Deterministic Nonlinear Systems Research Articles

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.

A Simple Model for Targeting Industrial Investments with Subsidies and Taxes

We consider an investor, whose capital is divided into an industrial investment xt and cash yt, and satisfy a nonlinear deterministic dynamical system. The investor fixes fractions of capital to be invested, withdrawn, and consumed, and also the production factor parameter. The government fixes a subsidy fraction for industrial investments and a tax fraction for the capital outflow. We study a Stackelberg game, corresponding to the asymptotically stable equilibrium (x∗,y∗) of the mentioned dynamical system. In this game, the government (the leader) uses subsidies to make incentives for the investor (the follower) to maintain the desired level of x∗, and uses taxes to achieve this with the minimal cost. The investor’s aim is to maximize the difference between the consumption and the price of the production factor at equilibrium. We present an explicit analytical solution of the specified Stackelberg game. Based on this solution, we introduce the notion of a fair industrial investment level, which is costless for the government, and show that it can produce realistic results using a case study of water production in Lahore.

Stability of Traffic Equilibria in a Day-to-Day Dynamic Model of Route Choice and Adaptive Signal Control

Adaptive traffic signal control has the potential to promote the efficient use of road intersections, thus contributing to the effectiveness of urban traffic management schemes. However, the reaction of drivers to repeatedly updated signal settings and the ensuing route choice dynamics may trigger the emergence of various kinds of network instability. In this study, the joint evolution of traffic flows and adaptive signal settings in a road network is investigated at the level of day-to-day dynamics with an explicit focus on the stability issue. We show how a Logit form signal control policy can be used, in interaction with route choice, to counter the emergence of instabilities possibly arising as a consequence of various behavioral factors and network conditions. After providing a general formulation of the model as a discrete time, deterministic nonlinear dynamical system, an explicit analysis of fixed-point stability is carried out for a simple network. Numerical results obtained from the implementation of the model on two example networks are presented in order to support the analytical findings of this study. We conclude that, in an integrated traffic management and information system, a properly calibrated adaptive signal control policy has the potential to offset the destabilizing effect of highly accurate driver information supplied by navigational aids. Our findings also suggest that the Logit-like control policy performs better than the Equisaturation signal setting method, in terms of average intersection delay at equilibrium, for all levels of driver information and travel demand tested in the experiment.

Cloud Security System for ECG Transmission and Monitoring Based on Chaotic Logistic Maps

Biomedical data or information must be transmitted securely via the internet for smart healthcare. The Electrocardiogram (ECG) signal is amongst the most essential clinical signals which must be delivered to hospital facilities. Prime focus of this research is on the encryption of ECG for secure transmission. Chaos theory is used for the development of deterministic nonlinear systems, that can be used to create random numbers for the Chaotic Logistic Map (CLM) based encryption. This study describes a cryptographic algorithm for encrypting ECG signals that uses a mix of the CLM and fingerprint data. The common factor between the patient section and monitoring section is the operation on sample data points of ECG. The choice of proper encryption and decryption theme can save more amount of time and is invulnerable both to noise-based attacks and hacking instances. The proposed framework is implemented on Dropbox based cloud storage and access is possible from any given locations. Simulation tests are used to assess the system performance in terms of Structural Similarity Index Matrix (SSIM), Histogram, Spectral Distortion (SD), Correlation and Log-Likelihood Ratio (LLR). The incorporation of complex layers of CLM encryption increases security.

Online estimation of objective function for continuous-time deterministic systems

We developed two online data-driven methods for estimating an objective function in continuous-time linear and nonlinear deterministic systems. The primary focus addressed the challenge posed by unknown input dynamics (control mapping function) in the expert system, a critical element for an online solution of the problem. Our methods leverage both the learner’s and expert’s data for effective problem-solving. The first approach, which is model-free, estimates the expert’s policy and integrates it into the learner agent to approximate the objective function associated with the optimal policy. The second approach estimates the input dynamics from the learner’s data and combines it with the expert’s input-state observations to tackle the objective function estimation problem. Compared to other methods for deterministic systems that rely on both the learner’s and expert’s data, our approaches offer reduced complexity by eliminating the need to estimate an optimal policy after each objective function update. We conduct a convergence analysis of the estimation techniques using Lyapunov-based methods. Numerical experiments validate the effectiveness of our developed methods.

