Dynamical systems and Chaos

The drainage of ice-dammed lakes produces floods that can pose hazards, waste water resources and modulate ice flow. In this thesis I investigate several aspects of ice-dammed lake drainage through the development and analysis of mathematical models.
After an introduction in the first chapter and a description of the mathematical background to the thesis in the second, the third chapter investigates the mechanisms behind observed variability in the size and timing of subglacial floods from ice-dammed lakes. In particular, I examine how environmental controls like the weather and the shape of glaciers affect floods.

In the next chapter, I quantify how well simple models can predict the dates of floods from an ice-marginal lake in Kyrgyzstan. I find that incorporating environmental controls into models improves their prediction ability.

Next I investigate the coupling between subglacial drainage and glacier motion during ice-dammed lake drainage by developing and analysing a model which couples a marginal lake, glacier sliding, subglacial drainage through a channel and subglacial drainage through a distributed system of cavities. I show how changes in lake level cause the rate at which a glacier slides to increase during the first half of floods and decrease during the second half.

The next two chapters are concerned with two lake-drainage scenarios that involve water flowing as an open stream: firstly, the subglacial open-channel flow that occurs after a marginal lake drains completely during a flood, and secondly, the drainage of supraglacial lakes across the surface of ice sheets.

I end the thesis with a summary of my findings and some suggestions of theoretical and field-based investigations that are worthwhile pursing in the future.

This paper addresses fault diagnosis in dynamic systems represented by discrete state-space models. The main idea of the paper is to propose a systematic way to implement a Petri net-based fault diagnosis system. This procedure consists of three main steps. In the first step, a fault diagnosis system is built based on the Luenberger observer. In the second step, the obtained fault diagnoser equations are transformed to a suitable format. Finally, in the third step, the obtained equations are implemented by a Petri net called continuous-time delay Petri net (CTDPN) that can realize difference equations. Based on this method, a systematic approach is proposed for realizing a classical fault diagnoser by CTDPN. By integrating the concept of state-space observers and PNs in this paper, new and effective methods are developed for the analysis and fault diagnosis of systemsknown as hybrid systems-that have both continuous and discrete variables. The performance of the proposed method is thoroughly investigated, and the obtained results show that the proposed CTDPN can precisely detect the occurred faults, their types and their occurrence time instances.

In this paper, the use of a chaotic circuit - namely, Chua’s Circuit - is explored as a possible method of random number generation. Using the chaotic nature of the output and the sensitive dependence on initial conditions, random binary bits were generated and compared against the output of known state-of-the-art Pseudo-Random-Number-Generators (PRNGs) and with those of a True-Random-Number-Generator (TRNG) using known statistical tests for random- ness. The results were used understand the randomness of the values generated, and to draw conclusions for the possible application of Chua Circuits as RNGs for encryption.

Perry, N. S., Baucom, K. J. W., Bourne, S., Butner, J., Crenshaw, A. O., Hogan, J. N., Imel, Z. E., Wiltshire, T. J., & Baucom, B. R. W.

Researchers commonly use repeated-measures actor–partner interdependence models (RM-APIM) to understand how romantic partners change in relation to one another over time. However, traditional interpretations of the results of these models do not fully or correctly capture the dyadic temporal patterns estimated in RM-APIM. Interpretation of results from these models largely focuses on the meaning of single-parameter estimates in isolation from all the others. However, considering individual coefficients separately impedes the understanding of how these associations combine to produce an interdependent pattern that emerges over time. Additionally, positive within-person, or actor, effects are commonly misinterpreted as indicating growth from one time point to the next when they actually represent decline. We suggest that change-as-outcome RM-APIMs and vector field diagrams (VFDs) can be used to improve the understanding and presentation of dyadic patterns of association described by standard RM-APIMs. The current article briefly reviews the conceptual foundations of RM-APIMs, demonstrates how change-as-outcome RM-APIMs and VFDs can aid interpretation of standard RM-APIMs, and provides a tutorial in making VFDs using multilevel modeling.

