Nonlinear Dynamical Systems Theory Research Articles

Dark energy and dark matter interaction: A nonlinear dynamical system study

This paper presents a methodical dynamical analysis of the Big Bang model, taking into account dark matter and an arbitrary form of dark energy. Nevertheless, why dynamic analysis? The primary advantage of the dynamical analysis approach is that it obviates the necessity to precisely solve the system for every individual dependent variable. Instead, it enables the prediction of the overall evolutions. In this regard, there are some powerful tools of non-dynamical systems (theorems, perturbation, etc.) which can help us a lot. The model under consideration is a nonlinear system with two dimensions (2D). In order to investigate this, we employ a sophisticated form of nonlinear dynamical systems theory in our research. We have delved into the full potential of nonlinear dynamical system theory known until now in our work. This analysis has produced very interesting solutions, which are in quite a contrast with the linear analysis approach. The stability of the system is considered at length, and the corresponding evolutions of the universe are also discussed consequently. A handful number of figures are given to visualize the behavior of the evolutions. The plots that are presented here are vector field plots and a new plotting technique initiated by us and used in our earlier papers. Interestingly, our work indicates the universe might resemble “phantom” evolution. Also, among the fixed points of our model, one is able to escape the singular situation called “Big Rip”.

Analyzing physical activity impact based on ubiquitous nonlinear dynamics and electroencephalography data.

Understanding complex systems is made easier with the tools provided by the theory of nonlinear dynamic systems. It provides novel ideas, algorithms, and techniques for signal processing, analysis, and classification. Presently, these ideas are being applied to the investigation of how physiological signals evolve. The study applies nonlinear dynamics theory to electroencephalogram (EEG) signals to better comprehend the range of alcoholic mental states. One of the main contributions of this paper is an algorithm for automatically distinguishing between sober and drunken EEG signals based on their salient features. The study utilized various entropy-based features, including ApEn, SampEn, Shannon and Renyi entropies, PE, TS, FE, WE, and KSE, to extract information from EEG signals. To identify the most relevant features, the study employed ranking methods like T-test, Wilcoxon, and Bhattacharyya, and trained SVM classifiers with the selected features. The Bhattacharyya ranking method was found to be the most effective in achieving high classification accuracy, sensitivity, and specificity. Classification accuracy of 95.89%, the sensitivity of 94.43%, and specificity of 96.67% are achieved by the SVM classifier with radial basis function (RBF) for polynomial Kernel using the Bhattacharyya ranking method. From the result, it is clear that the model serves as a cost-effective and accurate decision-support tool for doctors in diagnosing alcoholism and for rehabilitation centres to monitor the effectiveness of interventions aimed at mitigating or reversing brain damage caused by alcoholism.

Non-smooth dynamics of a fishery model with a two-threshold harvesting policy

The non-linear dynamical systems theory helps implement regulatory measures to control the growth and evolution of various populations. While invasion by alien fish species is an emerging threat to native fish species in marine ecosystems, a suitable fishery management protocol needs to be incorporated in marine protected areas (MPAs) to mitigate the problem. We propose a policy of selective harvesting whenever the density of a species crosses a predefined threshold. An ecosystem with such a harvesting policy is modelled by a piece-wise smooth prey-predator type fishery model, where a control function defined by two thresholds evokes signals to cease or commence harvesting. This work reports the dynamics and bifurcations of a Filippov system, including the possibility of sliding mode dynamics. We show that the variations in the growth rate and the harvesting rate of the invasive fish species can lead to hysteresis through saddle–node and transcritical bifurcations and the appearance of a stable homoclinic orbit. We obtain the conditions for sliding domains, regular or virtual equilibria, boundary equilibria, pseudo-equilibria, and tangent points. Depending on the parameter choice, the system can stabilize at a regular equilibrium, a pseudo-equilibrium, or a pseudo-attractor, exhibiting multistability. We report the occurrence of regular or virtual equilibrium bifurcation, boundary node bifurcation and pseudo-saddle–node bifurcation. Our findings indicate that proper harvesting strategies can prevent the proliferation of invasive fish species in MPAs.

