Nonlinear Dynamical Systems Theory Research Articles

Spatial chaos on surface and its associated bifurcation and Feigenbaum problem

The qualitative theory of nonlinear spatial dynamical systems has attracted increasing attention recently. In particular, we have studied construction of spatial periodic orbits, dynamical behaviors of spatial chaos in the sense of Li–Yorke–Marotto, spatial Lyapunov exponents, control and generalized synchronization of spatial chaotic systems. In this paper, we apply a special mathematical transform to obtain spatial chaos on surface and its associated bifurcation and Feigenbaum problem. The 2D Logistic system is used for illustration. In addition, we also illustrate the difference in essence about chaos on surface as high-dimensional spatial system and dimension in phase space and we also analyze the parallel and different characters of theory system between 1D and 2D Logistic system.

Bodily synchronization and ecological validity: a relevant concern for nonlinear dynamical systems theory.

Bodily synchronization and ecological validity: a relevant concern for nonlinear dynamical systems theory.

Empirical Forecasting of HF-Radar Velocity Using Genetic Algorithms

We present a coastal ocean current forecasting system using exclusively past observations of a high-frequency radar (HF-Radar). The forecast is made by developing a new approach based on physical and mathematical results of the nonlinear dynamical systems theory that allows to obtain a predictive equation for the currents. Using radial velocities from two HF-Radar stations, the spatiotemporal variability of the fields is first decomposed using the empirical orthogonal functions. The amplitudes of the most relevant modes representing their temporal evolution are then approximated with functions obtained through a genetic algorithm. These functions will be then combined to obtain the hourly currents at the area for the next 36 h. The results indicate that after 4 h and for a horizon of 24 h, the computed predictions provide more accurate current fields than the latest available field (i.e., persistent field).

Modified plasma–fluid equations from nonstandard Lagrangians with applications to nuclear fusion

Nonstandard Lagrangian dynamics have gained great interest recently, in particular within the theory of nonlinear differential equations and dissipative dynamical systems. In this paper, we address their implications in plasma–fluid dynamics. The mathematical settings are constructed starting from the modified Vlasov–Boltzmann transport equation, which is derived from modified Euler–Lagrange equations of motion. Far from giving a self-consistent nonstandard Lagrangian theory of plasma–fluid dynamics, in this paper we have just introduced the basic settings and discussed some illustrative examples that such a modified theory should have in plasma–fluid theory and nuclear fusion.

Human Brain Networks: Spiking Neuron Models, Multistability, Synchronization, Thermodynamics, Maximum Entropy Production, and Anesthetic Cascade Mechanisms

Advances in neuroscience have been closely linked to mathematical modeling beginning with the integrate-and-fire model of Lapicque and proceeding through the modeling of the action potential by Hodgkin and Huxley to the current era. The fundamental building block of the central nervous system, the neuron, may be thought of as a dynamic element that is “excitable”, and can generate a pulse or spike whenever the electrochemical potential across the cell membrane of the neuron exceeds a threshold. A key application of nonlinear dynamical systems theory to the neurosciences is to study phenomena of the central nervous system that exhibit nearly discontinuous transitions between macroscopic states. A very challenging and clinically important problem exhibiting this phenomenon is the induction of general anesthesia. In any specific patient, the transition from consciousness to unconsciousness as the concentration of anesthetic drugs increases is very sharp, resembling a thermodynamic phase transition. This paper focuses on multistability theory for continuous and discontinuous dynamical systems having a set of multiple isolated equilibria and/or a continuum of equilibria. Multistability is the property whereby the solutions of a dynamical system can alternate between two or more mutually exclusive Lyapunov stable and convergent equilibrium states under asymptotically slowly changing inputs or system parameters. In this paper, we extend the theory of multistability to continuous, discontinuous, and stochastic nonlinear dynamical systems. In particular, Lyapunov-based tests for multistability and synchronization of dynamical systems with continuously differentiable and absolutely continuous flows are established. The results are then applied to excitatory and inhibitory biological neuronal networks to explain the underlying mechanism of action for anesthesia and consciousness from a multistable dynamical system perspective, thereby providing a theoretical foundation for general anesthesia using the network properties of the brain. Finally, we present some key emergent properties from the fields of thermodynamics and electromagnetic field theory to qualitatively explain the underlying neuronal mechanisms of action for anesthesia and consciousness.

