This book reveals the French scientific contribution to the mathematical theory of nonlinear oscillations and its development. The work offers a critical examination of sources with a focus on the twentieth century, especially the period between the wars. Readers will see that, contrary to what is often written, France's role has been significant. Important contributions were made through both the work of French scholars from within diverse disciplines, and through the geographical crossroads that France provided to scientific communication at (...) the time. This study includes an examination of the period before the First World War which is vital to understanding the work of the later period. By examining literature sources such as periodicals on the topic of electricity from that era, the author has unearthed a very important text by Henri Poincaré, dating from 1908. In this work Poincaré applied the concept of limit cycle to study the stability of the oscillations of a device for radio engineering. The “discovery” of this text means that the classical perspective of the historiography of this mathematical theory must be modified. Credit was hitherto attributed to the Russian mathematician Andronov, from correspondence dating to 1929. In the newly discovered Poincaré text there appears to be a strong interaction between science and technology or, more precisely, between mathematical analysis and radio engineering. This feature is one of the main components of the process of developing the theory of nonlinear oscillations. Indeed it is a feature of many of the texts referred to in these chapters, as they trace the significant developments to which France contributed. Scholars in the fields of the history of mathematics and the history of science, and anyone with an interest in the philosophical underpinnings of science will find this a particularly engaging account of scientific discovery and scholarly communication from an era full of exciting developments. (shrink)

I take the phrase ''the theory of nonlinear oscillations'' to identify a historical phenomenon. Under this heading a powerful school in Soviet science, L. I. Mandelstam's school, developed its version of what was later called ''nonlinear dynamics''. The theory of nonlinear oscillations was formed around the concept of self-oscillations, which was elaborated by Mandelstam's graduate student A. A. Andronov. This concept determined the paradigm of the theory of nonlinear oscillations as well as its ideology, that is, (...) a set of characteristic ideas which, together with the corresponding examples and analogues, allowed the expansion of the theory into associated areas where it indicated new interesting phenomena and posed new problems. It was the ideology that made possible the broader application of the theory of nonlinear oscillations, whose domain was originally lumped systems, to continuous media and its subsequent progress toward synergetics. In the course of its ideological application, the concept of self-oscillations was greatly extended, became vague and diffuse, and related concepts such as self-waves and self-structures appeared. (shrink)

The aim of our commentary is to strengthen Cowan's proposal for an inherent capacity limitation in STM by suggesting a neurobiological mechanism based on competitive networks and nonlinear oscillations that avoids some of the shortcomings of the scheme discussed in the target article (Lisman & Idiart 1995).

In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka–Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched by the (...) multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincaré section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found. (shrink)

In this work we present a construction of coherent states based on ”complexifier” method for a special type of one dimensional nonlinear harmonic oscillator presented by Mathews and Lakshmanan. We will show the state quantization by using coherent states, or to build the Hilbert space according to a classical phase space, is equivalent to departure from real coordinates to complex ones.

In this study, complex dynamics of Briggs–Rauscher reaction system is investigated analytically and numerically. First, the Briggs–Rauscher reaction system is reduced into a new nonlinear parametric oscillator. The Melnikov method is used to derive the condition of the appearance of horseshoe chaos in the cases ω = Ω and ω ≠ Ω. The performed numerical simulations confirm the obtained analytical predictions. Second, the prediction of coexisting attractors is investigated by solving numerically the new nonlinear parametric ordinary differential (...) equation via the fourth-order Runge–Kutta algorithm. As results, it is found that the new nonlinear chemical system displays various coexisting behaviors of symmetric and asymmetric attractors. In addition, the system presents a rich variety of bifurcations phenomena such as symmetry breaking, symmetry restoring, period doubling, reverse period doubling, period-m bubbles, reverse period-m bubbles, intermittency, and antimonotonicity. On the contrary, emerging chaotic band attractors and period-1, period-3, period-9, and period-m bubbles routes to chaos occur in this system. (shrink)

A generalized third-order autonomous Duffing–Holmes system is proposed and deeply investigated. The proposed system is obtained by adding a parametric quadratic term m x 2 to the cubic nonlinear term − x 3 of an existing third-order autonomous Duffing–Holmes system. This modification allows the system to feature smoothly adjustable nonlinearity, symmetry, and nontrivial equilibria. A particular attention is given to the effects of symmetric and asymmetric nonlinearity on the dynamics of the system. For the specific case of m = (...) 0, the system is symmetric and interesting phenomena are observed, namely, coexistence of symmetric bifurcations, presence of parallel branches, and the coexistence of four and six symmetric attractors. For m ≠ 0, the system loses its symmetry. This favors the emergence of other behaviors, such as the coexistence of asymmetric bifurcations, involving the coexistence of several asymmetric attractors. All these phenomena have been numerically highlighted using nonlinear dynamic tools and an analog computer of the system. In fact, PSpice simulations of the latter confirm numerical results. Moreover, amplitude control and synchronization strategies are also provided in order to promote the exploitation of the proposed system in engineering. (shrink)