The effect of school size and class size on school preparedness.

The purpose of the present study was to understand students' school readiness as a function of student and teacher behaviors but also school size and class size using both linear and non-linear analytical approaches. Data came from 21,903 schools distributed across 80 countries as per the 2018 cohort of the PISA database. Results pointed to a preference for the Cusp model in that the relationship between school and class sizes with achievement proved to be best described by the non-linearity of the Cusp catastrophe model. The critical benchmarks were a school size of 801 students and a class size of 27 students for which increases beyond those thresholds were linked to nonlinearity and unpredictability in school readiness. For this reason, we suggest using the cusp catastrophe model from Nonlinear Dynamical Systems Theory (NDST) to understand more fully such complex phenomena.

Nonlinear Dynamic Systems Theory: Its Use in Understanding Psychological Development and the Process of Change in Psychoanalysis

ABSTRACT Nonlinear Dynamic Systems Theory has been used by researchers in development for over 25 years. This paper demonstrates how this theory applies to psychodynamic development throughout the life cycle and illustrates how it can be used to explain change in psychoanalytic treatment.

What Analysts Really Do: Psychoanalysis as a Nonlinear Dynamic System

ABSTRACT Of all current psychologies, psychoanalysis is the most engaged with the complex and chaotic dimensions of human development and suffering – intricate psychic processes, fluxing mental and physical states, unexpected shifts between clarity and confusion, big changes following small inputs, uncertainty about causes and locations – including whose thoughts belong to whom, and more. These crucial elements, however, have not always been taken up explicitly. Nonlinear dynamic systems theory (NLDS) provides an aesthetically pleasing and scientifically current framework for these factors and illuminates established clinical practice and theory. Through clinical examples, conceptual elaboration, and an account of the author’s growing interest in this field, this paper offers a user-friendly invitation to NLDS.

Introduction to the Section: Nonlinear Dynamic Systems Theory

ABSTRACT Nonlinear Dynamic Systems Theory is discussed by three child, adolescent, and adult analysts who use this theory in their thinking about development from infancy through old age and as a way of understanding the process of change in psychoanalysis. This theory explains how the systems of the brain, the body, the environment, and the culture interact and help determine character formation throughout the life cycle. Examples of development and psychoanalytic treatment are used to demonstrate the application of nonlinear dynamic systems.

The stabilization of uncertain dynamic systems involving the generalized Riemann-Liouville fractional derivative via linear state feedback control

The main objective of this paper is to investigate the stabilization problem of uncertain fractional-order dynamic systems (UFDSs) that involve the generalized Riemann-Liouville fractional derivative using a linear feedback controller. We establish several sufficient conditions for achieving Mittag-Leffler stabilization and asymptotic stabilization of UFDSs. The primary methodology employed in our study is Lyapunov's direct method (LDM), along with other inequality techniques, which have demonstrated significant efficacy in studying the stability theory of nonlinear dynamic systems. To demonstrate the effectiveness of the proposed approach, examples and corresponding simulations are provided.

Low-Power Fault-Tolerant Control for Nonideal High-Order Fully Actuated Systems

This article presents a novel observer-based fault-tolerant controller framework for a class of nonideal time-varying high-order fully actuated systems (HOFASs). The HOFAS theory is an emerging nonlinear dynamical system theory, which can yield global stability. In order to improve the ability of HOFAS theory to handle parameter uncertainties, actuator faults, and measurement noises, a nonlinear extended state observer and low-power fully actuated controller framework is established in this article. Particularly, the linear framework conforms to the generalized separation principle and highlights the advantages of HOFAS parametric design. Moreover, the work takes into account fault tolerance and noise suppression in engineering applications, reducing the dependence of HOFAS theory on model accuracy. The uniformly bounded stability and noise suppression performance are also proved theoretically and illustrated experimentally.

