Symmetry of chaotic attractors and kink-antikink structures in CMLs

In this paper the symmetries of coupled map lattices (CMLs) and their attractors are investigated by group and dynamical system theory, as well as numerical simulation, by means of which the kink-antikink patterns of CMLs in space-amplitude plots are discussed.

Multifractal dimension of chaotic attractors in a driven semiconductor superlattice

The multifractal dimension of chaotic attractors has been studied in a weakly coupled superlattice driven by an incommensurate sinusoidal voltage as a function of the driving voltage amplitude. The derived multifractal dimension for the observed bifurcation sequence shows different characteristics for chaotic, quasiperiodic, and frequency-locked attractors. In the chaotic regime, strange attractors are observed. Even in the quasiperiodic regime, attractors with a certain degree of strangeness may exist. From the observed multifractal dimensions, the deterministic nature of the chaotic oscillations is clearly identified.

Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboxes

We consider a model for rattling in single-stage gearbox systems with some backlash consisting of two wheels with a sinusoidal driving; the equations of motions are analytically integrated between two impacts of the gear teeth. Just after each impact, a mapping is used to obtain the dynamical variables. We have observed a rich dynamical behavior in such system, by varying its control parameters, and we focus on intermittent switching between laminar oscillations and chaotic bursting, as well as crises, which are sudden changes in the chaotic behavior. The corresponding transient basins in phase space are found to be riddled-like, with a highly interwoven fractal structure. (C) 2004 Elsevier Ltd. All rights reserved.

Predicting chaotic time-series with wavelet networks

A new technique, wavelet network, is introduced to predict chaotic time series. By using this technique, firstly, we make accurate short-term predictions of the time series from chaotic attractors. Secondly, we make accurate predictions of the values and bifurcation structures of the time series from dynamical systems whose parameter values are changing with time. Finally we predict chaotic attractors by making long-term predictions based on remarkably few data points, where the correlation dimensions of predicted attractors are calculated and are found to be almost identical to those of actual attractors.

Taming Chaotic Circuits

Control algorithms that exploit chaotic behavior can vastly improve the performance of many practical and useful systems. The program Perfect Moment is built around a collection of such techniques. It autonomously explores a dynamical system's behavior, using rules embodying theorems and definitions from nonlinear dynamics to zero in on interesting and useful parameter ranges and state-space regions. It then constructs a reference trajectory based on that information and causes the system to follow it. This program and its results are illustrated with several examples, among them the phase-locked loop, where sections of chaotic attractors are used to increase the capture range of the circuit.

Experimental identification of chaotic fibers

In a 2D parameter space, by using nine experimental time series of a Clitia`s circuit, we characterized three codimension-1 chaotic fibers parallel to a period-3 window. To show the local preservation of the properties of the chaotic attractors in each fiber, we applied the closed return technique and two distinct topological methods. With the first topological method we calculated the linking, numbers in the sets of unstable periodic orbits, and with the second one we obtained the symbolic planes and the topological entropies by applying symbolic dynamic analysis. (C) 2007 Elsevier Ltd. All rights reserved.

Bistable attractors in a model of convection-driven spherical dynamos

The range of existence and the properties of two essentially different chaotic attractors found in a model of nonlinear convection-driven dynamos in rotating spherical shells are investigated. A hysteretic transition between these attractors is established as a function of the rotation parameter t. The width of the basins of attraction is also estimated. © 2012 The Royal Swedish Academy of Sciences.

Interpersonal coordination tendencies, decision-making and information governing dynamics in rugby union

In this chapter, ideas from ecological psychology and nonlinear dynamics are integrated to characterise decision-making as an emergent property of self-organisation processes in the interpersonal interactions that occur in sports teams. A conceptual model is proposed to capture constraints on dynamics of decisions and actions in dyadic systems, which has been empirically evaluated in simulations of interpersonal interactions in team sports. For this purpose, co-adaptive interpersonal dynamics in team sports such as rubgy union have been studied to reveal control parameter and collective variable relations in attacker-defender dyads. Although interpersonal dynamics of attackers and defenders in 1 vs 1 situations showed characteristics of chaotic attractors, the informational constraints of rugby union typically bounded dyadic systems into low dimensional attractors. Our work suggests that the dynamics of attacker-defender dyads can be characterised as an evolving sequence since players' positioning and movements are connected in diverse ways over time.

Interpersonal pattern dynamics and adaptive behavior in multiagent neurobiological systems : conceptual model and data

Ecological dynamics characterizes adaptive behavior as an emergent, self-organizing property of interpersonal interactions in complex social systems. The authors conceptualize and investigate constraints on dynamics of decisions and actions in the multiagent system of team sports. They studied coadaptive interpersonal dynamics in rugby union to model potential control parameter and collective variable relations in attacker–defender dyads. A videogrammetry analysis revealed how some agents generated fluctuations by adapting displacement velocity to create phase transitions and destabilize dyadic subsystems near the try line. Agent interpersonal dynamics exhibited characteristics of chaotic attractors and informational constraints of rugby union boxed dyadic systems into a low dimensional attractor. Data suggests that decisions and actions of agents in sports teams may be characterized as emergent, self-organizing properties, governed by laws of dynamical systems at the ecological scale. Further research needs to generalize this conceptual model of adaptive behavior in performance to other multiagent populations.

Periodic window arising in the parameter space of an impact oscillator

In the bi-dimensional parameter space of an impact-pair system, shrimp-shaped periodic windows are embedded in chaotic regions. We show that a weak periodic forcing generates new periodic windows near the unperturbed one with its shape and periodicity. Thus, the new periodic windows are parameter range extensions for which the controlled periodic oscillations substitute the chaotic oscillations. We identify periodic and chaotic attractors by their largest Lyapunov exponents. (C) 2010 Elsevier B.V. All rights reserved.