Although its roots can be traced to the 19th century, progress in the study of nonlinear dynamical systems has taken off in the last 30 years. While pertinent source material exists, it is strewn about the literature in mathematics, physics, biology, economics, and psychology at varying levels of accessibility. A compendium research methods reflecting the expertise of major contributors to NDS psychology, Nonlinear Dynamical Systems Analysis for the Behavioral Sciences Using Real Data examines the techniques proven to be (...) the most useful in the behavioral sciences. The editors have brought together constructive work on new practical examples of methods and application built on nonlinear dynamics. They cover dynamics such as attractors, bifurcations, chaos, fractals, catastrophes, self-organization, and related issues in time series analysis, stationarity, modeling and hypothesis testing, probability, and experimental design. The analytic techniques discussed include several variants of the fractal dimension, several types of entropy, phase-space and state-space diagrams, recurrence analysis, spatial fractal analysis, oscillation functions, polynomial and Marquardt nonlinear regression, Markov chains, and symbolic dynamics. The book outlines the analytic requirements faced by social scientists and how they differ from those of mathematicians and natural scientists. It includes chapters centered on theory and procedural explanations for running the analyses with pertinent examples and others that illustrate applications where a particular form of analysis is seen in the context of a research problem. This combination of approaches conveys theoretical and practical knowledge that helps you develop skill and expertise in framing hypotheses dynamically and building viable analytic models to test them. (shrink)

A (to our knowledge) novel Generalized Nonlinear Schrödinger equation based on the modifications of Nottale-Cresson’s fractal-scale calculus and resulting from the noncommutativity of the phase space coordinates is explicitly derived. The modifications to the ground state energy of a harmonic oscillator yields the observed value of the vacuum energy density. In the concluding remarks we discuss how nonlinear and nonlocal QM wave equations arise naturally from this fractal-scale calculus formalism which may have a key role in the (...) final formulation of Quantum Gravity. (shrink)

With renewable energy being increasingly connected to power grids, pumped storage plants play a very important role in restraining the fluctuation of power grids. However, conventional control strategy could not adapt well to the different control tasks. This paper proposes an intelligent nonlinear model predictive control strategy, in which hydraulic-mechanical and electrical subsystems are combined in a synchronous control framework. A newly proposed online sequential extreme learning machine algorithm with forgetting factor is introduced to learn the dynamic behaviors of (...) the coupling system. Specifically, the initial learning parameters are optimized by prior-knowledge learning and a new self-adaptive adjustment strategy is also put forward. Subsequently, the stair-like control strategy and artificial sheep algorithm are used in rolling the optimization mechanism to replace the existing complex differential geometric solutions. Comparative experiments are carried out under different working conditions based on a PSP in China. The results show that the influence from coupling factors can be considerable and the proposed MPC strategy indicates superiority in voltage and load adjustment as well as the frequency oscillation suppression. (shrink)

Climate dynamics and trends have significant environmental and socioeconomic impacts; however, in the Benin Republic, they are generally studied with diverse statistical methods ignoring the nonstationarity, nonlinearity, and self-similarity characteristics contained in precipitation time series. This can lead to erroneous conclusions and an unclear understanding of climatic dynamics. Based on daily precipitation data observed in the six synoptic stations of Benin Republic, in the period from 1951 to 2010, we have proposed determining the local trends of precipitations, investigating precipitation (...) class='Hi'>nonlinear dynamics, and assessing climatic shift in the study period by Ensemble Empirical Mode Decomposition and Multifractal Detrended Fluctuation Analysis method. To overcome the detrending issue in the standard MFDFA method, the EEMD algorithm is embedded into the MFDFA. The study period is subdivided into three subperiods: 1951–1970, 1971–1990, and 1991–2010. Intrinsic Mode Functions are obtained according to the climatic region in which the stations are located. Results show that precipitation variation is significantly governed by the five first IMFs, in which oscillation periods vary from 1 to 25 days. The trend curves decrease at all the synoptic stations, and their slope values vary accordingly to the subperiods. Referring to the values of the multifractal spectrum parameters, α 0, and the width of the spectrum w, consistent changes are observed regardless of the subperiods and the concerned stations. The spatial and temporal variability of precipitation indicates that the multifractal properties are good indicators for assessing changes in precipitation dynamics, and the changes in their features could be explained by the global change than by the local climate variation. Despite the observed differences in multifractal spectra properties from the three subperiods, it is not possible to verify the subdivision of 1951–2010 in three subperiods as it is done by previous studies in West Africa. Our findings can be used in the validation of global and regional climate models since a valid model should explain empirically detected scaling properties in observed data. (shrink)