The Application of Random Noise Reduction By Nearest Neighbor Method To Forecasting of Economic Time Series

Abstract Since the deterministic chaos appeared in the literature, we have observed a huge increase in interest in nonlinear dynamic systems theory among researchers, which has led to the creation of new methods of time series prediction, e.g. the largest Lyapunov exponent method and the nearest neighbor method. Real time series are usually disturbed by random noise, which can complicate the problem of forecasting of time series. Since the presence of noise in the data can significantly affect the quality of forecasts, the aim of the paper will be to evaluate the accuracy of predicting the time series filtered using the nearest neighbor method. The test will be conducted on the basis of selected financial time series.

Multiscale Analysis of Acoustic Emission during Plastic Flow of Al and Mg Alloys: From Microseconds to Minutes

Recent studies of plastic deformation using high-resolution experimental techniques bear witness that deformation processes are often characterized by collective effects emerging on an intermediate scale between the scales describing the dynamics of individual crystal defects or the macroscopic plastic flow. In particular, the acoustic emission (AE) reveals intermittency of plastic deformation in various experimental conditions, which is manifested by the property of scale invariance, a characteristic feature of self-organized phenomena. Some materials, e.g., Al or Mg alloys, display a macroscopic discontinuity of plastic flow due to the Portevin-Le Chatelier effect or twinning. These materials are therefore of special interest for the study of collective effects in plasticity. The present work reviews the results of a multiscale investigation of AE accompanying plastic deformation of such model alloys. The AE is analyzed by methods borrowed from the theory of nonlinear dynamical systems, including statistical and multifractal analyses.

A novel Lie-group theory and complexity of nonlinear dynamical systems

The complexity of a dynamical system ẋ=f(x,t) is originated from the nonlinear dependence of f on x. We develop a full Lorentz type Lie-group for the augmented dynamical system in the Minkowski space. Depending on a signum function Sign≔sign(‖f‖2‖x‖2-2(f·x)2)=-sign(cos2θ), where θ is the intersection angle between x and f, we can derive a group-preserving scheme based on SOo(n,1), which has two branches: compact one and non-compact one. The barcode to show the complexity of nonlinear dynamical systems is obtained by plotting the value of Sign with respect to time. The Rössler equation possesses the most elementary chaotic mechanisms of stretching and folding, which is useful to validate the present theory of nonlinear dynamical system. Analyzing the barcode, we can point out the bifurcation of subharmonic motions and the chaotic range in the parameter space. The bifurcation diagram of the percentage of the first set of dis-connectivity A1-≔{sign(cosθ)=+1andsign(cos2θ)=+1} with respect to the amplitude of harmonic loading leads to a finer structure of devil staircase for the ship rolling oscillator, and the cascade of subharmonic motions to chaos for the Duffing oscillator.

Emotional inertia: A key to understanding psychotherapy process and outcome

The processes underlying psychotherapeutic change have increasingly been emphasized in both research and clinical practice. Nonlinear dynamical systems theory (NDS) offers a transdisciplinary scientific approach to the study of these processes. This paper introduces the NDS concept of “emotional inertia”, the property of human emotion by which it retains its course so long as it is not acted upon by an external force, as a key to understanding moment-by-moment and also longer-term change processes within psychotherapy. A testable mathematical model of emotional inertia is presented that represents specific impacts of psychotherapeutic processes on emotional dynamics over time. Emotional trajectories in phase space, treatment energy, and the interaction between them are the essential elements of the model, and a detailed explanation is provided. Procedures for testing this model are described, such as by tracking the movement of emotion in phase space within and across therapy sessions, along with clinical implications of the model, which can potentially help to make more clear the complementary roles of therapeutic force, timing, and leverage.

