Analysis Of Nonlinear Dynamical Systems Research Articles

Cross-nonnegativity and monotonicity analysis of nonlinear dynamical systems

Let K be a closed convex cone in R n and let F : R n → R n be a locally Lipschitz map, not necessarily differentiable. We revisit the problem of analyzing whether the flow generated by an autonomous dynamical system φ ˙ ( t ) = F ( φ ( t ) ) is order-preserving with respect to K . This issue has to do with the notion of cross-nonnegativity of a nonlinear map relative to a cone. We study this property in-depth and derive various characterizations of it. We deviate from the classical literature in at least two ways. First of all, K is allowed to be unpointed and nonsolid. These nontrivial relaxations are needed to cover some interesting examples arising in applications. On the other hand, taking into account the possible lack of differentiability of F , we bring the machinery of Clarke's nonsmooth analysis into the picture.

Extracting Nonlinear Dynamics from Psychological and Behavioral Time Series Through HAVOK Analysis

Analytical methods derived from nonlinear dynamical systems, complexity, and chaos theories offer researchers a framework for in-depth analysis of time series data. However, relatively few studies involving time series data obtained from psychological and behavioral research employ such methods. This paucity of application is due to a lack of general analysis frameworks for modeling time series data with strong nonlinear components. In this article, we describe the potential of Hankel alternative view of Koopman (HAVOK) analysis for solving this issue. HAVOK analysis is a unified framework for nonlinear dynamical systems analysis of time series data. By utilizing HAVOK analysis, researchers may model nonlinear time series data in a linear framework while simultaneously reconstructing attractor manifolds and obtaining a secondary time series representing the amount of nonlinear forcing occurring in a system at any given time. We begin by showing the mathematical underpinnings of HAVOK analysis and then show example applications of HAVOK analysis for modeling time series data derived from real psychological and behavioral studies.

Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

Abstract Reachability analysis of nonlinear dynamical systems is a challenging and computationally expensive task. Computing the reachable states for linear systems, in contrast, can often be done efficiently in high dimensions. In this paper, we explore verification methods that leverage a connection between these two classes of systems based on the concept of the Koopman operator. The Koopman operator links the behaviors of a nonlinear system to a linear system embedded in a higher dimensional space, with an additional set of so-called observable variables. Although the new dynamical system has linear differential equations, the set of initial states is defined with nonlinear constraints. For this reason, existing approaches for linear systems reachability cannot be used directly. We propose the first reachability algorithm that deals with this unexplored type of reachability problem. Our evaluation examines several optimizations, and shows the proposed workflow is a promising avenue for verifying behaviors of nonlinear systems.

Mechanical and mathematical modeling of the load movement on the thread wound on cylindrical drum with movable axis

A mechanical system, where the load in the form of material point is suspended on inextensible thread screwed on the rotating cylindrical drum, but the drum is connected to the boom rotating around fixed horizontal axis is considered. Using the Lagrange equation of the second kind, a mathematical model of the motion of the mechanical system is obtained. The system has three degrees of freedom, two of which are cylindrical. The investigation of the system motion is carried out using computer technology. As a result, the dependences of linear and angular coordinates and velocities in time at different values of the output data for two main modes of the system operation, namely – under the conditions of lifting and lowering the load are obtained. Appropriate graphs are constructed, including the trajectories of the cargo motion. The mathematical model takes into account nonlinearities of the system and allows you to find the amount of tension of the hoisting rope at any time. The analysis showed that vertical oscillations occur twice as fast as horizontal ones. The phase portrait of the generalized coordinate (angle of the rope with the vertical axis) is the focus, which is untwisted when lifting due to nonlinearity in the system, and when the load moves down, the focus, which twists and approaches the mathematical pendulum is obtained. The obtained results can be used in modeling of controlled pendulum motions for different mechanical systems. The methodology and program are recommended to the students and graduate students in terms of learning the principles of construction and analysis of complex nonlinear dynamical systems.

A generalization of linear positive systems with applications to nonlinear systems: Invariant sets and the Poincaré–Bendixson property

The dynamics of linear positive systems maps the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. What linear systems map the set of vectors with k sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called k-positive linear systems, that reduces to positive systems for k=1. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case k=2 establish the Poincaré–Bendixson property for any bounded trajectory.