Walter Freeman's theory of nonlinear neurodynamics has had a major impact on brain dynamics in modern cognitive neuroscience. Steven Bressler's theory of neurocognitive networks follows from Freeman's work, and his empirical evidence for the coordination of cortical areas by phase-coupled beta rhythms in large-scale cognitive brain networks supports Freeman's ideas on nonlinear brain dynamics. Bressler's work, taking Freeman's concepts into the realm of cognitive neurodynamics, also supports Scott Kelso's theory of metastability in coordination dynamics. The aims of the (...) present paper are threefold: 1) to explore relevant concepts in Freeman's theoretical framework, 2) unify it and provide empirical support for it, and 3) explore its import for the study of consciousness. We take as our point of departure Bressler and Kelso's 2016 publication on cortical coordination dynamics and the Freeman-Kelso dialogue noted in the literature on this topic. In our view, Freeman's conceptual framework has great explanatory power. Its grounding in a dynamical systems approach to the study of brain function, and its incorporation of embodiment, make it relevant to the study of consciousness. Thus, with Freeman, we argue that the scientific understanding of consciousness must move beyond notions of computationalism and mental representations, to employ concepts of nonlinear brain dynamics and embodiment. (shrink)

Recently, megastable systems have grabbed many researchers’ interests in the area of nonlinear dynamics and chaotic systems. In this paper, the oscillatory terms’ coefficients of the simplest megastable oscillator are forced to blink in time. The forced system can generate an infinitive number of hidden attractors without changing parameters. The behavior of these hidden attractors can be chaotic, tori, and limit cycle. The attractors’ topology of the system seems unique and looks like picture frames. Besides, the existence of (...) different coexisting attractors with different kinds of behaviors reflects the system's high sensitivity. Using the sample entropy algorithm, the system’s complexity for different initial values is assessed. In addition, the circuit of the introduced forced system is designed, and the possibility of implicating the system with analog elements is investigated. (shrink)

This work introduces a three-dimensional, highly nonlinear quadratic oscillator with no linear terms in its equations. Most of the quadratic ordinary differential equations such as Chen, Rossler, and Lorenz have at least one linear term in their equations. Very few quadratic systems have been introduced and all of their terms are nonlinear. Considering this point, a new quadratic system with no linear term is introduced. This oscillator is analyzed by mathematical tools such as bifurcation and Lyapunov (...) exponent diagrams. It is revealed that this system can generate different behaviors such as limit cycle, torus, and chaos for its different parameters’ sets. Besides, the basins of attractions for this system are investigated. As a result, it is revealed that this system’s attractor is self-excited. In addition, the analog circuit of this oscillator is designed and analyzed to assess the feasibility of the system’s chaotic solution. The PSpice simulations confirm the theoretical analysis. The oscillator’s time series complexity is also investigated using sample entropy. It is revealed that this system can generate dynamics with different sample entropies by changing parameters. Finally, impulsive control is applied to the system to represent a possible solution for stabilizing the system. (shrink)

The scope of the thermodynamic theory of nonlinear irreversible processes is widened to include the nonlinear stability analysis of system motion. The emphasis is shifted from the analysis of instantaneous energy flows to that of the average work performed by periodic nonlinear processes. The principle of virtual work separates dissipative and conservative forces. The vanishing of the work of conservative forces determines the natural period of oscillation. Stability is then determined by the variations of the dissipative forces (...) with amplitude of oscillation. If the work is a minimum, under certain conditions, the motion is stable. Reduction to linear analysis shows the coincidence with the impedance analysis of electrical circuit theory. The theory is applied to the analysis of temporal interactions of nonlinear irreversible processes in the particular cases of synchronization and hysteresis. Characteristic nonequilibrium phenomena of directional energy transfers, self-excitation, system passivity, wave modulation, and “beat” phenomena are observed. Possible relationships with biological processes are discussed. (shrink)

In this paper, some novel analytical and numerical techniques are introduced for solving and analyzing nonlinear second-order ordinary differential equations that are associated to some strongly nonlinear oscillators such as a quadratically damped pendulum equation. Two different analytical approximations are obtained: for the first approximation, the ansatz method with the help of Chebyshev approximate polynomial is employed to derive an approximation in the form of trigonometric functions. For the second analytical approximation, a novel hybrid homotopy with Krylov–Bogoliubov–Mitropolsky method (...) is introduced for the first time for analyzing the evolution equation. For the numerical approximation, both the finite difference method and Galerkin method are presented for analyzing the strong nonlinear quadratically damped pendulum equation that arises in real life, such as nonlinear phenomena in plasma physics, engineering, and so on. Several examples are discussed and compared to the Runge–Kutta numerical approximation to investigate and examine the accuracy of the obtained approximations. Moreover, the accuracy of all obtained approximations is checked by estimating the maximum residual and distance errors. (shrink)