Hydrological records can be used to reconstruct the resilience of watersheds to climatic extremes

Hydrologic resilience modeling is used in public watershed management to assess watershed ability to supply life-supporting ecoservices under extreme climatic and environmental conditions. Literature surveys criticize resilience models for failing to capture watershed dynamics and undergo adequate testing. Both shortcomings compromise their ability to provide management options reliably protecting water security under real-world conditions. We formulate an empirical protocol to establish real-world correspondence. The protocol applies empirical nonlinear dynamics to reconstruct hydrologic dynamics from watershed records, and analyze the response of reconstructed dynamics to extreme regional climatic conditions. We devise an AI-based early-warning system to forecast (out-of-sample) reconstructed hydrologic resilience dynamics. Application to the La Tejería (Spain) experimental watershed finds it to be a low dimensional nonlinear deterministic dynamic system responding to internal stressors by irregularly oscillating along a watershed attractor. Reconstructed and forecasted hydrologic resilience behavior faithfully captures monthly wet-cold/dry-warm weather patterns characterizing the Mediterranean region.

Transmission of COVID-19 in the presence of single-dose and double-dose vaccines with hesitancy: mathematical modeling and optimal control analysis

IntroductionThe unexpected emergence of novel coronavirus identified as SAR-CoV-2 virus (severe acute respiratory syndrome corona virus 2) disrupted the world order to an extent that the human activities that are core to survival came almost to a halt. The COVID-19 pandemic created an insurmountable global health crisis that led to a united front among all nations to research on effective pharmaceutical measures that could stop COVID-19 proliferation. Consequently, different types of vaccines were discovered (single-dose and double-dose vaccines). However, the speed at which these vaccines were developed and approved to be administered created other challenges (vaccine skepticism and hesitancy).MethodThis paper therefore tracks the transmission dynamics of COVID-19 using a non-linear deterministic system that accounts for the unwillingness of both susceptible and partially vaccinated individuals to receive either single-dose or double-dose vaccines (vaccine hesitancy). Further the model is extended to incorporate three time-dependent non-pharmaceutical and pharmaceutical intervention controls, namely preventive control, control associated with screening-management of both truly asymptomatic and symptomatic infectious individuals and control associated with vaccination of susceptible individuals with a single dose vaccine. The Pontryagin's Maximum Principle is applied to establish the optimality conditions associated with the optimal controls.ResultsIf COVID-19 vaccines administered are imperfect and transient then there exist a parameter space where backward bifurcation occurs. Time profile projections depict that in a setting where vaccine hesitancy is present, administering single dose vaccines leads to a significant reduction of COVID-19 prevalence than when double dose vaccines are administered. Comparison of the impact of vaccine hesitancy against either single dose or double dose on COVID-19 prevalence reveals that vaccine hesitancy against single dose is more detrimental than vaccine hesitancy against a double dose vaccine. Optimal analysis results reveal that non-pharmaceutical time-dependent control significantly flattens the COVID-19 epidemic curve when compared with pharmaceutical controls. Cost-effectiveness assessment suggest that non-pharmaceutical control is the most cost-effective COVID-19 mitigation strategy that should be implemented in a setting where resources are limited.DiscussionPolicy makers and medical practitioners should assess the level of COVID-19 vaccine hesitancy inorder to decide on the type of vaccine (single-dose or double-dose) to administer to the population.

Control of chaotic systems through reservoir computing

Chaos is an important dynamic feature, which generally occurs in deterministic and stochastic nonlinear systems and is an inherent characteristic that is ubiquitous. Many difficulties have been solved and new research perspectives have been provided in many fields. The control of chaos is another problem that has been studied. In recent years, a recurrent neural network has emerged, which is widely used to solve many problems in nonlinear dynamics and has fast and accurate computational speed. In this paper, we employ reservoir computing to control chaos in dynamic systems. The results show that the reservoir calculation algorithm with a control term can control the chaotic phenomenon in a dynamic system. Meanwhile, the method is applicable to dynamic systems with random noise. In addition, we investigate the problem of different values for neurons and leakage rates in the algorithm. The findings indicate that the performance of machine learning techniques can be improved by appropriately constructing neural networks.

Stability analysis of optimal control problems with time-dependent costs

We present stability conditions for deterministic time-varying nonlinear discrete-time systems whose inputs aim to minimize an infinite-horizon time-dependent cost. Global asymptotic and exponential stability properties for general attractors are established. This work covers and generalizes the related results on discounted optimal control problems to more general systems and cost functions.

