Simple Nonlinear System Research Articles

Global Sensitivity Study of a Duffing-Type Nonlinear Vibration System

Abstract An interesting field of studying nonlinear systems is their sensitivity study. With sensitivity study the most influential parameters on a system can be obtained and then the simplification and improvement of the model will be possible. In this paper the global sensitivity study of a Duffing-type vibration system is carried out with Sobol’s variance-based method taking the root mean square of acceleration and the maximum acceleration as output variables. With the sensitivity study it was observed that the parameters of the excitation signal like the amplitude and the angular velocity are the most influential. It was also found that a single parameter has less influence on the system than the parameter combinations. The aim of the research is to carry out the global sensitivity study of a relatively simple nonlinear system. The study is the basis for further research tasks in order to perform the sensitivity study of more complex systems.

On the predictability of possible storylines for forced complex systems

Abstract It is well-known that even for fairly simple deterministic nonlinear systems, exact prediction of future state is, on average, impossible beyond some small multiple of the Lyapunov time that quantifies the rate of separation of trajectories within an attractor. Nonetheless, it may be possible to find a physical measure that is the distribution of a trajectory within the attractor. In that sense, there can be a still weaker form of predictability. In this paper, we show that this can also fail but an even weaker form of predictability can appear for non-autonomous (i.e. forced) systems in the presence of tipping points. The {\em predictability of possible storylines} appears when one can interpret the frequencies of runs within an ensemble arriving at one of several possible future attractors (storylines) in a probabilistic manner.
As predictability is a major concern and a challenge in climate science, we illustrate this notion of predictability with two climate-related examples: a chaotic energy balance model and a global ocean model featuring a tipping point of the Atlantic Meridional Overturning Circulation.

Realization of logic gates in bi-directionally coupled nonlinear oscillators.

Implementation of logic gates has been investigated in nonlinear dynamical systems from various perspectives over the years. Specifically, logic gates have been implemented in both single nonlinear systems and coupled nonlinear oscillators. The majority of the works in the literature have been done on the evolution of single oscillators into OR/AND or NOR/NAND logic gates. In the present study, we demonstrate the design of logic gates in bi-directionally coupled double-well Duffing oscillators by applying two logic inputs to the drive system alone along with a fixed bias. The nonlinear system, comprising both bi-directional components, exhibits varied logic behaviors within an optimal range of coupling strength. Both attractive and repulsive couplings yield similar and complementary logic behaviors in the first and second oscillators. These couplings play a major role in exhibiting fundamental and universal logic gates in simple nonlinear systems. Under a positive bias, both the first and second oscillators demonstrate OR logic gate for the attractive coupling, while exhibiting OR and NOR logic gates, respectively, for the repulsive coupling. Conversely, under a negative bias, both the first and second oscillators display AND logic gate for the attractive coupling, and AND and NAND logical outputs for the repulsive coupling. Furthermore, we confirm the robustness of the bi-directional oscillators against moderate noise in maintaining the desired logical outputs.

Some properties of the basins of attraction of the Newton’s method for simple nonlinear geodetic systems

Abstract The research presented in this paper concerns the determination of the attraction basins of Newton’s iterative method, which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The simple observation systems corresponding to the intersections or linear and angular resections used in practice were considered. The main goal was to investigate the properties of the sets of convergent initial points of the applied iterative method. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness, or self-similarity were sought. The research also concerned the iterative structures of the basin: maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method: the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the significant results is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises. Their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases, it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points located far from the solution points.

Z-number based neural network structured inference system

Z-number based Neural Network structured Inference System (ZNIS) with rule-base consisting of linguistic Z-terms trainable with Differential Evolution with Constraints (DEC) optimization algorithm is suggested. The inference mechanism of the multi-layered ZNIS consists of a fuzzifier, fuzzy rule base, inference engine, and output processor. Due to the use of extended fuzzy terms, each processing layer implements appropriate extended fuzzy operations, including computation of fuzzy valued rule firing strengths, fuzzy Level-2 aggregate outputs, and two consecutive Center of Gravity (COG) defuzzification procedures. The experiments with different versions of ZNIS have demonstrated that it is a universal modeling tool suitable for dealing with both approximate reasoning and functional mapping tasks. Random experiments on benchmark examples (among which are simple functional mapping, Parkinson disease, and non-linear system identification) have shown that ZNIS performance is equivalent to or better than FLS Type 2 and far superior to FLS Type 1, showing on average 2–3 times lower MSE. Along with this, the main advantages of ZNIS over other inference systems are better semantic expressing power, higher degree of perception and interpretability of the linguistic rules by humans, and a higher confidence in the reliability of achieved decision due to the transparency of the underlying decision-making mechanism.