Defense mechanisms are very important to all animal life. Predators in every biome must eat to survive. With predators being top on the food chain and always on the lookout for a meal, prey must constantly avoid being eaten. In this paper, we have proposed and analyzed a tri-trophic predator–prey model of one prey and two predator exhibiting group defense mechanism. We have assumed Monod-Haldane functional response for interaction between species due to group defense ability of prey and middle predator. We have performed Kolmogorov and Hopf bifurcation analysis for the model system. Linear and global stability of the model system have been analyzed. Lyapunov exponents are computed numerically and 2D scan for different parameters of the model have performed to characterize the complex behavior of the model system. The numerical simulations shows the chaotic and periodic oscillations of the model system for certain range of parameter. We have drawn bifurcation diagrams for different parameter values which shows the complex dynamical behavior of model system. This work is an attempt to study the effect of group defense mechanism of prey in predator population and effect of immigration within top predator population is investigated. It is also observed that in the presence of group defense, the model system stabilizes after adding a small amount of constant immigration within top predator population.

This textbook, now in its second edition, provides a broad introduction to the theory and practice of both continuous and discrete dynamical systems with the aid of the Mathematica software suite. Taking a hands-on approach, the reader is guided from basic concepts to modern research topics. Emphasized throughout are numerous applications to biology, chemical kinetics, economics, electronics, epidemiology, nonlinear optics, mechanics, population dynamics, and neural networks. The book begins with an efficient tutorial introduction to Mathematica, enabling new users to become familiar with the program, while providing a good reference source for experts. Working Mathematica notebooks will be available at:

http://library.wolfram.com/infocenter/Books/9563/

The author has focused on breadth of coverage rather than fine detail, with theorems and proofs being kept to a minimum, though references are included for the inquisitive reader. The book is intended for senior undergraduate and graduate students as well as working scientists in applied mathematics, the natural sciences, and engineering. Many of the chapters are especially useful as reference material for senior undergraduate independent project work. New to the second edition:

Since the first printing of this book in 2007, Mathematica has evolved from version 6.0 to version 11.2 in 2017. Accordingly, the second edition has been thoroughly updated and new material has been added. There are many more applications, examples and exercises, all with solutions, and new sections on series solutions of ordinary differential equations and Newton fractals, have been added. There are also new chapters on delay differential equations, image processing, binary oscillator computing, and simulation with Wolfram SystemModeler. Praise for the first edition:

“[This book’s] content and presentation style convey the excitement that has drawn many students and researchers to dynamical systems in the first place.”

—Dynamical Systems Magazine

“This book presents an original, cheap and powerful solution to the problem of analysis of large data sets.”

—Studia Universitatis Babes’-Bolyai Mathematica

“The one-liner programs come to life when typed in, and the growing programming skill lends itself to inventing [one's] own extensions to the supplied problems.”

—Datafile, The Journal of the HPCC

This book describes the birth of the new theory of Chaos. This is a difficult new concept that is still evolving but it popularized the term: Butterfly Effect and introduced new concepts to a popular audience, such as fractals and introduced pioneering thinkers, such as Feigenbaum and Mandelbrot; it inspired the novel and movie Jurassic Park. This concept opens up a new view of nature: where previously randomness had to be forced in to explain the unpredictable variations, now chaos is seen as spanning both order (patterns) and disorder. Now, this phenomenon helps explain the shape of clouds, smoke, water eddies, mountain ranges and coastlines. Implicitly, it shows how Newtonian mathematics has constrained physics (and science in general) to make simplifying assumptions that enable  the calculus to become the universal tool-set of the scientific viewpoint. The book describes how this tough problem was cracked by five theoreticians described herein with a novelist's eye.  Key to the solution was the early use of computers to repeat simple calculations, very many times. The viewpoint changed from static 'state' to dynamic process: becoming rather than being. Chaos is everywhere, it is switching the simple mathematical models of classical physics. It is the science of the global nature of systems. I show here (but not in the book or Wiki) that this is the start of the Death of Newtonian Physics and the Calculus: a TRUE REVOLUTION.

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the
field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is
a three-dimensional phenomenon.