This work utilizes examples from chemical sciences to present fundamentals of dialectics and synergetics. The laws of dialectics remain appropriate at the level of atoms, at the level of molecules, at the level of the reactions, and at the level of ideas. The law of the unity and conflict of opposites is seen, for instance, in the relationships between the ionization energy and electron affinity of atoms, between the forward and back reactions, as well as in the differentiation and integration (...) between the various areas of chemistry. The law of the passage of quantitative changes into qualitative changes describes the transformation properties of the compounds when the number and arrangement of atoms in the molecule undergoes changes. According to this law, the development is accompanied by breaks and jumps. This paper suggests equations for the description of these relationships. The law of the negation of the negation is manifested in the Periodic Law, in the evolution of ideas about the mutual transformation of chemical elements, in the development the concept of triads of elements, etc. Small changes in the conventional Periodic Table on the basis of previously rejected versions allow reflecting secondary and additional periodicity. Synergetics, similarly to dialectics, is dedicated to the studies of general laws of evolution. Synergetics includes highly advanced and specified ideas of dialectics. The cornerstone of synergetics is the principles of self-organization and nonlinearity. Mathematical development of these concepts was substantially facilitated by considering oscillating chemical reactions as an example. These reactions are quite complex and therefore provide adequate models for self-organization. Oscillations may exist only upon execution of specific thermodynamic, mathematical, and chemical conditions. Mechanisms of several oscillatory reactions are briefly reviewed. The construction of novel biomimetic or smart materials based on oscillating reactions is described. (shrink)

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly by obtaining the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobian elliptic functions. We solve the cubic-quintic Duffing equation under arbitrary initial conditions. Physical applications are provided. The solution to the mixed parity Duffing oscillator is also formally derived. We illustrate the (...) obtained results with concrete examples. We give high accurate trigonometric approximations to the Jacobian function cn. (shrink)

Meditation is an umbrella term for a number of mental training practices designed to improve the monitoring and regulation of attention and emotion. Some forms of meditation are now being used for clinical intervention. To accompany the increased clinical interest in meditation, research investigating the neural basis of these practices is needed. A central hypothesis of contemplative neuroscience is that meditative states, which are unique on a phenomenological level, differ on a neurophysiological level. To identify the electrophysiological correlates of meditation (...) practice, the electrical brain activity of highly skilled meditators engaging in one of six meditation styles was recorded. A mind-wandering task served as a control. Lempel–Ziv complexity showed differences in nonlinear brain dynamics during meditation compared with mind wandering, suggesting that meditation, regardless of practice, affects neural complexity. In contrast, there were no differences in power spectra at six different frequency bands, likely due to the fact that participants engaged in different meditation practices. Finally, exploratory analyses suggest neurological differences among meditation practices. These findings highlight the importance of studying the electroencephalography correlates of different meditative practices. (shrink)

Recent work by Robert Batterman and Alexander Rueger has brought attention to cases in physics in which governing laws at the base level “break down” and singular limit relations obtain between base- and upper-level theories. As a result, they claim, these are cases with emergent upper-level properties. This paper contends that this inference—from singular limits to explanatory failure, novelty or irreducibility, and then to emergence—is mistaken. The van der Pol nonlinear oscillator is used to show that there can (...) be a full explanation of upper-level properties entirely in base-level terms even when singular limits are present. Whether upper-level properties are emergent depends not on the presence of a singular limit but rather on details of the ampliative approximation methods used. The paper suggests that focusing on explanatory deficiency at the base level is key to understanding emergence in physics. (shrink)

A signal development in contemporary physics is the widespread use, in explanatory contexts, of highly idealized models. This paper argues that some highly idealized models in physics have genuine explanatory power, and it extends the explanatory role for such idealizations beyond the scope of previous philosophical work. It focuses on idealizations of nonlinear oscillator systems.

Due to the introduction of memristors, the memristor-based nonlinear oscillator circuits readily present the state initial-dependent multistability, i.e., coexisting multiple attractors. The dimensionality reduction modeling for a memristive circuit is carried out to realize accurate prediction, quantitative analysis, and physical control of its multistability, which has become one of the hottest research topics in the field of information science. Based on these considerations, this paper briefly reviews the specific multistability phenomenon generating from the memristive circuit in the voltage-current (...) domain and expounds the multistability control strategy. Then, this paper introduces the accurate flux-charge constitutive relation of memristors. Afterwards, the dimensionality reduction modeling method of the memristive circuits, i.e., the incremental flux-charge analysis method, is emphatically introduced, whose core idea is to implement the explicit expressions of the initial conditions in the flux-charge model and to discuss the feasibility and effectiveness of the multistability reconstitution of the memristive circuits using their flux-charge models. Furthermore, the incremental integral transformation method for modeling of the memristive system is reviewed by following the idea of the incremental flux-charge analysis method. The theory and application promotion of the dimensionality reduction modeling and multistability reconstitution are proceeded, and the application prospect is prospected by taking the synchronization application of the memristor-coupled system as an example. (shrink)