Multilayer adaptive critic design with digital twin for data-driven optimal tracking control and industrial applications

In this paper, an optimal trajectory tracking control problem for general nonlinear systems is investigated. An adaptive critic control method with the digital twin (DT) theory is developed. Divergent from the existing tracking control methods, the advantages of adaptive dynamic programming (ADP) and the theory of DT are combined in this paper, and the novel multilayer artificial system structure is constructed. The action-critic structure is employed by each artificial system to obtain an approximate optimal control policy. The model network (MN) is built by using the actual input and output data sets of the controlled system, which means the dependence on the dynamics of the system is overcame. Then, the weights of the trained action network (AN) and MN are passed to the real system to realize the optimal tracking control. The feasibility of the algorithm is proved by theoretical analysis. Finally, the algorithm is applied to a simple nonlinear torsional pendulum system and an industrial wastewater treatment system (WWTS), and the effectiveness of the algorithm is verified. The algorithm effectively realizes the tracking control of nonlinear systems.

Modelling the Use of the Trace-Test-Isolate-Treat Strategy for Controlling the Spread of COVID-19

During the COVID-19 pandemic that ravaged the entire world between 2019 and 2021, the Trace-Test-Isolate-Treat Strategy was devised as an emergency way of managing the spread of the disease. As the name implies, the Trace-Test-Isolate-Treat Strategy involves identifying those who had contact with an infected person through contact tracing, and subsequent isolation and treatment if confirmed to be infected with the disease. This paper aims to model the transmission dynamics of COVID-19, with the Trace-Test-Isolate-Treat Strategy as a control strategy. To do this, we propose a simple nonlinear system of ordinary differential equations that models COVID-19 dynamics and incorporates the Trace-Test-Isolate-Treat strategy as a way of controlling the spread of the disease. The analysis of the model shows that the disease-free equilibrium is locally asymptotically stable if the reproduction number, R_eff is less than one. Furthermore, the model is shown to possess a unique and stable endemic equilibrium if, R_eff>1. This confirms the global asymptotic stability of the disease-free equilibrium and the absence of backward bifurcation in the model. Numerical plots show the effectiveness of isolation and treatment of infected persons in reducing the spread of the disease.

Practice-oriented controller design of a simple nonlinear system

The paper analyses and illustrates the design of a constrained gain scheduling for a simple non-linear hydraulic system based on automatic-reset controller. The main objective is to explain the basics of gain-scheduling control in a simple way so that it can be understood by computer programmers working in the field of automatic control. The study is based on small deviations from the selected operating points that allow the use of linear design methods. In contrast to linearization-based design methods, where the control action is calculated by integrating small control increments, the new control output is obtained by adding the calculated small increments to the operating point values. Another innovative aspect is the introduction of a gain-scheduled automatic-reset controller. This is the oldest and the simplest control solution including a disturbance observer based on ultra-local models and accurate even in steady-states.

PI Tracking Control of a Nonlinear plant

The tracking control design for nonlinear plants by model inversion, following the Linear Algebra Based methodology, usually assumes a proportional to the error approaching of the process variables to the references evolution as a design criterion for the control. In this paper, by using the well-known properties of the integral action, a PI-like control is assumed, countering constant disturbances and model parameters uncertainty. The application to a simple mass-spring nonlinear system illustrates the procedure.