Message survival and decision dynamics in a class of reactive complex systems subject to external fields

In this study, the dynamics of decisions in complex networks subject to external fields are studied within a Markov process framework using nonlinear dynamical systems theory. A mathematical discrete-time model is derived using a set of basic assumptions regarding the convincement mechanisms associated with two competing opinions. The model is analyzed with respect to the multiplicity of critical points and the stability of extinction states. Sufficient conditions for extinction are derived in terms of the convincement probabilities and the maximum eigenvalues of the associated connectivity matrices. The influences of exogenous (e.g., mass media-based) effects on decision behavior are analyzed qualitatively. The current analysis predicts: (i) the presence of fixed-point multiplicity (with a maximum number of four different fixed points), multi-stability, and sensitivity with respect to the process parameters; and (ii) the bounded but significant impact of exogenous perturbations on the decision behavior. These predictions were verified using a set of numerical simulations based on a scale-free network topology.

Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit: analytic extension and numerical validation

In the research field of nonlinear dynamical system theory, it is well known that a homoclinic/heteroclinic point leads to unpredictable motions, such as chaos. Melnikov’s method enables us to judge whether the system has a homoclinic/heteroclinic orbit. Therefore, in order to assess a vessel's safety with respect to capsizing, Melnikov’s method has been applied for investigations of the chaos that appears in beam sea rolling. This is because chaos is closely related to capsizing incidents. In a previous paper (Maki et al. in J Mar Sci Technol 15:102–106, 2010), a formula to predict the capsizing boundary by applying Melnikov’s method to analytically obtain the non-Hamiltonian heteroclinic orbit was proposed. However, in that paper, only limited numerical investigation was carried out. Therefore, further comparative research between the analytical and numerical results is conducted, with the result being that the formula is validated.

Self-Organization and Explanatory Pluralism: Avoiding the Snares of Reductionism in Developmental Science

Over the last 20 years, the concept of self-organization has played a central role in efforts to establish a relational metatheoretical approach to the study of development. Yet the notion of self-organization that predominates in scientific discussions today has undergone a conceptual narrowing relative to the broader use and interpretation of self-organization that characterized the biologically-oriented systems thinking of Weiss and von Bertalanffy and was grounded in Kant's teleological account of organisms as natural ends or purposes. In this article, the author discusses how our conceptual use of self-organization in developmental science has changed under the influence of nonlinear dynamical systems theory and how these changes could actually foster rather than discourage new forms of reductionism in our thinking about development.

Robust Methods for the Prediction of Customer Demands based on Nonlinear Dynamical Systems

This paper investigates the forecasting accuracy of different prediction methods based on the theory of nonlinear dynamical systems and phase space reconstruction. In particular, a locally linear regression method is described. Different regularization methods to achieve better predictions in case of short time series as well as to increase the robustness against noise are specified. An evaluation by means of synthetic and real time series indicates high forecasting accuracy compared to established methods used as benchmarks.

A Nonlinear Lens on Berta Bornstein’s “Frankie”

This reading of Berta Bornstein's case of "Frankie" half a century after its publication focuses on the knowing attitude that pervades her description of child analysis and child development, and this attitude is used to explain interventions that appear harsh and anti-analytic to today's reader. Using concepts from nonlinear dynamic systems theory and, particularly, network theory help to both understand what is troubling in the case description and how these problematic features came to be part of the case description. The mistaken view that development is linear leads to attempts to get development "on track" and a view that the goal of analysis is well-defined psychological maturity, as opposed to the ongoing freedom to explore the psychological world in new and creative ways. Bornstein's authoritative style was not only coercive of her young patient but also of the reader who is invited to uncritically agree with her formulations. It is suggested that the appeal of this way of writing is best understood in the broad, historical context within which the work with Frankie was undertaken.

Dynamics of the biological system in turbellarians inhabiting the littoral water bodies of Lake Baikal (Phagocata sibirica) subjected to stress: Model of coupled bistable oscillators

An experimental investigation into the behavior of turbellarians inhabiting the water bodies of the basin of Lake Baikal (Phagocata sibirica (Zabussov, 1903)) under the influence of light, toxicants, and both these factors allows one to find biosystem features under effects other than homeostasis. It is revealed that, when the system is subjected to increasing light intensity, it demonstrates turbulent oscillation with the effects of synchronization and desynchronization. An analysis of the experimental results within the framework of the theory of nonlinear dynamical systems shows that the behavior of the turbellarian biosystem corresponds to the model of coupled bistable oscillators, the synchronization of which is induced by internal biosystem noise.