In 1963 Lorenz published his seminal paper Deterministic Non-­‐‑ periodic flow in the Journal of Atmospheric Sciences. The philosophical ramifications of the unpredictability of phenomenon in nature noted in this work were profound and the implications have fueled an incredible development in dynamical systems. In this paper, we explore this system and its enigmatic strange attractor, by looking into the dynamics of the Lorenz equations, defining its chaotic attributes thru both an analytic and visual approach, and ultimately showing that the Lorenz system does indeed support the existence of this strange attractor.

В данной работе делается попытка показать, с помощью методов математического моделирования, при каких условиях процессы самоорганизации в рабочей среде могут приводить к взрывному характеру конфликтов, к нелинейным эффектам в их развитии в силу действия внутренних факторов. Первый раздел статьи включает краткое рассмотрение роли внешних и внутренних факторов в развитии волн стачечной динамики. Во  втором разделе предлагается нелинейная модель стачечной динамики, проводится ее анализ; в частности, выявляются условия, в которых эта динамика может иметь неустойчивый, хаотический характер, когда малые (возможно, случайные) воздействия приводят к резким изменениям стачечной активности. Именно такие процессы изучает синергетика.
      In this paper an attempt is made to show by means of mathematical modeling, the conditions under which self-organization processes in the working environment can lead to explosive nature of the conflict, to nonlinear effects in their development by virtue of endogeneous and exogeneous factors. The first section of the article includes a brief review of the role of external and internal factors in the development of the strike wave dynamics. The second section provides a non-linear model of the strike dynamics and its analysis; in particular, identifies the conditions under which these dynamics can be unstable or chaotic, when small (possibly random) effects lead to rapid changes in strike activity. It is these processes synergetics studies.

O pêndulo simples é um sistema de fácil análise que é governado por equações diferenciais nãolineares. Diversos problemas físicos podem ser modelados por pêndulos simples e geralmente nos cursos de física
básica é feito um processo de linearização do modelo obtido para produzir o que se chama de pêndulo harmônico
simples, cujas equações que regem o sistema podem ser resolvidas de forma analítica. Todavia o modelo linearizado não produz resultados satisfatórios para grandes amplitudes, não condizendo com o que é observado na
prática. O modelo não-linear fornece diversas informações qualitativas importantes que não aparecem no linear,
todavia possui a desvantagem de não permitir uma solução analítica simples, se fazendo necessário recorrer ao
uso de métodos numéricos para obter soluções. Este trabalho tem como objetivo analisar a dinâmica do sistema
mecânico que constitui um pêndulo simples composto por uma barra rígida de massa desprezível e comprimento
L com um corpo puntiforme de massa m em sua extremidade, forças dissipativas não serão consideradas. Será
utilizado o MATLAB na construção de retratos de fase para discutir as soluções não-lineares e compará-las com
as obtidas por linearização. Foi empregado o método numérico de Runge-Kutta de quarta ordem para resolver os
sistemas de equações diferenciais.

IPB doctrine states that all four mission variables—including civil considerations—and their interactions must be analyzed if the process is to be effective. Staffs must “determine how the interactions of friendly forces, enemy forces, and indigenous populations affect each other.” However, in practice, the process tends to emphasize the enemy and not holistically account for the civil considerations.

"The idea of this didactical apparatus was born during a discussion of one of the authors (G.T.) with prof. A. Loria after the 30th Int. Physics Olympiad held in Padua in 1999, about the problem on inverted torsion pendulum assigned to the students as part of the experimental competition. The very simple device given to the students allowed either horizontal or vertical rotation axis, and the torque of the eccentric mass could be adjusted by screwing the threaded pendulum rod, thus allowing to vary the system behaviour and showing bifurcation.
Prof. Loria suggested us to setup an apparatus that could make possible both real time data acquisition of the system evolution and controlled forced oscillations, in order to exploit the full range of educational use of this interesting device.
Our first step was to prepare a prototype on which S. Lasic’s thesis was based,demonstrating essentially all the features. The numerical simulations, were performed within the thesis later assigned to P. Bedendo. The result is an experiment that, using simple and cheap tools, allows to fully investigate both linear and non-linear oscillators with a complete control of all the interesting parameters. Both approximate-analytical and exact-numerical solutions of the Duffing oscillator may be tested in comparison with experimental measurement, leading to a deeper understanding of complex phenomena."

