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

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

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 Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized

Nonlinear Dynamics In Musical Interactions

This thesis examines nonlinear dynamical processes in musical tools, identifying certain roles that they play in creative interactions with existing tools, and investigates the roles they might play in digital tools. Nonlinear dynamical processes are fundamental in the everyday physical world. They lie at the core of many acoustic instruments, playing a particularly significant role in bowed and blown instruments. Two major studies are presented that approach these issues from different perspectives. Firstly a set of comparative studies explore the ways in which musicians engage with systems that do and do not incorporate nonlinear dynamical processes. Secondly, interviews with a range of musicians engaged in contemporary musical practices — particularly free improvisation — are used to investigate the role of nonlinear dynamical processes in instrumental interactions in relation to unpredictability and creative exploration. Evidence is presented demonstrating that nonlinear dynamical processes can be drawn on as resources for exploration over long time periods. An approach to creative interaction that explicitly draws on the properties of nonlinear dynamical processes is uncovered and connected to material-oriented notions of creative processes. Nonlinear dynamics are shown to facilitate a productive ‘‘sweet spot’’ between unpredictability and complexity on the one hand, and detailed, sensitive, deterministic control, coupled with the potential to repeat and develop particular actions on the other. The importance of timing in interactions with nonlinear dynamical processes is highlighted as being significant in creating explorable interactions, particularly close to critical thresholds. A distinction is raised between instantaneous unpredictabilities that emerge from the interaction with the tool (interactional ), and unpredictabilities that result from the unexpected implications of the conjunction of otherwise anticipated elements (combinatorial). While the usefulness of the latter in creative interactions is frequently acknowledged in HCI research, the former is often excluded, or seen as a hinderance or obstruction. Engagements with nonlinear dynamical processes in existing musical instruments and practices provide clear evidence of the utility of both nonlinear dynamics, and interactional surprises more generally, suggesting that they can be of use in other domains where creative exploration is a concern

On the Existence of Solutions for Impulsive Duffing Dynamic Equations on Time Scales with Dirichlet Boundary Conditions

By using critical point theory, some new sufficient conditions for the existence of solutions of impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions are obtained. Some examples are also given to illustrate our results