The relevance of nonlinear dynamics to calcium metabolism led us to reevaluate the role of Ca-regulating hormones in Ca homeostasis. We suggest that, firstly, the main Ca metabolic functions in rat-bone and gut - are organized as dynamic entities able to generate various temporal expressions, including self-oscillating patterns and, secondly, Ca homeostasis results from interaction between both metabolic and hormonal oscillators. Following this schema, a major role for the hormonal system, with its circadian pattern, could be to act directly (...) on metabolic functions or indirectly through feeding behaviour, in order to optimize, coordinate and synchronize the Ca fluxes at ECF level. (shrink)

By emphasizing nonlinear dynamics between appraisal and emotions, Lewis's model provides a valuable platform for integrating psychological and neural perspectives on the emotion-cognition interface. In this commentary, I discuss the role of neuroscience in shaping new conceptualizations of emotion and the putative role of theta oscillation within frontocingulate pathways in depression, a syndrome in which emotion-cognition relations are dysfunctional.

A system of nonlinear integro-differential equations is derived for the motion of the classical electron with a rigid and spherically symmetric 3D gaussian distribution of charge. The equations are analyzed for stability around the state of rest and of uniform rectilinear motion with velocity small with respect to the velocity of light. The extremely high-frequency and radiationless micro-oscillations that the electron executes when disturbed from the equilibrium states show the inconsistency of the Abraham-Lorentz equation and of all concepts associated (...) with this equation, like the notion that the electron may have a mass of electromagnetic origin. (shrink)

Throughout the twentieth century, women's magazines in the United States have socialized their readers to the “proper” expression of love and anger in marriage. Our analysis of a random sample of marital advice articles from 1900 to 1979 examines this cultural convergence of gender, marriage, and emotion. A qualitative analysis identifies techniques for socializing readers to the emotional culture of marriage and shows a historical change toward equating love with self-fulfillment and advocating the expression of anger. A quantitative analysis then (...) specifies the timing of changes in emotion norms and showed an oscillation between modern and traditional norms that is related to waves of political liberation versus oppression. We discuss the relation between emotion norms and behavior and explore the effect of group conflict on the production of emotional culture. Emotion norms have become less rigid and more tolerant of diversity; but gender differences persist, and women are still responsible for maintaining intimate relationships. Historical trends in love and anger norms are nonlinear, not a continuous shift toward individualism, self-development, and free expression, as suggested by recent cultural theories. (shrink)

Patterns resulting from the sole interplay between reaction and diffusion are probably involved in certain stages of morphogenesis in biological systems, as initially proposed by Alan Turing. Self-organization phenomena of this type can only develop in nonlinear systems (i.e. involving positive and negative feedback loops) maintained far from equilibrium. We present Turing patterns experimentally observed in a chemical system. An oscillating chemical reaction, the CIMA reaction, is operated in an open spatial reactor designed in order to obtain a pure (...) reaction-diffusion system. The two types of Turing patterns observed, hexagonal arrays of spots and parallel stripes, are characterized by an intrinsic wavelength. We identify the origin of the necessary difference of diffusivity between activator and inhibitor. We also describe a pattern growth mechanism by spot splitting that recalls cell division. (shrink)

Analogue Gravity Phenomenology is a collection of contributions that cover a vast range of areas in physics, ranging from surface wave propagation in fluids to nonlinear optics. The underlying common aspect of all these topics, and hence the main focus and perspective from which they are explained here, is the attempt to develop analogue models for gravitational systems. The original and main motivation of the field is the verification and study of Hawking radiation from a horizon: the enabling feature (...) is the possibility to generate horizons in the laboratory with a wide range of physical systems that involve a flow of one kind or another. The years around 2010 and onwards witnessed a sudden surge of experimental activity in this expanding field of research. However, building an expertise in analogue gravity requires the researcher to be equipped with a rather broad range of knowledge and interests. The aim of this book is to bring the reader up to date with the latest developments and provide the basic background required in order to appreciate the goals, difficulties and success stories in the field of analogue gravity. Each chapter of the book treats a different topic explained in detail by the major experts for each specific discipline. The first chapters give an overview of black hole spacetimes and Hawking radiation before moving on to describe the large variety of analogue spacetimes that have been proposed and are currently under investigation. This introductory part is then followed by an in-depth description of what are currently the three most promising analogue spacetime settings, namely surface waves in flowing fluids, acoustic oscillations in Bose-Einstein condensates and electromagnetic waves in nonlinear optics. Both theory and experimental endeavours are explained in detail. The final chapters refer to other aspects of analogue gravity beyond the study of Hawking radiation, such as Lorentz invariance violations and Brownian motion in curved spacetimes, before concluding with a return to the origins of the field and a description of the available observational evidence for horizons in astrophysical black holes. (shrink)