Can groundwater storage in turn affect the cryospheric variables? A new perspective from nonlinear dynamic causality detection

Despite that the supplying role of cryosphere (glaciers, permafrost, and snow) in groundwater storage (GWS) in Tibetan Plateau (TP) is well-known by comparing their long-term linear trends, the question whether GWS could in turn affect the variation of cryospheric variables remains controversial, since long-term trend analysis fails to distinguish the direction of their interplay. To find evidence of GWS causally affecting cryosphere, this research resorts to the causal inference community and investigates a novel causal interaction between GWS and cryosphere in TP: nonlinear dynamic causality (NDC), based on the Nonlinear Dynamic System (NDS) theory. The specific method applied is called Convergent Cross-mapping (CCM), which detects NDC between two targeted variables X and Y from both directions (X → Y, Y → X). Important findings are summarized as follows: (1) With CCM, NDCs with similar strengths are found from glaciers retreat, snowmelt, and permafrost thaw to GWS, respectively; (2) Also in the form of NDC, GWS is proven to reversely affect permafrost, but not to glacier and snow; (3) NDCs are also found between GWS and other hydrometeorological variables in TP, including lakes, soil moisture, precipitation, and temperature; (4) Some “nontraditional” NDCs from glaciers and lakes towards GWS are identified. Overall, using CCM, our new findings about NDC answer the controversial question of whether GWS could in turn affect cryosphere, completing previous conclusions about how GWS interplays with cryosphere in TP, and more importantly, this research would shed light on future causality detection in hydrology.

Transient Synchronization Stability of Grid-Forming Converter During Grid Fault Considering Transient Switched Operation Mode

With the increasing penetration of renewable energy generation, grid-forming (GFM) converters that simulate the dynamics of synchronous generator (SG) and actively participate in grid voltage and frequency regulation are increasingly employed. In GFM converter dominated power system, similar transient synchronization stability that is common in conventional power system also exists, which is closely related with the control dynamics of GFM converters. In this paper, the transient synchronization stability of GFM converters with virtual synchronous generator (VSG) control is focused on, with the influence of transient switched operation mode (TSOM) introduced by current saturation limitation considered especially. First, a novel transient equivalent motion model of VSG is developed. The model is a switched system and TSOM significantly changes the power angle characteristics. Based on power angle curve analysis and equal area method, the transient stability analysis is carried out from non-existence of equilibrium point and non-matching of accelerating-decelerating area. Analysis indicates that TSOM imposes another constraint to equilibrium point and reduces deceleration area, which worsens the transient stability. Further, employing the stability region theory of nonlinear dynamic systems, the stability region of VSG is visually displayed in phase plane. It is identified that the TSOM reduces the stability region and worsens the stability. Finally, the influence factors including power reference, grid voltage sag and grid impedance are discussed.

Detecting nonlinear information about drought propagation time and rate with nonlinear dynamic system and chaos theory

Current approaches for calculating propagation time and rate from meteorological to hydrological drought mainly focus on the time series of SPI and SRI, and most of these methods are conducted in time domain (e.g., correlation analysis) and frequency domain (e.g., wavelet analysis), which are regarded as linear statistic methods. Inaccuracy could emerge when the complexity and nonlinearity of drought propagation is addressed by such linear methods. In light of this problem, this research adopts Nonlinear Dynamic System (NDS) conducted in phase domain, which provides a nonlinear and systematic perspective on drought propagation. NDS may conditionally generate chaos (commonly known as “butterfly” phenomenon), which was not covered by previous studies on drought propagation. Assuming an underlying NDS for drought events, we demonstrate the NDS of drought propagation could generate chaos, and the nonlinear information about the propagation from meteorological to hydrological droughts can be detected within such chaotic NDS. The propagation time of 1–5 months and the propagation rate of 0.686 were found in the Pearl River Basin. In addition, the propagation time of 3, 7 and 11 months was also found in the Wei River Basin, to prove the broad applicability of our method. The results for these two basins are verified by previous studies.