Integer-fractional decomposition and stability analysis of fractional-order nonlinear dynamic systems using homotopy singular perturbation method

Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2.

Enhancing subdivision technique with an adaptive interpolation sampling method for global attractors of nonlinear dynamical systems

The cell mapping method is a prominent one for global analysis of nonlinear dynamical systems, with which multiple invariant sets can be obtained. However, it is a continuous challenge to enhance the efficiency of the cell mapping method, especially when dealing with high-dimensional nonlinear dynamical systems. In this paper, the subdivision technique commonly used in the cell mapping method is incorporated with an interpolation sampling method, which can further enhance the efficiency over the set-oriented method with subdivision for global attractors of nonlinear dynamical systems. In the present method, a new lattice of interpolating nodes is adopted to fit into the nesting structures of the subdivided cells, portions of which are sequentially removed when resolution goes from low to high. An improved interpolation method is developed to obtain one-step sample mappings, instead of integrations when error bounds are met, in the process of subdivision iterations. Furthermore, a Hash table is introduced in order to fast search and locate the coordinates of cells that cover the invariant sets. Three examples of nonlinear dynamical systems are presented to demonstrate the enhancement in efficiency and effectiveness of the proposed method, indicating that a computational cost is one half down to one sixth of the previous methods.

S-synchronization Structural Identifiability and Identification of Nonlinear Dynamic Systems

An approach to the structural identifiability analysis of nonlinear dynamic systems under uncertainty is proposed. We have shown that S-synchronization is the necessary condition for the structural identifiability of a nonlinear system. Conditions are obtained for the design of a model which identifies the nonlinear part of the system. The method is proposed for the obtaining of a set which contains the information on the nonlinear part. A class of geometric frameworks which reflect the state of the system nonlinear part is introduced. Geometrical frameworks are defined on the synthesized set. The conditions are given for the structural indistinguishability of geometric frameworks on the set of S-synchronizing inputs. Local identifiability conditions are obtained for the nonlinear part. We are shown that a non-synchronizing input gives an insignificant geometric framework. This leads to a structural non-identifiability of the system nonlinear part. The method is proposed for the estimation of the structural identifiability the nonlinear part of the system. Conditions for parametric identifiability of the system linear part are obtained. We show that the structural identifiability is the basis for the structural identification of the system. The hierarchical immersion method is proposed for the estimation of nonlinear system structural parameters. The method is used for the structural identification of a system with Bouc-Wen hysteresis.

A novel chaotic hyperjerk circuit with bubbles of bifurcation: mixed-mode bursting oscillations, multistability, and circuit realization

This contribution proposes a novel chaotic hyperjerk circuit with the new phenomenon of mixed-mode bursting oscillations (MMBOs). The bifurcation analysis of this specific oscillator reveals a relatively rare and complex phenomenon of various coexisting two full Feigenbaum remerging trees. Common nonlinear dynamic systems analysis tools such as bifurcation diagram, phase portraits, time series and Lyapunov exponent are used to uncover the complex behaviours of the circuit with respect to its parameters and initial conditions as well. As one of the major contributions of this work, we introduce a novel hyperjerk circuit with rare phenomena such as the coexistence of a new phenomenon named bubbles of bifurcation (referred as BsB hereafter), the coexistence of two or three asymmetric mixed-mode bursting oscillations, just to name a few. All these phenomena are found simultaneously in a single circuit and thus diverse dissemination. Experimental results support the theoretical analysis.

Global Analysis of Stochastic Systems by the Digraph Cell Mapping Method Based on Short-Time Gaussian Approximation

The digraph cell mapping method is popular in the global analysis of stochastic systems. Traditionally, the Monte Carlo simulation is used in finding the image cells of one-step mapping, and it is notably costly in the computation time. In this paper, a novel short-time Gaussian approximation (STGA) scheme is incorporated into the digraph cell mapping method to study the global analysis of nonlinear dynamical systems under Gaussian white noise excitations. In order to find out all the active image cells in one-step cell mapping quickly, the STGA scheme together with a probability truncation method is introduced for systems without periodic excitation, and then in the case with periodic excitation. The global structures, such as the stochastic attractors, stochastic basins of attraction and stochastic saddles, are calculated by the digraph analysis algorithm. The proposed methodology has been applied to three typical stochastic dynamical systems. For each system, the effectiveness and superiority of the proposed STGA scheme are verified by checking the image cells of one-step mapping and comparing with the results of Monte Carlo simulation. It is found in the global analysis that the change of the amplitude of periodic excitation induces stochastic bifurcations in the stochastic Duffing system. Moreover, a stochastic bifurcation occurs in the stochastic Lorenz system with the increase of noise intensities.