The elliptical orbits resulting from Newtonian gravitation are generated with a multifaceted symmetry, mainly resulting from their conservation of both angular momentum and a vector fixing their orientation in space—the Laplace or Runge-Lenz vector. From the ancient formalisms of celestial mechanics, I show a rather counterintuitive behavior of the classical hydrogen atom, whose orbits respond in a direction perpendicular to a weak externally-applied electric field. I then show how the same results can be obtained more easily and directly from the (...) intrinsic symmetry of the Kepler problem. If the atom is subjected to an oscillating electric field, it enjoys symmetry in the time domain as well, which is manifest by quasi-energy states defined only modulo ħω. Using the Runge-Lenz vector in place of the radius vector leads to an exactly-solvable model Hamiltonian for an atom in an oscillating electric field—embodying one of the few meaningful exact solutions in quantum mechanics, and a member of an even more exclusive set of exact solutions having a time-dependent Hamiltonian. I further show that, as long as the atom suffers no change in principal quantum number, incident radiation will produce harmonic radiation with polarization perpendicular to the incident radiation. This unusual polarization results from the perpendicular response of the wavefunction, and is distinguished from most usual harmonic radiation resulting from a scalar nonlinear susceptibility. Finally, I speculate on how this radiation might be observed. (shrink)

Introduction to complexity and complex systems -- Introduction to large linear systems -- Introduction to biochemical oscillators and nonlinear biochemical systems -- Modularity, redundancy, degeneracy, pleiotropy and robustness in complex biological systems -- The evolution of biological complexity; invertebrate immune systems -- Irreducible and specified complexity in living systems -- The complex adaptive and innate human immune systems -- Complexity in quasispecies : microRNAs -- Introduction to complexity in economic systems -- Complexity in quasispecies : micrornas -- Dealing with (...) complexity. (shrink)

A Quantum Effect in the Classical Limit: Non-equilibrium Tunneling in the Duffing OscillatorAlec Maassen van den BrinkRCAS, Academia Sinica, Taiwanemail: [email protected] Duffing model is an oscillator with weak near-resonant driving, damping, and nonlinearity. For certain parameters, the stationary amplitude and phase bifurcate depending on initial conditions, and vary widely from one stable branch to the other. Due to this sensitivity, the system can be used for constructing detection devices.In recent years, an implementation using superconducting devices—the so-called Josephson bifurcation amplifier (...) (JBA)—has been successfully used experimentally for superconducting qubit readout. In the experimental literature, the JBA is often taken as classical. However, for e.g. understanding how the stability of the stationary states is modified by tunneling, a proper quantum analysis is necessary. Such tunneling transitions would be an error process from the point of view of detecto. (shrink)

We present a classical hydrodynamic analog of free relativistic quantum particles inspired by de Broglie’s pilot wave theory and recent developments in hydrodynamic quantum analogs. The proposed model couples a periodically forced Klein–Gordon equation with a nonrelativistic particle dynamics equation. The coupled equations may represent both quantum particles and classical particles driven by the gradients of locally excited Faraday waves. Exact stationary solutions of the coupled system reveal a highly nonlinear mechanism responsible for the self-propulsion of free particles, leading (...) to the onset of unsteady motion. Although the model is essentially nonrelativistic, a stabilizing mechanism for any particle traveling close to the wave signaling speed emerges through the coupling with the wavefield. Consequently, inline particle oscillations comparable to de Broglie’s wavelength are realized through this fully-classical model, suggesting a new classical interpretation for the motion of relativistic quantum particles. (shrink)

This major work covers almost all that has been learned about the acoustics of stringed instruments from Helmholtz's 19th-century theoretical elaborations to recent electroacoustic and holographic measurements.Many of the results presented here were uncovered by the author himself over a 20-year period of research on the physics of instruments in the violin family. Lothar Cremer is one of the world's most respected authorities on architectural acoustics and, not incidentally, an avid avocational violinist and violist.The book - which was published in (...) German in 1981 - first of all meets the rigorous technical standards of specialists in musical acoustics. But it also serves the needs and interests of two broader groups: makers and players of stringed instruments are expressly addressed, since the implications of the mathematical formulations are fully outlined and explained; and acousticians in general will find that the work represents a textbook illustration of the application of fundamental principles and up-to-date techniques to a specific problem.The first - and longest - of the book's three parts investigates the oscillatory responses of bowed strings. The natural nonlinearities that derive from considerations of string torsion and bending stiffness are deftly handled and concisely modeled.The second part deals with the body of the instrument. Special attention is given to the bridge, which transmits the oscillations of the strings to the wooden body and its air cavity. In this case, linear modeling proves serviceable for the most part - a simplification that would not be possible with lute - like instruments such as the guitar.The radiation of sound from the body into the listener's space, which is treated as an extension of the instrument itself, is the subject of the book's final part.Lothar Cremer is Professor Emeritus at the Technical University of Berlin, where he served as director of the Institute for Acoustical Engineering. (shrink)

