Nonlinear Systems Theory Research Articles

Geometric decomposition, potential-based representation and integrability of non-linear systems

Abstract This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

A Global Interconnected Observer for Attitude and Gyro Bias Estimation with Vector Measurements.

This paper proposes a novel interconnected observer to get good estimates of attitude and gyro bias from high-noise vector measurements. The observer is derived based on the theory of nonlinear and linear cascade systems, and its error dynamics have the properties of global exponential stability and robustness to bounded noise. These properties ensure the convergence and boundedness of the attitude and gyro bias estimation errors. To obtain higher estimation accuracy, an approach to calculate time-varying gains for the proposed auxiliary observer is designed under the premise of considering noise terms in the rate gyro and vector sensors. The simulation results show that when the vector sensors’ outputs contain high-level noise, the proposed observer with time-varying gains yields better performance in both the transient and steady-state phases.

Global models for 222Rn and CO2 concentrations in the Cave of Altamira

The purpose of this paper is to model numerically the dynamics of CO2 and 222Rn in cave atmospheres, particularly in the noteworthy Cave of Altamira (Spain). We aim to get a better understanding of the nature of these dynamics and their couplings with climatic controls, more specifically the soil water content, which role in the said dynamics poses some questions. For the first time, we apply the global modeling technique in the field of cave microclimate and atmospheric composition. The global modeling technique is a methodology based on the theory of nonlinear systems and designed to extract mathematical models directly from observational time series. We were able to extract four global models from our data. These models represent a step forward from the existent conceptual ones. They also show that CO2 and 222Rn dynamics can be approximated by low-dimensional, deterministic systems, which can be chaotic or, at least, close to chaos; this has decisive methodological consequences for future research. Moreover, the global modeling technique was used for the first time in a non-autonomous formulation; this enabled the possibility of studying the influence of the external forcing (soil water content) on the gas concentration in different scenarios.

Dynamical systems and complex systems theory to study unsteady combustion

Reacting flow fields are often subject to unsteadiness due to flow, reaction, diffusion, and acoustics. Further, flames can also exhibit inherent unsteadiness caused by various intrinsic instabilities. Interaction between various unsteady processes across multiple scales often makes combustion dynamics complex. Characterizing such complex dynamics is essential to ensure the safe and reliable operation of high efficiency combustion systems. Tools from nonlinear dynamics and complex systems theory provide new perspectives to analyze and interpret the data from real systems. They could also provide new ways of monitoring and controlling combustion systems. We discuss recent advances in studying unsteady combustion dynamics using the tools from dynamical systems theory and complex systems theory. We cover a range of problems involving unsteady combustion such as thermoacoustic instability, flame blowout, fire propagation, reaction chemistry and flow flame interaction.

Review article: Hilbert problems for the climate sciences in the 21st century – 20 years later

Abstract. The scientific problems posed by the Earth's atmosphere, oceans, cryosphere – along with the land surface and biota that interact with them – are central to major socioeconomic and political concerns in the 21st century. It is natural, therefore, that a certain impatience should prevail in attempting to solve these problems. The point of a review paper published in this journal in 2001 was that one should proceed with all diligence but not excessive haste, namely “festina lente”, i.e., “to hurry in a measured way”. The earlier paper traced the necessary progress through the solutions of 10 problems, starting with “What can we predict beyond 1 week, for how long, and by what methods?” and ending with “Can we achieve enlightened climate control of our planet by the end of the century?” A unified framework was proposed to deal with these problems in succession, from the shortest to the longest timescale, i.e., from weeks to centuries and millennia. The framework is that of dynamical systems theory, with an emphasis on successive bifurcations and the ergodic theory of nonlinear systems, on the one hand, and on pursuing this approach across a hierarchy of climate models, from the simplest, highly idealized ones to the most detailed ones. Here, we revisit some of these problems, 20 years later,1 and extend the framework to coupled climate–economy modeling.

A Necessary and Sufficient Condition for Stability of a Class of Planar Positive Nonlinear Systems

In this letter, the stability problem of a class of planar positive nonlinear systems is investigated. Based on homogeneous nonlinear system theory and positive system theory, a necessary and sufficient condition for stability is established. To prove the necessity, a new method called special solution method is provided, and this method can be efficiently used to analyze instability. Three examples are provided to demonstrate the efficiency of the obtained results.