Poincaré maps for multiscale physics discovery and nonlinear Floquet theory

Poincaré maps are an integral aspect to our understanding and analysis of nonlinear dynamical systems. Despite this fact, the construction of these maps remains elusive and is primarily left to simple motivating examples. In this manuscript we propose a method of data-driven discovery of Poincaré maps based upon sparse regression techniques, specifically the sparse identification of nonlinear dynamics (SINDy) algorithm. This work can be used to determine the dynamics on and near invariant manifolds of a given dynamical system, as well as provide long-time forecasting of the coarse-grained dynamics of multiscale systems. Moreover, the method provides a mathematical formalism for determining nonlinear Floquet theory for the stability of nonlinear periodic orbits. The methods are applied to a range of examples including both ordinary and partial differential equations that exhibit periodic, quasi-periodic, and chaotic behavior.

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Stability analysis of the homogeneous nonlinear dynamical systems using Lyapunov function generation based on the basic functions

In this paper, based on Lyapunov functions candidates, a new approach in the stability analysis of homogeneous nonlinear systems is proposed in which instead of concentrating on the positive definiteness of the Lyapunov candidate functions, we stress on the negative definiteness of its derivative. Having ensured of negative definiteness of the derivative function, based on sign assignment of the primitive function, the stability of the equilibrium is analyzed wherein the necessary and sufficient conditions are declared simultaneously. Selecting the trend of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions whose unknown coefficients in the candidate function structure are computed based on negative definiteness of the derivative function. Afterward, using these determined coefficients in the Lyapunov function, the sign of the primitive function in the state space is argued. Therefore, the triple sign attitudes of the candidate function can be used to deduce the stability/instability of the equilibrium point. Moreover, in the process of the negative definiteness of the derivative function, the coefficients are obtained using two independent methods. Numerical simulations support the proposed theoretical results and show their effectiveness.

Physics-Informed Probabilistic Learning of Linear Embeddings of Nonlinear Dynamics with Guaranteed Stability

The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural networks in a measure-theoretic framework. Our approach induces two types of models: differential and recurrent form, the choice of which depends on the availability of the governing equations and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference (MFVI) on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements.

Analytical-numerical method of analysis and behavioral modelling of non-linear dynamic systems

An analytical-numerical method for solving non-linear integro-differential equation systems, that is based on the application of generalized functions, whose regular components are described by the Taylor series, is proposed. These non-linear integro-differential equation systems are applied for modelling of non-linear electrical circuits. The procedure for determining the parameters of oscillatory regimes in deterministic, non-linear, non-autonomous mathematical models with lumped parameters is described. Within the framework of the “black box” concept on establishing the input-output relationship of an object the behavioural models of non-linear compensators for a power amplifier are constructed and the results of the mathematical models comparison are represented.

Dynamical system analysis of Hessence scalar field in teleparallel gravity: Invariant manifold technique

This paper investigates the cosmological dynamics of the Hessence scalar field coupled with the dark matter in the background of the teleparallel gravity. We have assumed that the potential of the scalar field is exponential in nature whereas the [Formula: see text] appearing in the teleparallel theory has the form [Formula: see text]. The field equations of this system reduce to a nonlinear autonomous system and dynamical system analysis is then performed. Due to the nonlinearity and the existence of multiple zero eigenvalues, the traditional procedures of analysis break down. So some novel technique is required. One of the latest such techniques is the invariant manifold theory. By the application of this theory, one projects the variables linked with the zero eigenvalues onto the variables linked with the nonzero eigenvalues to compute the center manifolds and the reduced systems associated with the critical points. These reduced systems reflect the nature of the whole dynamical systems. They also have less dimension and are often simple in nature. Hence, it is possible to solve them directly. In this paper, we work exactly in this spirit and find the center manifolds and solve the corresponding reduced system for some of the critical points associated with the dynamical system. We discover some interesting results namely that there are certain bounds on the interaction term [Formula: see text] which asserts the stability of the systems. We also present various stability diagrams of the reduced systems. An asymptotic analysis is then done for the critical points at infinity. Finally, we discuss the cosmological interpretation of our results.