It is the contention of this volume that only on the basis of generalised technology can we begin to approach the phenomenon of language in its broadest sense. Integrating conceptual approaches from both continental & analytic philosophy, LITERATE TECHNOLOGIES demonstrates that any system of sign operations, in which an event of transmission or transcription can be said to take place, is technological & that all technology is poetic before it is technical. At the same time, this universalising aspect of a ‘technopoetics’ appears as a constellation-effect, bridging the entire field of scientific discourse–from atomic and molecular structures to the transcriptive coding and decoding processes of DNA; from the evolving neural structures of the human brain to computing programmatics and artificial intelligence; from simple binary procedures to the most complex topologies–which is thus also to say, the entire 'textual' field.

Lucretius was the first philosopher of immanence. It is he and not Democritus or Epicurus who holds this title. If we want to understand the historical emergence of the concept of immanence, we should start by distinguishing its precursors in Greek atomism from its first complete incarnation in Lucretius. This way, we can see exactly what first defined and distinguished immanence from its past. Therefore in what follows I would like to make three, perhaps controversial, claims about the emergence of philosophical immanence. 1) Lucretius was not an atomist, 2) Greek atomism reintroduced transcendence, and 3) It is the primacy of motion in Lucretius that defines his philosophical immanence. Lucretius was not an atomist This thesis is as counterintuitive as it is straightforward. The first major difference between Lucretius and the earlier Greek atomists is precisely that—the atom. For Leucippus, Democritus, and Epicurus atoms are always in motion, but the atom itself remained fundamentally unchanged, indivisible, and thus internally static—even as it moved. Thus instead of positing discrete atoms as ontologically primary as both ancient Greek and later modern theories do, one of Lucretius's greatest novelties was to posit the movement or flow of matter as primary. Lucretius did not simply " translate Epicurus, " as the Greco-centric story goes; rather, he introduced the first immanent kinetic materialism in the West. For example, although the Latin word atomus (smallest particle) was available to Lucretius to use in his poem, he intentionally did not use it, nor did he use the Latin word particula or particle to describe matter. The English translations of " atom, " " particle, " and others have all been added to the text in translation based on a certain historical interpretation of it. The idea that Lucretius subscribed to a world of discrete particles called atoms is therefore both a projection of Epicureanism, who used the Greek word atomos, and a retroaction of modern scientific mechanism of the fifteenth century onto De Rerum Natura. Lucretius rejected entirely the notion that things emerged from discrete particles. To believe otherwise is to distort the original meanings of the Latin text as well as the absolutely enormous poetic apparatus he summoned to describe the flowing, swirling, folding, and weaving of the flux of matter. Although Lucretius rejected the term atomus, he remained absolutely true to one aspect of the original Greek meaning of the word, τομος (átomos, " indivisible "), from-(a-, " not ") + τέμνω (témnō, " I cut "). Being is not cut up into discrete particles, but is composed of continuous flows, folds, and weaves. Discrete " things " (rerum) are composed of corporeal flows (corpora) that move together (conflux) and fold over themselves (nexus) in a woven knot work (contextum). For Lucretius, things only emerge and have their being within and immanent to the flow and flux of matter in motion. Discreteness is an apparent product of continuous folded matter, uncut, undivided, and in motion and not the other way around.

The International Journal of Chaos, Control, Modeling and Simulation is a Quarterly open access peer-reviewed journal that publishes articles which contribute new results in all areas of Chaos Theory, Control Systems, Scientific Modeling and Computer Simulation. In the last few decades, chaos theory has found important applications in secure communication devices and secure data encryption. Control methods are very useful in several applied areas like Chaos, Automation, Communication, Robotics, Power Systems, and Biomedical Instrumentation. The journal focuses on all technical and practical aspects of Chaos, Control, Modeling and Simulation. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on chaotic systems and applications, control theory, process control, automation, modeling concepts, computational methods, computer simulation and establishing new collaborations in these areas.