Discussion: Putting our Heads Together; Mentalizing Systems

This discussion reviews the psychoanalytic and developmental psychological theories that utilize nonlinear dynamic systems theory to situate Sperry's paper in its theoretical and historical context. The power of such nonlinear models for clinical work and theories of mind is argued. The essay then argues that the shift in theory entails a shift in ways of clinical writing and that the clinical discussion in the paper does not always feel consonant with the theory. Moving to a model of nonlinearity, intersubjectivity, and uncertainty is likely to entail shifts in conceptualizing and conveying clinical material.

A control theoretic framework for modular analysis and design of biomolecular networks

Control theory has been instrumental for the analysis and design of a number of engineering systems, including aerospace and transportation systems, robotics and intelligent machines, manufacturing chains, electrical, power, and information networks. In the past several years, the ability of de novo creating biomolecular networks and of measuring key physical quantities has come to a point in which quantitative analysis and design of biological systems is possible. While a modular approach to analyze and design complex systems has proven critical in most control theory applications, it is still subject of debate whether a modular approach is viable in biomolecular networks. In fact, biomolecular networks display context-dependent behavior, that is, the input/output dynamical properties of a module change once this is part of a network. One cause of context dependence, similar to what found in many engineering systems, is retroactivity, that is, the effect of loads applied on a module by downstream systems. In this paper, we focus on retroactivity and review techniques, based on nonlinear control and dynamical systems theory, that we have developed to quantify the extent of modularity of biomolecular systems and to establish modular analysis and design techniques.

Approach to Forecasting Occurrence of Slope Failure with Nonlinear Dynamical System

Landslides constitute a major geologic hazard because they are widespread and commonly occur in connection with other major natural disasters such as earthquakes, rainstorms, wildfires and floods. Nonlinear dynamical system (NDS) techniques have been developed to analyze chaotic time series data. According to NDS theory, the correlation dimension and predictable time scale are evaluated from a single observed time series. The Xintan landslide case study is presented to demonstrate that chaos exists in the evolution of a landslide and the predictable time scale must be considered. The possibility for long-term, medium-term and short-term prediction of landslide is discussed.

Foreword to the special issue on nonlinear economic dynamics

The theory of dynamical systems has been one of the most developed mathematical areas in the last 50years with a wide spectrum of applications ranging from physics to biology to economics. During the last decades, economic theory in particular has witnessed an important shift in methodology. The classical approach that describes economic outcomes as resulting from the choices of fully rational and identical economic agents has failed to explain many important features of economic complexity in the real world. Fueled by its inability to predict sudden changes and (financial and economic) crises, it has been criticized on a number of grounds. As a consequence, a growing interest in alternative approaches based on concepts like bounded rationality and heterogeneity, social interaction, and learning has emerged. Following this new paradigm, economics and finance are viewed as complex evolving systems characterized by emerging non-equilibrium situations. The corresponding mathematical modeling approaches require a wide range of analytical and numerical techniques, some of which are new and innovative. These techniques consist of a mix of theories, among them the qualitative theory of nonlinear dynamical systems, optimal control theory, game theory, and the theory of stochastic processes. The current status and the upcoming trends of these new modeling techniques and numerical toolswere presented at the InternationalConferenceonNonlinearEconomic

Action Recognition Based on Motion Representing and Reconstructed Phase Spaces Matching of 3D Joint Positions

This paper presents an efficient and novel framework for human action recognition based on representing the motion of human body-joints and the theory of nonlinear dynamical systems. Our work is motivated by the pictorial structures model and advances in human pose estimation. Intuitively, a collective understanding of human joints movements can lead to a better representation and understanding of any human action through quantization in the polar space. We use time-delay embedding on the time series resulting of the evolution of human body-joints variables along time to reconstruct phase portraits. Moreover, we train SVM models for action recognition by comparing the distances between trajectories of human body-joints variables within the reconstructed phase portraits. The proposed framework is evaluated on MSR-Action3D dataset and results compared against several state-of-the-art methods.