Multiscale Modelling Tool: Mathematical modelling of collective behaviour without the maths

Collective behaviour is of fundamental importance in the life sciences, where it appears at levels of biological complexity from single cells to superorganisms, in demography and the social sciences, where it describes the behaviour of populations, and in the physical and engineering sciences, where it describes physical phenomena and can be used to design distributed systems. Reasoning about collective behaviour is inherently difficult, as the non-linear interactions between individuals give rise to complex emergent dynamics. Mathematical techniques have been developed to analyse systematically collective behaviour in such systems, yet these frequently require extensive formal training and technical ability to apply. Even for those with the requisite training and ability, analysis using these techniques can be laborious, time-consuming and error-prone. Together these difficulties raise a barrier-to-entry for practitioners wishing to analyse models of collective behaviour. However, rigorous modelling of collective behaviour is required to make progress in understanding and applying it. Here we present an accessible tool which aims to automate the process of modelling and analysing collective behaviour, as far as possible. We focus our attention on the general class of systems described by reaction kinetics, involving interactions between components that change state as a result, as these are easily understood and extracted from data by natural, physical and social scientists, and correspond to algorithms for component-level controllers in engineering applications. By providing simple automated access to advanced mathematical techniques from statistical physics, nonlinear dynamical systems analysis, and computational simulation, we hope to advance standards in modelling collective behaviour. At the same time, by providing expert users with access to the results of automated analyses, sophisticated investigations that could take significant effort are substantially facilitated. Our tool can be accessed online without installing software, uses a simple programmatic interface, and provides interactive graphical plots for users to develop understanding of their models.

Impact of environmental tax on green development: A nonlinear dynamical system analysis

With green development becoming a global movement, environmental tax has been adopted by many governments to promote green development. This study analyzes the impact of environmental tax on green development by using a four-dimension dynamical system. The establishment of the system is based on the complex and dynamic interactions among economic development, pollution emissions, resources consumption, and environmental tax, where roles of environmental tax are reflected by the linear parameters. A theoretic analysis shows the complexity of the behavior of the system. Mainly, the existence of chaos is inferred by Lyapunov exponent spectrum and bifurcation diagram, then verified by the presence of a chaotic attractor. An empirical study of the green development dynamical system in China demonstrates the particular evolution paths of economic growth, pollution intensity, and resource intensity under different environmental tax parameters. Results indicate a robust beneficial role of environmental tax on green development. Furthermore, when an environmental tax is imposed, a firm government control, an active consumer awareness, an advanced technology level can stimulate economic growth, decrease pollution intensity, and control the resource intensity. But the government control has a stronger effect. This study provides a viable and promising approach to analyze the role of imposing an environmental tax on green development and may have potential application in other areas and countries.

Dynamic ecological system analysis: A holistic analysis of compartmental systems

This article develops a new mathematical method for holistic analysis of nonlinear dynamic compartmental systems in the context of ecology. The method is based on the novel dynamic system and subsystem partitioning methodologies through which compartmental systems are decomposed to the utmost level. The dynamic system and subsystem partitioning enable tracking the evolution of the initial stocks, environmental inputs, and intercompartmental system flows, as well as the associated storages derived from these stocks, inputs, and flows individually and separately within the system. Moreover, the transient and the dynamic direct, indirect, acyclic, cycling, and transfer (diact) flows and associated storages transmitted along a given flow path or from one compartment, directly or indirectly, to any other are analytically characterized, systematically classified, and mathematically formulated. Further, the article develops a dynamic technique based on the diact transactions for the quantitative classification of interspecific interactions and the determination of their strength within food webs. Major concepts and quantities of the current static network analyses are also extended to nonlinear dynamic settings and integrated with the proposed dynamic measures and indices within the proposed unifying mathematical framework. Therefore, the proposed methodology enables a holistic view and analysis of ecological systems. We consider that this methodology brings a novel complex system theory to the service of urgent and challenging environmental problems of the day and has the potential to lead the way to a more formalistic ecological science.