Clara Grácio

Symbolic Dynamics and Chaotic Synchronization in Coupled Duffing Oscillators

Abstract 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 Poincaré 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.

On the Embedding of a (p-1)-Dimensional Noninvertible Map into a p-Dimensional Invertible Map (p=2,3)

This paper concerns the description of some properties of p-dimensional invertible real maps Tb, turning into a (p-1)-dimensional noninvertible 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.

TOPOLOGICAL ENTROPY IN THE SYNCHRONIZATION OF PIECEWISE LINEAR AND MONOTONE MAPS: COUPLED DUFFING OSCILLATORS

In this paper is presented a relationship between the synchronization and the topological entropy. We obtain the values for the coupling parameter, in terms of the topological entropy, to achieve synchronization of two unidirectional and bidirectional coupled piecewise linear maps. In addition, 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 unidirectional and bidirectional coupled identical chaotic Duffing equations is given. We discuss the complete synchronization of two identical double-well Duffing oscillators, from the point of view of symbolic dynamics. Working with Poincare cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized.

Boundary Maps and Fenchel–Nielsen–Maskit Coordinates

We consider a genus 2 surface, M, of constant negative curvature and we construct a 12-sided fundamental domain, where the sides are segments of the lifts of closed geodesics on M (which determines the Fenchel–Nielsen–Maskit coordinates). Then we study the linear fractional transformations of the side pairing of the fundamental domain. This construction gives rise to 24 distinct points on the boundary of the hyperbolic covering space. Their itineraries determine Markov partitions that we use to study the dependence of the Lyapunov exponent and length spectrum of the closed geodesics with the Fenchel–Nielsen coordinates.

Conditions for the formation of clusters depending on the conductance and the coefficient of clustering

One of the ultimate goals of researches on complex networks is to understand how the structure of complex networks affects the dynamical process taking place on them, such as traffic flow, epidemic spread, cascading behavior, and so on. In previous works [1] and [2], we have studied the synchronizability of a network in terms of the local dynamics, supposing that the topology of the graph is fixed. Now, we are interested in studying the effects of the structure of the network, i.e., the topology of the graph on the network synchronizability. The synchronization interval is given by a formula relating the first non zero and the largest eigenvalue of the Laplacian matrix of the graph with the maximum Lyapunov exponent of the local nodes. Our goal is to understand under what conditions can ensure the formation of clusters depending on the conductance and the coefficient of clustering.

LAPLACIANS AND THE CHEEGER CONSTANTS FOR DISCRETE DYNAMICAL SYSTEMS

Our subject is the study of discrete dynamical systems arising from the iterates of a map in the interval and our main concern is to search for invariants which could differentiate systems with the same topological entropy. In this paper is made a strong connection between a geometric point of view and a discrete dynamical system point of view. We explore the connections Cheeger constant/conductance, Laplace-Beltrami operator/discrete laplacian and systoles and isoperimetric inequalities in both contexts. We begin with the geometric motivation, see Grácio-Sousa Ramos. The idea of a Riemann surface is a central one in mathematics, and appears in such seemingly diverse areas as low dimensional topology, algebraic, differential and hyperbolic geometry, complex analysis, group theory and even number theory. This paper concerns compact Riemann surfaces endowed with a hyperbolic metric. It is possible to describe it in some ways, these include its representation as an algebraic curve; as a period matrix; as a Schottky group; as a Fuchsian group and as a hyperbolic manifold, in particular, using Fenchel-Nielsen (F-N) coordinates. It is exactly this last way that we will use, Riemann surface theory based on closed geodesics using Fenchel-Nielsen (F-N) coordinates. This geometry has always played

Networks Synchronizability, Local Dynamics and Some Graph Invariants

The synchronization of a network depends on a number of factors, including the strength of the coupling, the connection topology and the dynamical behaviour of the individual units. In the first part of this work, we fix the network topology and obtain the synchronization interval in terms of the Lyapounov exponents for piecewise linear expanding maps in the nodes. If these piecewise linear maps have the same slope ± s everywhere, we get a relation between synchronizability and the topological entropy. In the second part of this paper we fix the dynamics in the individual nodes and address our work to the study of the effect of clustering and conductance in the amplitude of the synchronization interval.

CHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL

We consider a discrete-time economic model which is a particular case of the Kaldor-type business cycle model and it is described by a two-dimensional dynamical system. Under certain conditions the map can be reduced to a skew map whose components, the base and the fiber map, both have entropy. Our proposal is to study and measure the complexity of the system using symbolic dynamics techniques and the topological entropy.

Strong generalized synchronization with a particular relationship R between the coupled systems

The question of the chaotic synchronization of two coupled dynamical systems is an issue that interests researchers in many fields, from biology to psychology, through economics, chemistry, physics, and many others. The different forms of couplings and the different types of synchronization, give rise to many problems, most of them little studied. In this paper we deal with general couplings of two dynamical systems and we study strong generalized synchronization with a particular relationship R between them. Our results include the definition of a window in the domain of the coupling strength, where there is an exponentially stable solution, and the explicit determination of this window. In the case of unidirectional or symmetric couplings, this window is presented in terms of the maximum Lyapunov exponent of the systems. Examples of applications to chaotic systems of dimension one and two are presented.

Window of Chaotic Delayed Synchronization

We consider a general coupling of two chaotic dynamical systems and we obtain conditions that provide delayed synchronization. We consider four different couplings that satisfy those conditions. We define Window of Delayed Synchronization and we obtain it analytically. We use four different free chaotic dynamics in order to observe numerically the analytically predicted windows for the considered couplings.