On the embedding of a (p-1)-dimensional non invertible map into a p-dimensional invertible map

This paper concerns the description of some properties of p-dimensional invertible real maps Tb, turning into a (p - 1)-dimensional non invertible ones T0, p = 2, 3, when a parameter b of the first map is equal to a critical value, say b=0. Then it is said that the noninvertible map is embedded into the invertible one. More particularly properties of the stable, and the unstable manifolds of a saddle fixed point are considered in relation with this embedding. This is made by introducing the notion of folding as resulting from the crossing through a commutation curve when p = 2, or a commutation surface when p = 3

Geodesic length spectrum on compact Riemann surfaces

In this paper we use techniques linking combinatorial structures (symbolic dynamics) and algebraic-geometric structures to study the variation of the geodesic length spectrum, with the Fenchel-Nielsen coordinates, which parametrize the surface of genus τ = 2. We explicitly compute length spectra, for all closed orientable hyperbolic genus two surfaces, identifying the exponential growth rate and the first terms of growth series

The first eigenvalue of the Laplacian and the Conductance of a Compact surface

We present some results with the central theme of is the phenomenon of the first eigenvalue of the Laplacian and conductance of the dynamical system. Our main tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under deformation of the surface. With this model, we show variation of the first eigenvalue of the laplacian and the conductance of the dynamical system, with the Fenchel–Nielsen coordinates, that characterize the surface

Rigidity and flexibility of surface groups

The aim of this work is the °exibility of the hyperbolic surfaces. The results are about °exibility and geometrical boundedness. Bers are stated the universal property for all hyperbolic surface of ¯nite area where introduced the constant of boundedness. We determine this constant, using symbolic dynamics

Measuring complexity in a business cycle

The purpose of this paper is to study the dynamical behavior of a family of two- dimensional nonlinear maps associated to an economic model. Our objective is to measure the complexity of the system using techniques of symbolic dynamics in order to compute the topological entropy. The analysis of the variation of this im- portant topological invariant with the parameters of the system, allows us to distin- guish different chaotic scenarios. Finally, we use a another topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy. This work provides an illustration of how our under- standing of higher dimensional economic models can be enhanced by the theory of dynamical systems

Symbolic Dynamics and Chaotic Synchronization in Coupled Duffing Oscillators

In this work we discuss the complete synchronization of two identical double-well Duffing oscillators unidirectionally coupled, from the point of view of symbolic dynamics. Working with Poincar´e cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized. We obtained analytically the threshold value of the coupling parameter for the synchronization of two unimodal and two bimodal piecewise linear maps, which by semi-conjugacy, under certain conditions, gives us information about the synchronization of the Duffing oscillators

The first eigenvalue of the Laplacian and the ground flow of a compact surface

We present some results whose central theme is that the phenomenon of the first eigenvalue of the Laplacian and the ground flow of the compact surface (bitorus). Our main tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under deformation of the surface. With this model, we show that there are variation of the first eigenvalue of the Laplacian and the ground flow with the Fenchel-Nielsen coordinates, that characterize the surface

Chaotic synchronization of Piecewise Linear Maps

We derive a threshold value for the coupling strength in terms of the topological entropy, to achieve synchronization of two coupled piecewise linear maps, for the unidirectional and for the bidirectional coupling. We prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the bidirectional coupling of two identical chaotic Duffing equations is given

Kneading theory analysis of the Duffing equation

The purpose of this paper is to study the symmetry effect on the kneading theory for symmetric unimodal maps and for symmetric bimodal maps. We obtain some properties about the kneading determinant for these maps, that implies some simplifications in the usual formula to compute, explicitly, the topological entropy. As an application, we study the chaotic behaviour of the two-well Duffing equation with forcing

Symbolic Dynamics and chaotic synchronization

Chaotic communications schemes based on synchronization aim to provide security over the conventional communication schemes. Symbolic dynamics based on synchronization methods has provided high quality synchronization [5]. Symbolic dynamics is a rigorous way to investigate chaotic behavior with finite precision and can be used combined with information theory [13]. In previous works we have studied the kneading theory analysis of the Duffing equation [3] and the symbolic dynamics and chaotic synchronization in coupled Duffing oscillators [2] and [4]. In this work we consider the complete synchronization of two identical coupled unimodal and bimodal maps. We relate the synchronization with the symbolic dynamics, namely, defining a distance between the kneading sequences generated by the map iterates in its critical points and defining n-symbolic synchronization. We establish the synchronization in terms of the topological entropy of two unidirectional or bidirectional coupled piecewise linear unimodal and bimodal maps. We also give numerical simulations with coupled Duffing oscillators that exhibit numerical evidence of the n-symbolic synchronization