Exploring the Impact of Nonlinearities in Police Recruitment and Criminal Capture Rates: A Population Dynamics Approach

The interplay between criminal activity and crime control/prevention measures is inherently dynamic. This paper presents a simple nonlinear dynamical system in which criminal activity levels are coupled to policing effort. Through the process of non-dimensionalisation and sensitivity analysis, policing efficiency and the responsiveness of policing effort are identified as key parameter groupings. An analysis of the system shows that bi-stability is a feature of the dynamics. When there is no feedback between criminal activity and police recruitment, a saddle-node bifurcation occurs and threshold levels of criminal activity are required for the activity to be maintained. When feedback is permitted, we also find a backward bifurcation and criminal activity can be contained for policing efficiency below its threshold level. We demonstrate proof of concept for how the model might be used as a predictive tool with real data.

Stability and Resilience—A Systematic Approach

Stability and resilience are two crucial concepts to the proper functioning and understanding of the behavior of both natural and man-made systems exposed to perturbations and change. However, although the two have covered a similar territory within dynamic systems, the terminology and applications differ significantly. This paper presents a critical analysis of the two concepts by first collating the wealth of modern stability concept literature within dynamics systems and then linking it to resilience thinking, defined as adaptation where the system has the ability to respond perturbations and change through passive and active feedback structures. A lumped mass and simple pendulum, two simple linear and nonlinear dynamic systems following a state-space approach from modern control systems theory, are used to support the analysis and application. The research findings reveal that the two overarching categories of engineering resilience and socio-ecological resilience (extended ecological resilience) are in fact a reinvention of a closed-loop system dynamic stability with different types of active feedback mechanisms. Additionally, structural stability describes some vital aspects of social–ecological resilience such as critical thresholds where, under change, a system loses the ability to return to the starting form or move to another suitable form through active feedback mechanisms or direct management actions.

A topological perspective on weather regimes

It has long been suggested that the mid-latitude atmospheric circulation possesses what has come to be known as ‘weather regimes’, loosely categorised as regions of phase space with above-average density and/or extended persistence. Their existence and behaviour has been extensively studied in meteorology and climate science, due to their potential for drastically simplifying the complex and chaotic mid-latitude dynamics. Several well-known, simple non-linear dynamical systems have been used as toy-models of the atmosphere in order to understand and exemplify such regime behaviour. Nevertheless, no agreed-upon and clear-cut definition of a ‘regime’ exists in the literature, and unambiguously detecting their existence in the atmospheric circulation is stymied by the high dimensionality of the system. We argue here for an approach which equates the existence of regimes in a dynamical system with the existence of non-trivial topological structure of the system’s attractor. We show using persistent homology, an algorithmic tool in topological data analysis, that this approach is computationally tractable, practically informative, and identifies the relevant regime structure across a range of examples.

Machine-learning potential of a single pendulum.

Reservoir computing offers a great computational framework where a physical system can directly be used as computational substrate. Typically a "reservoir" is comprised of a large number of dynamical systems, and is consequently high dimensional. In this work, we use just a single simple low-dimensional dynamical system, namely, a driven pendulum, as a potential reservoir to implement reservoir computing. Remarkably we demonstrate, through numerical simulations as well as a proof-of-principle experimental realization, that one can successfully perform learning tasks using this single system. The underlying idea is to utilize the rich intrinsic dynamical patterns of the driven pendulum, especially the transient dynamics which has so far been an untapped resource. This allows even a single system to serve as a suitable candidate for a reservoir. Specifically, we analyze the performance of the single pendulum reservoir for two classes of tasks: temporal and nontemporal data processing. The accuracy and robustness of the performance exhibited by this minimal one-node reservoir in implementing these tasks strongly suggest an alternative direction in designing the reservoir layer from the point of view of efficient applications. Further, the simplicity of our learning system offers an opportunity to better understand the framework of reservoir computing in general and indicates the remarkable machine-learning potential of even a single simple nonlinear system.