Robust Multimode Flight Framework Based on Sliding Mode Control for a Rotary UAV

SUMMARYThis work presents a multimode flight framework control scheme for a quadrotor based on the super twisting algorithm. The controller design stages for six flight control modes are presented. The stability proof for each flight mode is carried out by means of Lyapunov functions, while the stability analysis for the complete control scheme, when a transition from one flight mode to another occurs, is demonstrated using the switching nonlinear systems theory. The performance of the proposed framework is shown in a 3D simulation environment considering a forest fire detection task, which takes into account external disturbances.

Complexity Analysis on Functional-Near Infrared Spectroscopy Time Series: a Preliminary Study on Mental Arithmetic.

It is well known that physiological systems show complex and nonlinear behaviours. In spite of that, functional near-infrared spectroscopy (fNIRS) is usually analyzed in the time and frequency domains with the assumption that metabolic activity is generated from a linear system. To leverage the full information provided by fNIRS signals, in this study we investigate topological entropy in fNIRS series collected from 10 healthy subjects during mental mental arithmetic task. While sample entropy and fuzzy entropy were used to estimate time series irregularity, distribution entropy was used to estimate time series complexity. Our findings show that entropy estimates may provide complementary characterization of fNIRS dynamics with respect to reference time domain measurements. This finding paves the way to further investigate functional activation in fNIRS in different case studies using nonlinear and complexity system theory.

Preface

4th International Conference on Mathematical Methods & Computational Techniques in Science & Engineering (MMCTSE 2020)This volume is from proceedings of the 4th International Conference on Mathematical Methods & Computational Techniques in Science & Engineering. The main objective of the Conference was to provide a platform for researchers, engineers, academicians as well as industrial professionals from all over the world to present their research results and development activities in Mathematical Methods & Computational Techniques as well as their applications in Science & Engineering. Participants met to discuss topics such as Numerical Methods, Mathematical Models, ODEs and PDES, Linear and Multilinear Algebra, Ordinary Differential Equations, Partial Differential Equations, Numerical Behaviour of Optimization Algorithms, High Complexity Numerical Methods, Non-Linear Systems Theory, Evolutionary Computation, Linear Programming, Software Development, Heuristic Algorithms, Randomized Algorithms, Equational logic programming, Functional Languages, Parallel and Distributed Computation, Theoretical Computer Science, Coding and Information Theory, Classification Mathematical Physics, Image Compression, Complexity Theory and Neural Networks, Signal Processing, Communications and Automation, etc. The Conference was held February 22-24, 2020 in London, UK. All chapters are self-contained and peer reviewed.List of The Editors are available in this pdf.

Integral vector field control for three-dimensional path following of autonomous underwater vehicle

A new control method for autonomous underwater vehicle (AUV) is developed to achieve three-dimensional (3D) path following under ocean current disturbances when AUV lacks lateral and vertical driving forces. To make the control system more convenient for practical application and reduce the energy consumption of AUV, the controller is designed with relative velocities. To avoid the singularity problem of curve path following, the Serret–Frenet frame is introduced as virtual target and the error model is defined. The adaptive law of the virtual target and an integral vector field (IVF) guidance law are designed in the kinematic controller. The back-stepping method and adaptive dynamical sliding mode control (BADSMC) technology are applied in the design of dynamic controller. The Lyapunov theory and nonlinear cascade system theory are applied to prove the closed-loop stability. Finally, the performances of the IVF + BADSMC and ILOS + PID algorithm are compared through simulation. Simulation results demonstrate that the proposed controller can realize the path following of AUV under disturbances of ocean current and can improve the following quality.

Existence and qualitative theory for nonlinear elliptic systems with a nonlinear interface condition used in electrochemistry

We study a nonlinear elliptic system with prescribed inner interface conditions. These models are frequently used in physical system where the ion transfer plays the important role for example in modelling of nano-layer growth or Li-on batteries. The key difficulty of the model consists of the rapid or very slow growth of nonlinearity in the constitutive equation inside the domain or on the interface. While on the interface, one can avoid the difficulty by proving a kind of maximum principle of a solution, inside the domain such regularity for the flux is not available in principle since the constitutive law is discontinuous with respect to the spatial variable. The key result of the paper is the existence theory for these problems, where we require that the leading functional satisfies either the delta-two or the nabla-two condition. This assumption is applicable in case of fast (exponential) growth as well as in the case of very slow (logarithmically superlinear) growth.