Durante un periodo più o meno lungo di relativa stabilità, il sistema riesce a neutralizzare le perturbazioni, ma ad un certo punto elementi pregressi provocano una crisi di instabilità durante la quale i vari elementi del sistema sono sottoposti a tensione, come un gas surriscaldato in cui aumenti progressivamente il movimento delle molecole. La rottura, o biforcazione catastrofica, è preceduta da uno stato caotico in cui ogni minima fluttuazione può essere estremamente amplificata da fenomeni di feedback positivo. Il futuro del sistema diventa imprevedibile se non si conosce la storia delle condizioni al contorno che hanno provocato lo stato attuale (René Thom, determinista; gli indeterministi sostengono invece che il sistema diventa imprevedibile e basta). In tale stato, una fluttuazione più ampia o una sincronia di condizioni catapultano il sistema ad uno stadio superiore il quale procede in un nuovo stato stabile.

Un sistema come quello capitalistico può rimanere stabile, cioè in equilibrio omeostatico, mediante l'autoregolazione dei propri flussi di energia (retroazione negativa, come nel termostato: keynesismo, patto sociale) per un certo periodo, fino al momento in cui i cicli stabilizzanti non vanno in crisi; la retroazione, da negativa diventa positiva (ad esempio sovrapproduzione e speculazione), le perturbazioni si accumulano, ed il sistema esplode in una biforcazione catastrofica verso un nuovo livello di stabilità. L'importante è ciò che succede nel periodo di fluttuazione caotica: in esso si osserva un fenomeno detto di nucleazione, vale a dire che una regione del sistema improvvisamente si auto-organizza in un nuovo ordine fino a raggiungere, via catastrofe, il nuovo assetto, il quale viene di nuovo stabilizzato da una serie di cicli di retroazione.

This article surveys and synthesizes dynamic systems models of development from biology, neuroscience, and psychology in order to propose an integrated account of growth, learning, and behavior. Key to this account is the concept of self-differentiation or symmetry-breaking. I argue that development can be understood as a cascade of symmetry-breaking events brought about by the ongoing interactions of multiple, nested, nonlinear dynamic systems whose self-organizing behaviors gradually alter their own anatomical conditions.

In this paper, we will study dynamics of an important physiological control system—human gait in disease and aging. The investigation of fluctuations overlying periodic motion in human walking may provide valuable information about neuromuscular system generating both normal and pathological walking patterns. Using nonlinear dynamics analysis, we analyzed walking gaits of the young, elderly and aged subjects with Parkinson’s disease. This inquiry demonstrates that nonlinear time-series analysis methods based on time-delay embedding may provide useful insight into the neuromuscular control of human locomotion.

From his earliest work forward, Merleau-Ponty attempted to develop a new ontology of nature that would avoid the antinomies of realism and idealism by showing that nature has its own endogenous sense which is prior to reflection. The key to this new ontology was the concept of form, which he appropriated from Gestalt psychology. However, Merleau-Ponty struggled to give a positive characterization of the phenomenon of form which would clarify its ontological status. Evan Thompson has recently taken up Merleau-Ponty’s ontology as the basis for a new, “enactive” approach to cognitive science, synthesizing it with concepts from dynamic systems theory and Francisco Varela’s theory of autopoiesis. However, Thompson does not quite succeed in resolving the ambiguities in Merleau-Ponty’s account of form. This article builds on an indication from Thompson in order to propose a new account of form as asymmetry, and of the genesis of form in nature as symmetry-breaking. These concepts help us to escape the antinomies of Modern thought by showing how nature is the autoproduction of a sense which can only be known by an embodied perceiver.

IT'S HERE NOW!!! 
IT’S GOING TO BE REAL STRANGE.  PEOPLE MUST NOT BE AFRAID, IN FACT, ALL PEOPLE NEED TO BE REAL HAPPY ABOUT THIS EVENT AND BE READY TO GET OUTSIDE. GROUNDED.  THIS IS MEANT TO CLEAN ALL OF THE POLLUTION AND PROBABLY EVEN THE RADIATION FROM THE PLANET.