Reframing Cognitive Science as a Complexity Science

Complexity science is an investigative framework that stems from a number of tried and tested disciplines-including systems theory, nonlinear dynamical systems theory, and synergetics-and extends a common set of concepts, methods, and principles to understand how natural systems operate. By quantitatively employing concepts, such as emergence, nonlinearity, and self-organization, complexity science offers a way to understand the structures and operations of natural cognitive systems in a manner that is conceptually compelling and mathematically rigorous. Thus, complexity science both transforms understandings of cognition and reframes more traditional approaches. Consequently, if cognitive systems are indeed complex systems, then cognitive science ought to consider complexity science as a centerpiece of the discipline.

The Genesis of Uncertainty: Structural Analysis of Stochastic Chaos in Finance Markets

The presented article is methodological in nature and is devoted to the analysis of observation series of financial asset quotation changes in capital markets. The most important feature of these processes is their instability, which manifests itself in high sensitivity to seemingly minor disturbing factors. This phenomenon is well-studied in the theory of nonlinear dynamical systems and is described by models of deterministic chaos. However, for the processes considered in the article, the dynamic instability of the immersion environment is exacerbated by stochastic uncertainty caused by random fluctuations in the pricing process. As a result, describing observation series of quotations of financial assets is difficult because it involves stochastic chaos. This article analyzes and classifies chaotic series of observations to help model and forecast related processes.

A creative new normal: Who can we be in 20 years?

Can we live better—with greater health, joy, purpose, and creative vision—toward a new type of “normality”? Picture a one-size-fits-all conformist norm. Now contrast a dynamic evolving process-picture, different for each of us, honoring a diversity of ways we can all creatively—while harmoniously—grow and contribute together. This article, from a new 2022 talk, continues discussion of Creativity, Chaos and Complexity (or NDS, nonlinear dynamical systems theory) from a 2021 keynote at the SOU Creativity Conference. Using five examples, it expands on one key feature, a Creative New Normal—with potential challenges and benefits, now and in 20 years. Drawn from chaos and complexity constructs, five colorful examples are used to paint one portrait of the future. A person, when creating, says “I am...” Alive!, Visionary, On a Path, Containing Worlds, and Empathetic. A shift could affect (a) our worldview—furthering a long awaited change in both worldview and view of self-in-world, consistent with an NDS paradigm shift, now 50 years old in the “hard sciences” but only beginning in social sciences; plus (b) educational practice—with an opportunity, given dramatic new interest in our universal “everyday creativity” (versus limitation to an eminent elite in arts and sciences) to open floodgates for creative development. Development of chaos and complexity intuition can boost further this facility. Although some still insist “I'm not creative!” too many have lacked resources, teachers, mentorship and confidence for creative self-development. Opening this creative path as a birthright and cultural expectation can help us all survive and thrive—for self, society, and our endangered planet.

Nonlinear dynamical systems to study epileptic seizures and extract average amount of mutual information from encephalographs – Part I

In this study, we apply the non-linear dynamical systems theory for the assessment built on recurrence-quantification analysis technique for characterizing—differentiating non-linear electro encephalograph (EEG) signals dynamics. The technique offers convenient quantifiable data plus information over normal, tumultuous, or probability and statistical stochastic properties of inherent systems dynamics theory. The R.Q.A-established processes as the quantifiable mathematical features of non-linear electroencephalograph signals dynamics. Average amount of mutual information (AAMI) applied to compute highly applicable feature-manifestation sub-sets out of R.Q.A-built centered-features. The chosen features were then fed into the computer using artificial intelligence based neural net-works for clustering the data of encephalograph-signals to identify ictic(i.e.,ictal), inter ictal, followed by state of controls. The study is implemented by validating R.Q.A with a data base for various issues of categorization. Results showed that the combination of five selected features created on AAMI attained the precision of100% and proves dominance of R.Q.A. Nonlinear dynamical control systems theory and analysis techniques centered on R.Q.A can be used as an appropriate methodology for distinguishing the non-linear systems dynamics of encephalograph signals data also epileptic seizures tracing.