A Novel Method for Performance Improvement of Chaos-Based Substitution Boxes

Symmetry plays an important role in nonlinear system theory. In particular, it offers several methods by which to understand and model the chaotic behavior of mathematical, physical and biological systems. This study examines chaotic behavior in the field of information security. A novel method is proposed to improve the performance of chaos-based substitution box structures. Substitution box structures have a special role in block cipher algorithms, since they are the only nonlinear components in substitution permutation network architectures. However, the substitution box structures used in modern block encryption algorithms contain various vulnerabilities to side-channel attacks. Recent studies have shown that chaos-based designs can offer a variety of opportunities to prevent side-channel attacks. However, the problem of chaos-based designs is that substitution box performance criteria are worse than designs based on mathematical transformation. In this study, a postprocessing algorithm is proposed to improve the performance of chaos-based designs. The analysis results show that the proposed method can improve the performance criteria. The importance of these results is that chaos-based designs may offer opportunities for other practical applications in addition to the prevention of side-channel attacks.

The Impact of Service and Channel Integration on the Stability and Complexity of the Supply Chain

This paper constructs a supply chain consisting of a manufacturer and a retailer. Considering channel integration and service cooperation, two dynamic Stackelberg game models are established: one without unit profit allocation M and the other one with unit profit allocation Mε. In two dynamic models, we analyze the influence of relevant parameters on the stability and complexity of the dynamic system and system profit by nonlinear system theory and numerical simulation. We find that the higher adjustment parameters can cause the system to lose stability, showing double period bifurcation or wave-shape chaos. The stable region becomes larger with increase in service value and value of unit profit sharing. Besides, when the system is in chaotic state, we find that the profit of the system will fluctuate or even decline sharply; however, keeping the parameters in a certain range is helpful in maintaining the system stability and is conducive to decision-makers to obtain steady profits. In order to control the chaos phenomenon, the state feedback method is employed to control the chaotic system well. This study provides some valuable significance to supply chain managers in channel integration and service cooperation.

Fuzzy-based Sliding Mode Control of Chaotic Oscillation in Power System

Power system will occur chaotic oscillation when it is exist periodic load disturbances. As for it, the simplified mathematical model of interconnected two- machine second-order power system is established based on the modelling and analysis of nonlinear system theory, and the dynamic behaviour of chaotic oscillation in power system is analysed. According to the sliding mode control and fuzzy control theory, the fuzzy-based sliding mode control is proposed, it is also analysed and proved theoretically. Simulations results show that the proposed method can suppress the chaotic oscillation in power system effectively. It not only reduce the convergence time, but also reduced the chattering of the system, and has strong robustness.

Formation Control with Orientation Alignment and Constrained Input

We design a formation controller for Multi Agent Systems such that the agents can form into the desired shape and track a given reference trajectory. The main feature of the proposed design is that not only the orientation of individual agents, but also the orientation of the whole formation is considered and is designed to be aligned with the moving direction of the reference trajectory, which helps the tracking movement to be smoother compared with the common tracking results. Moreover, the control inputs are designed in predefined input ranges to reflect the practical system. System stability is proved based on nonlinear system theory and some simulations are given to validate the proposed results.

Leonov’s nonlocal reduction technique for nonlinear integro-differential equations

Starting from pioneering works by Lur’e, Popov and Zames, global stability theory for nonlinear control systems has been primarily focused on systems with only one equilibrium. Global stability criteria for other kinds of attractors (such as e.g. infinite sets of equilibria) are not well studied and typically require special tools, primarily based on the Lyapunov method. Analysis of stability becomes especially complicated for infinite-dimensional dynamical systems with multiple equilibria, e.g. systems described by delay or more general convolutionary equations. In this paper, we propose novel stability criteria for infinite-dimensional systems with periodic nonlinearities, which have infinite sets of equilibria and describe dynamics of phase-locked loops and other synchronization circuits. Our method combines Leonov’s nonlocal reduction technique with the idea of Popov’s “integral indices” and allows to obtain new frequency-domain conditions, ensuring the convergence of every solution to one of the equilibria points.

Релаксационная мода макроскопической пластической деформации металлов

The relaxation of elastic energy during macroscopic plastic deformation in a strict formulation is determined by the solutions of the system of nonlinear equations of mechanics of a deformable solid. Using the methods of the theory of nonlinear systems, a nonlinear parabolic equation is obtained for the amplitude of an unstable mode, which describes plastic deformation at large spatial and temporal scales.