SOLID EVIDENCE OF THE INCOMING WAVE

In the last decade, chaos has emerged as a new promising candidate for cryptography because many chaos fundamental characteristics such as a broadband spectrum, ergodicity, and high sensitivity to initial conditions are directly connected with two basic properties of good ciphers: confusion and diffusion. In this chapter we recount some of the saga undergone
by this field; we review the main achievements in the field of chaotic cryptography, starting with the definition of chaotic systems and their properties and the difficulties it has to outwit. According to their intrinsic dynamics, chaotic cryptosystems are classified depending on whether the system is discrete or continuous. Due to their simplicity and rapidity the discrete chaotic systems based on iterative maps have received a lot of attention. In spite of the significant achievements accomplished in this field, there are still many problems, basically speed, that restrict the application of existing encoding/decoding algorithms to real systems. The major advantages and drawbacks of the most popular chaotic map ciphers in terms of security and computational cost are analyzed. The most significant cryptanalytic techniques are considered and applied for testing the security of some chaotic algorithms. Finally, future trends in the development of this topic are discussed.

In recent years, there has been a rising interest in authenticated encryption with associated data (AEAD) which combines encryption and authentication into a unified scheme. AEAD schemes provide authentication for a message that is divided into two parts: associated data which is not encrypted and the plaintext which is encrypted. However, there is a lack of chaos-based AEAD schemes in recent literature. This paper introduces a new 128-bit chaos-based AEAD scheme based on the single-key Even-Mansour and Type-II generalized Feistel structure. The proposed scheme provides both privacy and authentication in a single-pass using only one 128-bit secret key. The chaotic tent map is used to generate whitening keys for the Even-Mansour construction, round keys, and random s-boxes for the Feistel round function. In addition, the proposed AEAD scheme can be implemented with true random number generators to map a message to multiple possible ciphertexts in a nondeterministic manner. Security and statistical evaluation indicate that the proposed scheme is highly secure for both the ciphertext and the authentication tag. Furthermore, it has multiple advantages over AES-GCM which is the current standard for authenticated encryption.

Chaos and complexity theory are nonlinear dynamics within a dynamical system which changes over time (Warren et al.,1998). Teaching teenagers (students) about chaos and complexity gives them a broader understanding of biological and human social systems. Relationships are vital when considering both chaos and complexity theory within teenager groups and for teachers. Points of interested within this presentation address areas of chaos theory, complexity theory, background into theoretical thinking, system thinking, the butterfly effect, open and closed systems and leveraging in system thinking. It is important to understand areas of teaching and outcomes within secondary schools and the adaptability within teen groups. Aspects of chaos and complexity theory help to address system thinking among teenagers and within the subculture of the educational system. Future research within other subgroup cultures and geographic regions would be beneficial.

In this work methods for performing time series prediction on complex real
world time series are examined. In particular series exhibiting non-linear or
chaotic behaviour are selected for analysis. A range of methodologies based
on Takens’ embedding theorem are considered and compared with more
conventional methods. A novel combination of methods for determining the
optimal embedding parameters are employed and tried out with multivariate
financial time series data and with a complex series derived from an
experiment in biotechnology. The results show that this combination of
techniques provide accurate results while improving dramatically the time
required to produce predictions and analyses, and eliminating a range of
parameters that had hitherto been fixed empirically. The architecture and
methodology of the prediction software developed is described along with
design decisions and their justification. Sensitivity analyses are employed to
justify the use of this combination of methods, and comparisons are made
with more conventional predictive techniques and trivial predictors showing
the superiority of the results generated by the work detailed in this thesis.

To obtain a deep and clear understanding of dynamical systems of cardiovascular- respiratory system the interesting way is to investigate delay models [30]. The problem was that the presence of multi-delays causes the complexity of determining the bifurcation points. This work was undertaken in order to determine the Hopf bifurcation points of the cardiovascular respiratory mathematical model with two delays, where in the numerical simulation we consider a 30 years old woman in the physical activities such as walking, jogging, running fast. The algorithm was developed to overcome the difficulties of finding the Hopf bifurcation points of multiple delay models. We focused on linearization of two delays mathematical model around the equilibrium points and the numerical simulations based on the theory of alpha -dense curve we tried to find out the general algorithm. The findings results show that a small perturbation of Hopf bifurcation parameters allows a state to pass from stable state by passing through on an intermediate transition phase to unstable state in all these three considered physical activities.

This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of determin‐ istic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.