Existence and location of internal resonance of two-mode nonlinear conservative oscillators

Internal resonances can be widely observed in nonlinear systems; even a simple nonlinear system can exhibit intricate internal resonances when vibrating at large amplitudes. In this study, the existence and locations of internal resonances of a general two-mode system with an arbitrary eigenfrequency ratio are considered. This is achieved by first considering the symmetric case, where the internal resonances are found to be approximately captured by the Mathieu equation. It is shown that the bifurcations can exist in pairs; and, for each pair, the bifurcated solution branches capture modal interactions with the same commensurate frequency relationship but different phase relationships. To determine the existence and locations of internal resonances, the divergence and convergence for correlated bifurcation pairs are then considered. Lastly, the internal resonances in asymmetric cases are analytically derived, where the asymmetry induced bifurcation splitting is captured by a non-homogeneous extended Mathieu equation. This work explores the mechanism underpinning internal resonances, and explains their topological features, such as which internal resonances are observed as amplitude increases. A graphical method is also proposed for efficient determination of the existence and locations of internal resonances.

Dual Control by Reinforcement Learning Using Deep Hyperstate Transition Models

In dual control, the manipulated variables are used to both regulate the system and identify unknown parameters. The joint probability distribution of the system state and the parameters is known as the hyperstate. The paper proposes a method to perform dual control using a deep reinforcement learning algorithm in combination with a neural network model trained to represent hyperstate transitions. The hyperstate is compactly represented as the parameters of a mixture model that is fitted to Monte Carlo samples of the hyperstate. The representation is used to train a hyperstate transition model, which is used by a standard reinforcement learning algorithm to find a dual control policy. The method is evaluated on a simple nonlinear system, which illustrates a situation where probing is needed, but it can also scale to high-dimensional systems. The method is demonstrated to be able to learn a probing technique that reduces the uncertainty of the hyperstate, resulting in improved control performance.

Extended dynamic mode decomposition for cyclic macroeconomic data

<abstract><p>We apply methods from the Koopman operator theory, Extended Dynamic Mode Decomposition and machine learning in the study of business cycle models. We use a simple non-linear dynamical system whose main merit is that in the appropriate parameter space sector predicts intrinsically business cycles which in the phase space are structurally stable limit cycles. Our objective is to approximate this system with a finite dimensional linear model which is defined on some augmented state space. We approximate so the trajectories of the system and obtain an alternative non-perturbative description of the system which can be used for prediction and control. This approach can also be applied to other models as well as to real data.</p></abstract>

Chevron's Salt Lake City refinery starts up ISOALKY unit

Massive amounts of data, accumulated in real time and over decades, are spread over a wide variety of the modern automation in chemical plants. However, several standard process monitoring algorithms, such as kernel-based algorithms, do not easily scale to such orders of magnitude. Particularly in their running time and memory complexity increasing dramatically with the increase in the size of the data set. This paper proposes a scalable approximate kernel based on the multilevel maximum variance unfolding (MLMVU) technique that uses low rank kernel-based MVU approximations for reducing and distributing the computational load among parallel multilevel machines to achieve time efficiency. Theoretically, it is guaranteed that the performance of the proposed MLMVU can approximate that of the centralized MVU. The mathematical framework is presented for the development of MLMVU and various aspects of their computational complexity and approximation ability are discussed. The greater time efficiency and scalability in the computation are achievable. The effectiveness of the proposed algorithm is confirmed through a simple nonlinear system and the industrial Tennessee Eastman process.

Return time approximation in planar nonlinear systems

We present a method for the calculation of the series expansion of the exact return time/period of two-dimensional nonlinear systems. The practical applicability of the theorem is demonstrated through simple periodic nonlinear systems such as the Duffing oscillator, an oscillator with quadratic nonlinearity, the nonlinear pendulum, and an oscillator with velocity-dependent stiffness. The approximate periods are compared to series expansions of exact periods. The results are in accordance with the series expansion of the exact period even for the system with velocity-dependent stiffness.

Iterative parameter estimation of nonlinear systems

This paper discusses the identification methods for nonlinear systems. Using the negative gradient search, the gradient based iterative algorithm is derived to determine the parameters of the nonlinear system. In addition, in order to improve the computational efficiency, the gradient-based iterative algorithm based on the model transformation is derived. The basic principle is to transform a complex nonlinear system into a linear or simple nonlinear system. Finally, a numerical example is given to verify the effectiveness of the proposed methods.

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.