence is aperiodic, it is shown that the zeta functions have a natural boundary along its radius of convergence, beyond which the function lacks analytic continuation. We make a detailed study of the function ∏∞ n=...

In this note we consider a collection C of one parameter families of unimodal maps of [0, 1]. Each family in the collection has the form {μf} where μ ∈ [0, 1]. Denoting the kneading sequence of μf by K(μf), we will prove that for each member of C, the map μ 7→ K(μf) is monotone. For interest, μf(x) = 4μx(1− x) and μf(x) = μ sin(πx) are shown to belong to C. This extends the work of Masato Tsuji...

A numerical algorithm to compute the topological entropy of multimodal maps is proposed. This algorithm results from a closed formula containing the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, it simplifies a previous algorithm, also based on min-max symbols, which was originally proposed for twice differentiable multimodal maps. The new algorithm ...

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transforma...

The supramolecular structure of the inclusion complex of β-cyclodextrin (β-CD) with 1,1',2,2'-tetramethyl-3,3'-(p-phenylenedimethylene) diimidazolium dibromide (TetraPhimBr), a dicationic ionic liquid, has been investigated. The inclusion complex with 1:1 molar ratio was prepared by a kneading method. Fourier transform infrared spectroscopy (FTIR), powder X-ray diffraction (XRD) analysis, (1)H ...

While motor skill acquisition process is regarded as development of coordination, typically regarded as synchronisation among joint movements, we found another phenomenon which we call differentiation as a consequence of synchronisation. The synchronised movement established is decomposed into several sections or modulated to be executed on different timings without breaking the coordination am...