El artículo introduce el paradigma de la complejidad y las nuevas corrientes corporeizadas y enactivas de las ciencias cognitivas y discute cómo estos aportes pueden ser útiles como soporte teórico y heurístico para la investigación en terapia gestalt. Se articula la teoría del self en relación con la visión autopoiética y enactivista, la cual nos permite describir la relación terapéutica como un sistema complejo y auto-organizado. Palabras clave: Enacción, Terapia gestalt, Investigación, Validez, Campo, Sentido, Complejidad Introducción: Tras el IV Congreso Internacional de Investigación en Terapia Gestalt, el entusiasmo por la investigación en psicoterapia sigue in crescendo. La apertura, entusiasmo y también el eclecticismo que caracteriza la terapia gestalt se traslada a las prácticas de investigación, dando pie a una diversidad de acercamientos a la actividad científica y también, un replanteamiento de los mismos principios y prácticas de investigación. De entre la amalgama de visiones, teorías y capacidades, se hacen salientes dos líneas u objetivos generales de la investigación:

In this paper, a Fractional Sliding Mode Control strategy is introduced to realize the control of fractional stochastic chaotic system. Besides modeling stochastic chaos by an Ito stochastic differential form,  fractional-order  of  this  chaos  system  is  studied.  Based  on  the  Lyapunov  stability  theory,  the stability  of  the  closed-loop  system  is  guaranteed.  The  Chaos  control  is  also  obtained  between  two fractional-order stochastic and deterministic systems in two cases: 1) A control task; to behave period doubling  form,  2)  Synchronization  job;  both  in  different  order.  This  is  the  first  research  to apply  the proposed sliding mode controller to control of this type system. The simulation results and the lyapunov exponents demonstrate the feasibility of the proposed control method.

Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems' behavior.

Introduzione-Di cosa parliamo quando parliamo di caos?
LO SCRIGNO DI PROMETEO
COLLANA DI DIDATTICA, DIVULGAZIONE E STORIA DELLA FISICA
Direttore
Ettore GADIOLI
Università degli Studi di Milano
Piero Caldirola International Centre for the Promotion of Science
Comitato scientifico
Sigfrido BOFFI
Università degli Studi di Pavia
Giovanni FIORENTINI
Università degli Studi di Ferrara
Marco Alessandro Luigi GILIBERTI
Università degli Studi di Milano

—Everything in this universe is, to some extent, nonlinear in nature. For certain parameter values, a nonlinear dynamical system exhibits erratic behavior called chaos. In this paper, a novel yet simple method is presented for implementing nonlinear systems using operational amplifiers (op-amps) based circuits like integrators and subtractors. The non-linear function sin 2 (x) has been written in terms of polynomial equation using Taylor series expansion and that polynomial equation has been implemented using op-amps. The transition to chaos has been observed by varying the control parameters in simulations and in experiments.

In this paper I will argue that: (i) the term ‘harmony’ cannot characterize the system of law, (ii) the term ‘chaos’ is a better description here, (iii) but still, despite some recently fashionable claims, the latter is rather a distant metaphor than a useful concept for understanding and analyzing the law. I am aware of the fact that most readers will find all of these conclusions downright perplexing. On the one hand, according to common sense ‘harmony’ is certainly considered as a more appropriate term than ‘chaos’ in the context of legal analysis. On the other hand, those readers who don’t share the common intuition just mentioned are usually sympathetic with the new, dynamic approach, where the law is described in terms of the complex systems theory. I will argue that this approach is also completely mistaken.

Can stable regularities be explained without appealing to governing laws or any other modal notion? In this paper, I consider what I will call a ‘Humean system’—a generic dynamical system without guiding laws—and assess whether it could display stable regularities. First, I present what an be interpreted as an account of the rise of stable regularities, following from Strevens [2003], which has been applied to explain the patterns of complex systems (such as those from meteorology and statistical mechanics). Second, since this account presupposes that the underlying dynamics displays deterministic chaos, I assess whether it can be adapted to cases where the underlying dynamics is not chaotic but truly random—that is, cases where there is no dynamics guiding the time evolution of the system. If this is so, the resulting stable, apparently non-accidental regularities are the fruit of what can be called statistical necessity rather than of a primitive physical necessity.