A tight control of intracellular [Ca2+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2+}$$\end{document}] is essential for the survival and normal function of cells. In this study we investigate key mechanistic steps by which calcium is regulated and calcium oscillations could occur using in silico modeling of membrane transporters. To do so we give a deterministic description of intracellular Ca2+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2+}$$\end{document} dynamics using nonlinear dynamics in order to understand Ca2+\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2+}$$\end{document} signaling. We first present the ordinary differential equations system for cell calcium kinetics and make a preliminary work on Sobol indices. We then describe and analyze complex transporters action. Besides, we analyze the whole system. We finally perform numerical simulations and compare our results to real data. (shrink)

Various complex dynamics in ecologic-economic systems are presented with an emphasis upon models of global warming dynamics and fishery dynamics. Chaotic and catastrophic dynamic patterns are shown to be possible, along with other complex dynamics arising from nonlinearities in such combined systems. Problems associated with amplified oscillations due to these nonlinear interactions in the combined interactions of human economic decisionmaking with ecological dynamics are identified and discussed. Implications for policy are examined with strong recommendations for greater emphasis in particular (...) upon the precautionary principle to avoid catastrophic collapses beyond critical thresholds and the scale-matching principle to ensure that efforts to manage complex nonlinear dynamics are directed at the appropriate levels of ecologiceconomic interaction. Key Words: complex dynamics, chaos, catastrophe, fisheries, global warming.. (shrink)

This paper presents a stylized overview of the process of transition from planned command socialism to mixed market capitalism in stages, each involving nonlinear complex dynamical phenomena. The end of the command form arises out of a chaotic hysteretic long wave investment cycle. After the former institutional structure disappears a coordination failure brings about macroeconomic collapse. As recovery emerges various complex fluctuations of employment appear as government labor policies oscillate.

This article develops a theory of serial reading rethought as a theory of serial unreading, a processual form of attention directed toward virtual principles of production, whether a procedure, formula, operation, algorithm, or even an obsessive compulsion. The material specificity of the series’ parts matter but only insofar as the concrete detail suggests its own unwinding, the virtual and nonlinear path to its current state. In other words, the value or facility of serial unreading includes rediscovering the rules governing (...) the production of the work. Thus, unreading is not necessarily an end in itself. In fact, by dispersing and expanding our attentional field, acts of unreading make us better readers of nonserial works. In drawing our attention to their governing rules, operations, or patterns of production, serial works generate a primary principle of criticism: the impulse to reverse engineer the object. To show how this works, I examine three twentieth-century items—one series of minimalist drawings and two poetic works. By choosing a small cluster of examples I aim to mimic the oscillations of serial unreading, the way audiences are required to move from one member of the series to the transtextual techne that produced it. By withholding or abjecting the single finished object (often projecting divergent endlessness), serial art projects its governing intention in terms of techne—which subsists virtually along with the actual work as its formal and efficient cause—rather than deep expression. Moving quickly among individual works, I try to show how the feeling of navigating their expanded, multidirectional fields leads us on a cognitive path (whether we take it or not) toward criticism. (shrink)

This article presents a novel control approach, hybrid neuro-fuzzy, for the load frequency control of a four-area interconnected power system. The advantage of this controller is that it can handle nonlinearities, and at the same time, it is faster than other existing controllers. The effectiveness of the proposed controller in increasing the damping of local and inter-area modes of oscillation is demonstrated in a four-area interconnected power system. Areas 1 and 2 consist of a thermal reheat power plant, whereas Areas (...) 3 and 4 consist of a hydropower plant. Performance evaluation is carried out by using fuzzy, artificial neural network, adaptive neuro-fuzzy inference system, and conventional proportional and integral control approaches. Four different models with different controllers are developed and simulated, and performance evaluations are carried out with said controllers. The result shows that the intelligent HNF controller has improved dynamic response and is at the same time faster than ANN, fuzzy, and conventional PI controllers. (shrink)

The Belousov-Zhabotinsky reaction is a far-from-equilibrium reaction involving bromine and citric acid that results in the establishment of a nonlinear chemical oscillator. In addition to providing a paradigmatic chemical model of nonequilibrium biological phenomena, the mathematical models of the BZ reaction are theoretically interesting. The present article concentrates on the personal trajectory of Boris Pavlovich Belousov ; it attempts to explain what, in his routine work in applied and industrial chemistry, could have induced his discovery. The second section (...) describes Belousov’s reaction as presented in his short paper in the proceedings of the institute where he was employed , this being the only paper on his reaction that he was able to publish during his lifetime. The third section presents Belousov’s biography. I then discuss Belousov’s biography in the light of the paper representing his reaction. I have tried to find the prerequisites of Belousov’s discovery in his research in analytical chemistry and in toxicology and radiation medicine . The sixth section attempts an explanation of the ideological underpinning of Belousov’s reaction—the dialectics of continuity and discontinuity popular among Soviet scholars at that time. (shrink)