Coupled economic oscillations — synchronization dynamical model

Purpose of this work is the research of the dynamical processes and in particular the phenomenon of the synchronization in an ensemble of coupled chaotic economic oscillators. Methods. The research methods are the qualitative and numerical methods of the theory of nonlinear dynamical systems and the theory of the bifurcations. Results. The nonlinear model of economic oscillator as the system of automatic control are considered. Such kind of general economic models are unsuitable for getting some concrete economic estimations and recommendations. But such kind models are very useful for a development the theory of the economic cycles, theory of the generation, interactions, synchronization of the cycles and so on. Our numerical experiments demonstrated a good enough qualitative similarity of an chaotic economic oscillations in our model and real economic cycles. The phenomen of the synchronization of the chaotic oscillations in the ensemble of coupled economic oscillators are considered, however the accuracy of the synchronization depends with couplings essentially.

Расчет оптимальных параметров систем управления нелинейными динамическими процессами импульсных преобразователей напряжения

A technique for choosing the optimal control system parameters for DC voltage converters is proposed, based on the joint application of the theory of linear automatic control systems and the theory of nonlinear dynamic systems. A small-signal structural dynamic model of an open loop of an automatic control system based on a direct step-up voltage converter with a control system for nonlinear dynamic processes based on a delayed feedback method is considered. Applying this model makes it possible to carry out a scientifically substantiated choice of the control system parameters for nonlinear dynamic processes using the methods of linear automatic control system theory. Bode diagrams of an open loop system are calculated without additional control of nonlinear dynamic processes and with additional control. The efficiency of using control systems for nonlinear dynamic processes is shown, which allows eliminating undesirable dynamic modes without additional parametric synthesis of the controller and, consequently, without reducing the system performance as a whole. In addition, applying these methods allows adjusting the controller parameters to increase the system performance without switching the system to undesirable dynamic modes. The results obtained can be applied at the stage of designing wide-class DC voltage pulse converters

Influence of arginine vasopressin on the ultradian dynamics of Hypothalamic-Pituitary-Adrenal axis.

Numerous studies on humans and animals have indicated that the corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP) stimulate both individually and synergistically secretion of adrenocorticotropic hormone (ACTH) by corticotropic cells in anterior pituitary. With aim to characterize and better comprehend the mechanisms underlying the effects of AVP on Hypothalamic-Pituitary-Adrenal (HPA) axis ultradian dynamics, AVP is here incorporated into our previously proposed stoichiometric model of HPA axis in humans. This extended nonlinear network reaction model took into account AVP by: reaction steps associated with two separate inflows of AVP into pituitary portal system, that is synthesized and released from hypothalamic parvocellular and magnocellular neuronal populations, as well as summarized reaction steps related to its individual and synergistic action with CRH on corticotropic cells. To explore the properties of extended model and its capacity to emulate the effects of AVP, nonlinear dynamical systems theory and bifurcation analyses based on numerical simulations were utilized to determine the dependence of ultradian oscillations on rate constants of the inflows of CRH and AVP from parvocellular neuronal populations, the conditions under which dynamical transitions occur due to their synergistic action and, moreover, the types of these transitions. The results show that under certain conditions, HPA system could enter into oscillatory dynamic states from stable steady state and vice versa under the influence of synergy reaction rate constant. Transitions between these dynamical states were always through supercritical Andronov-Hopf bifurcation point. Also, results revealed the conditions under which amplitudes of ultradian oscillations could increase several-fold due to CRH and AVP synergistic stimulation of ACTH secretion in accordance with results reported in the literature. Moreover, results showed experimentally observed superiority of CRH as a stimulator of ACTH secretion compared to AVP in humans. The proposed model can be very useful in studies related to the role of AVP and its synergistic action with CRH in life-threatening circumstances such as acute homeostasis dynamic crisis, autoimmune inflammations or severe hypovolemia requiring instant or several-days sustained corticosteroid excess levels. Moreover, the model can be helpful for investigations of indirect AVP-induced HPA activity by exogenously administered AVP used in therapeutic treatment.

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.