Therapeutic Chaos.

The conventional view on interventions as mechanistically causing interchangeable clients to get better has come under attack. Group-based and linear approaches fall short in adequately describing the idiosyncratic and dynamic nature of treatment processes. Non-linear dynamic system theories in contrast hold great potential to better conceptualize and understand the generalities and idiosyncrasies of psychotherapeutic change processes. The aim of this study was to examine whether we can detect markers of complex dynamical systems behavior in two single-case therapies. All sessions from both therapies were coded with sequential plan analysis using a 10s sampling frequency. The coding system incorporates verbal and non-verbal behaviors and allows for the representation of contextualized interactive behaviors. The high sampling frequency results in long time series, which allowed us to apply non-linear analysis techniques. We found strong support for complex behavior and the existence of a butterfly effect, i.e., a relatively short prediction horizon in which reliable predictions about the system’s future behavior could be made. Further, critical fluctuations as a marker for phase-transitions were detected that were accompanied with different interactional patterns in both therapies. Finally, there was strong support for self-organized pattern formation, with a few interactional patterns dominating the interaction. Considering that we are intervening on complex dynamical systems means that we have to (1) acknowledge the principal individuality of change processes, (2) accept the fundamental limitations of the mechanistic input-output model of treatment effects and (3) appreciate the impossibility of long-term predictions of treatment responses.

Studying State Convergence of Input-to-State Stable Systems with Applications to Power System Analysis

In stability studies, the response of a system enforced by external, known or unknown, inputs is of great importance. Although such an analysis is quite easy for linear systems, it becomes a cumbersome task when nonlinearities exist in the system model. Nevertheless, most of the real-world systems are externally enforced nonlinear systems with nonzero equilibriums. Representative examples in this category include power systems, where studies on stability and convergence to equilibrium constitute crucial objectives. Driven by this need, the aim of the present work is twofold: First, to substantially complete the theoretical infrastructure by establishing globally valid sufficient conditions for externally enforced nonlinear systems that converge to nonzero equilibriums and, second, to deploy an efficient method easily applicable on practical problems as it is analyzed in detail on a typical power system example. To that end, in the theoretical first part of the paper, a rigorous nonlinear analysis is developed. Particularly, starting from the well-established nonlinear systems theory based on Lyapunov techniques and on the input-to-state stability (ISS) notion, it is proven after a systematic and lengthy analysis that ISS can also guarantee convergence to nonzero equilibrium. Two theorems and two corollaries are established to provide the sufficient conditions. As shown in the paper, the main stability and convergence objectives for externally enforced systems are fulfilled if simple exponential or asymptotic converging conditions can be proven for the unforced system. Then, global or local convergence is established, respectively, while for the latter case, a novel method based on a distance-like measure for determining the region of attraction (RoA) is proposed. The theoretical results are examined on classic power system generation nonlinear models. The power system examples are suitably selected in order to effectively demonstrate the proposed method as a stability analysis tool and to validate all the particular steps, especially that of evaluating the RoA. The examined system results clearly verify the theoretical part, indicating a rather wide range of applications in power systems.

Exploring the Triple Helix Synergy in Chinese National System of Innovation

Sustainable economic growth is closely linked to synergy in a national system of innovation. Although the dynamic synergy mechanism of the triple helix relations is essential to technology innovation, there are limited research methodologies to study or estimate the synergy effect accurately. This paper introduces a new approach in non-linear complex systems theory to offer steps towards a possible solution to this conundrum. Based on the pattern formation of the Belousov-Zhabotinsky’s reaction, the paper constructs a simulation equation to explore the evolution mechanism by comparing the ideal state with the current state in China. The research finds that (1) under the ideal balanced condition of industrial absorptive capacity and academic knowledge transfer capability, the stronger incentive policies would play much more important roles than weak policies; (2) the performance of collaborative innovation is not optimal under current situation in China, but the industrial absorptive capacity, especially in private enterprises, has exceeded the capability of knowledge transfer in academia, and it has become the main driving force to promote future innovation. If the innovation policy can be focused on the high-level balance between the knowledge network and innovation network to promote synergy in China, the innovation performance will be accelerated more efficiently.