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua`s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua`s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincar, sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems.

Chaos rules : an exploration of the work of instructional designers in distance education

This thesis provides an examination of the work of instructional designers in distance education, through the conceptual lens of chaos theory. Chaos theory was chosen as an analytical tool because of its ability to reveal the patterns and processes of complex systems as they move between order and turbulence. Recent work in the social sciences, specifically literary theory, has provided impetus for applications of chaos theory to educational settings. Specifically, chaos theory is used to analyse eight case studies of projects volunteered by instructional designers working in five institutions in Hong Kong and Australia. Data were gathered over a period of months with each participant, chiefly through interviews, but also involving diary accounts, electronic mail and letters. The methodology was thus qualitative, specifically informed by Eisner's vision of the ‘critical connoisseur’. Eisner equates an ‘enlightened eye’ with attainment of the skills of a critical connoisseur. First, an effective qualitative researcher must develop connoisseurship, the art of appreciation. On its own, though, connoisseurship is not enough; it is a private act, and thus needs a public face or presence. Criticism is this link, criticism being the art of disclosure. The critical connoisseur aims to help others to increase perception and deepen understanding of an educational situation or event. In addition to the empirical work, a parallel strand of this thesis investigates the theory and reported practice of instructional design. A brief history of instructional design is presented, along with discussion of acknowledged deficiencies of current theory and approaches. Recent reported investigations of both theory and practice are analysed from the viewpoint of chaos theory. Examination of key contributions in the literature of instructional design and distance education reveals considerable resonance between these contributions and the fundamental properties of chaotic systems. Links are made, in both the theoretical and empirical strands, between instructional design and the behaviour of dissipative structures, attractors and the process of bifurcation. Use is also made of the time-dependent nature of chaos theory as a theory of becoming, rather than one of being. The thesis comprises eight chapters, two appendices and a references section. The introductory chapter explains the research problem, and outlines the structure of the thesis. Methodological considerations are left until after an assessment of instructional design literature and (reported) practice. This deliberately theoretical investigation (Chapters 2 and 3) comprises the first of the parallel strands that are presented. The basic conclusions are that instructional design theory has not been particularly helpful to or used by instructional designers, and that chaos theory might provide an alternative way of viewing instructional design practice. The other parallel strand is the empirical work, which for four chapters outlines the methodology and my findings concerning the role of instructional designers in distance education. The methodology is detailed in Chapter 4. Chapter 5 establishes the contexts of the participants, by examining their backgrounds and introductions to their roles. It also investigates their views on their role and status within their institutions and with working colleagues. Chapter 6 is an exploration of the major issues that influenced the work of the instructional designers. These are the issues that arose naturally in the interviews as the participants outlined the development and interactions that took place on a day to day basis. Time emerges as a key influence in their work, and its effects on the projects are outlined and analysed. The ways that instructional designers give advice to those with whom they work is also investigated. The next chapter continues consideration of their work, but this time as they reflect on their role and its demands. This includes their reactions to the various metaphors that have appeared in the literature, along with those that they introduced into our discussions. The links that are established between the two parallel strands are drawn more explicitly in the final chapter, Chapter 8, which is a notion of what a model of instructional design based on my conclusions might resemble. It summarises the evidence that it is not necessarily by striving for order—in fact quite the opposite — during key periods of course development, that leads to creative outcomes. The introduction of uncertainty and turbulence does, in some cases and under some conditions, move the system to a higher level. The image that is offered from chaos theory is that of time-bound dissipative structures, interacting with their open environment at far-from-equilibrium conditions, and transforming themselves from disorder to order through bifurcation. The role of strange or chaotic attractors is highlighted in the process. The first appendix gives background information in terms of the methodology. The second is the heart of the data upon which the thesis draws. That is, the second appendix outlines the case studies of the participants. Most are short summaries, but the final one is a detailed study, tracing the progress of the design and development of a subject in distance education.

Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system

In this paper, by using the Poincare compactification in R(3) we make a global analysis of the Lorenz system, including the complete description of its dynamic behavior on the sphere at infinity. Combining analytical and numerical techniques we show that for the parameter value b = 0 the system presents an infinite set of singularly degenerate heteroclinic cycles, which consist of invariant sets formed by a line of equilibria together with heteroclinic orbits connecting two of the equilibria. The dynamical consequences related to the existence of such cycles are discussed. In particular a possibly new mechanism behind the creation of Lorenz-like chaotic attractors, consisting of the change in the stability index of the saddle at the origin as the parameter b crosses the null value, is proposed. Based on the knowledge of this mechanism we have numerically found chaotic attractors for the Lorenz system in the case of small b > 0, so nearby the singularly degenerate heteroclinic cycles.

Basins of attraction changes by amplitude constraining of oscillators with limited power supply

We investigate the dynamics of a Duffing oscillator driven by a limited power supply, such that the source of forcing is considered to be another oscillator, coupled to the first one. The resulting dynamics come from the interaction between both systems. Moreover, the Duffing oscillator is subjected to collisions with a rigid wall (amplitude constraint). Newtonian laws of impact are combined with the equations of motion of the two coupled oscillators. Their solutions in phase space display periodic (and chaotic) attractors, whose amplitudes, especially when they are too large, can be controlled by choosing the wall position in suitable ways. Moreover, their basins of attraction are significantly modified, with effects on the final state system sensitivity. (c) 2005 Elsevier Ltd. All rights reserved.

Dissipation and its consequences in the scaling exponents for a family of two-dimensional mappings

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)