The brain, rather than being homogeneous, displays an almost infinite topological genus, since it is punctured with a high number of “cavities”. We might think to the brain as a sponge equipped with countless, uniformly placed, holes. Here we show how these holes, termed topological vortexes, stand for nesting, non-concentric brain signal cycles resulting from the activity of inhibitory neurons. Such inhibitory spike activity is inversely correlated with its counterpart, i.e., the excitatory spike activity propagating throughout the whole brain tissue. (...) We illustrate how Pascal’s triangles and linear and nonlinear arithmetic octahedrons are capable of describing the three-dimensional random walks generated by the inhbitory/excitatory activity of the nervous tissue. In case of nonlinear 3D paths, the trajectories of excitatory spiking oscillations can be depicted as the operation of filling the numbers of octahedrons in the form of “islands of numbers”: this leads to excitatory neuronal assemblies, spaced out by empty areas of inhibitory neuronal assemblies. These mathematical procedures allow us to describe the topology of a brain of infinite genus, to represent inhibitory neurons in terms of Betti numbers and to highlight how spike diffusion in neural tissues is generated by the random activation of tiny groups of excitatory neurons. Our approach suggests the existence of a strong mathematical background subtending the intricate oscillatory activity of the central nervous system. (shrink)

The temporal behaviour of the nonlinear compartmental model we have developed for rat calcium metabolism is discussed with respect to the theoretical properties of the self-oscillating autocatalytic subunit around which the model is constructed. Depending on the approximations made, this subunit is described by a minimal two-variable model, SU2, or by a three-variable one, SU3. The diversity of the theoretical dynamic behaviours possible with SU2 is greatly increased with SU3. But the identification of SU3 parameter values in three different (...) experimental situations reveals that biological constraints efficiently preserve a simple circadian rhythm for bone metabolism. This analysis indicates the significant contribution of the available bone crystal pool to the dynamic organization of this tissue, and hence to extracellular calcium homeostasis. (shrink)

Travelling waves crossing the nervous networks at mesoscopic/macroscopic scales have been correlated with different brain functions, from long-term memory to visual stimuli. Here we investigate a feasible relationship between wave generation/propagation in recurrent nervous networks and a physical/chemical model, namely the Belousov–Zhabotinsky reaction. Since BZ’s nonlinear, chaotic chemical process generates concentric/intersecting waves that closely resemble the diffusive nonlinear/chaotic oscillatory patterns crossing the nervous tissue, we aimed to investigate whether wave propagation of brain oscillations could be described in terms (...) of BZ features. We compared experimentally detected oscillations during the spontaneous activity of the brain with BZ-like concentric waves simulated by a recently introduced artificial network. The observed overlap and agreement between simulated and measured oscillatory patterns suggests that changes in cortical areas’ neural activity might be described in terms of a recognizable diffusion pattern. We describe biological plausibility, benefits and limits of our approach and discuss the relationship among BZ-like networks, Pandemonium-like architectures and the spontaneous activity of the brain. ER -. (shrink)

Two classic learning situations are critically reviewed and interpreted from a synergetic point of view: human learning of complex skills, and animal discrimination learning. Both show typical characteristics of nonlinear phase transitions: instability, fluctuations, critical slowing down and reorganisation. Plateaus in the acquisition curves of complex skills can be viewed as phases of arrested progress in which a reorganisation of simple skills is necessary before their integration into complex units is possible. Fluctuations and critical slowing down are expressed in (...) instances of "vicarious trial-and-error," which describe the oscillating behavior of rats at a choice point that is shown only just before discrimination learning is completed. It is concluded that education might pay more attention to the role of individual learning rhythms. (shrink)

This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy. The Riccati equation, a nonlinear equation that can be linearized, has the potential to link (...) these two worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in a consistent mathematical description of physical laws. (shrink)

The nonlinearity of a composite system, whereby certain of its features (including powers and behaviors) cannot be seen as linear or other broadly additive combinations of features of the system's composing entities, has been frequently seen as a mark of metaphysical emergence, coupling the dependence of a composite system on an underlying system of composing entities with the composite system's ontological autonomy from its underlying system. But why think that nonlinearity is a mark of emergence, and moreover, of metaphysical rather (...) than merely epistemological emergence? Are there diverse ways in which nonlinearity might enter into an account of properly metaphysical emergence? And what are the prospects for there actually being phenomena that are metaphysically emergent in any available sense? Here I explore the mutual bearing of nonlinearity and metaphysical emergence, with an eye towards answering these and related questions. (shrink)