Adaptive finite-time neural network control for nonlinear stochastic systems with state constraints

This paper focuses on the adaptive finite-time neural network control problem for nonlinear stochastic systems with full state constraints. Adaptive controller and adaptive law are designed by backstepping design with log-type barrier Lyapunov function. Radial basis function neural networks are employed to approximate unknown system parameters. It is proved that the tracking error can achieve finite-time convergence to a small region of the origin in probability and the state constraints are confirmed in probability. Different from deterministic nonlinear systems, here the stochastic system is affected by two random terms including continuous Brownian motion and discontinuous Poisson jump process. Therefore, it will bring difficulties to the controller design and the estimations of unknown parameters. A simulation example is given to illustrate the effectiveness of the designed control method.

Псевдоспектральный метод поиска оптимального управления пучками траекторий на базе мультиагентных алгоритмов оптимизации

<p>A class of problems of optimal control of nonlinear continuous deterministic systems under conditions of uncertainty is considered. To solve the problem, a numerical algorithm for finding the optimal control is formed, in which the parameterization of the control law is used, which depends on time and a set of coordinates of the state vector available for measurement. This approach is based on the approximation of the control law by a series using a system of basis functions with unknown coefficients. The search for unknown coefficients in the expansion of the control law is implemented using multi-agent optimization methods: a hybrid multi-agent interpolation search algorithm and a multi-agent algorithm based on the use of linear controllers for controlling the movement of agents. A software has been developed and two model examples and an applied problem of stabilizing a satellite with the help of engines installed on it have been solved.</p>

Onset of chaotic gravitational lensing in non-Kerr rotating black holes with quadrupole mass moment* *Supported by the National Natural Science Foundation of China (NNSFC, 12005077) and Guangdong Basic and Applied Basic Research Foundation (2021A1515012374). We also gratefully acknowledge the financial support from Brazilian agencies Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa

In the electromagnetic channel, chaotic gravitational lensing is a peculiar phenomenon in strong gravitational lensing. In this study, we analyze the properties and emergence of chaotic gravitational lensing in the Manko-Novikov black hole spacetime. Aiming to better understand the underlying physics, we elaborate on the boundaries of the accessible region through analyses of the contours of the effective potentials. The latter is associated with the two roots of a quadratic equation. In particular, we explore its interplay with an ergoregion, which leads to specific features of the effective potentials, such as the emergence of a cuspy edge and the formation of a pocket, which serve as static constraints on the geodesics. Additionally, we investigate the properties of the radial and angular accelerations at the turning points in photon trajectories. The accelerations are further examined and may provide kinematic constraints on the geodesics, as argued herein. It is concluded that the onset of the chaotic lensing is significantly related to both constraints; as a result, an arbitrary slight deviation in the incident photon is significantly amplified during evolution through an extensive period, demonstrating the complexity in the highly nonlinear deterministic gravitational system.

A Convex Approach to Data-Driven Optimal Control via Perron–Frobenius and Koopman Operators

This article is about the data-driven computation of optimal control for a class of control affine deterministic nonlinear systems. We assume that the control dynamical system model is not available, and the only information about the system dynamics is available in the form of time-series data. We provide a convex formulation for the optimal control problem (OCP) of the nonlinear system. The convex formulation relies on the duality result in the dynamical system’s stability theory involving density function and Perron–Frobenius operator. We formulate the OCP as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the Koopman operator’s data-driven computation, which is dual to the Perron–Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Model-based inverse reinforcement learning for deterministic systems

This paper focuses on the development of an online data-driven model-based inverse reinforcement learning (MBIRL) technique for linear and nonlinear deterministic systems. Input and output trajectories of an agent under observation, attempting to optimize an unknown reward function, are used to estimate the reward function and the corresponding unknown optimal value function, online and in real-time. To achieve MBIRL using limited data, a novel feedback-driven approach to MBIRL is developed. The feedback policy and the dynamic model of the agent under observation are estimated from the measured data and the estimates are used to generate synthetic data to drive MBIRL. Theoretical guarantees for ultimate boundedness of the estimation errors in general, and convergence of the estimation errors to zero in special cases, are derived using Lyapunov techniques. Proof of concept numerical experiments demonstrates the utility of the developed method to solve linear and nonlinear inverse reinforcement learning problems.

Application of the modified self-organizing migration algorithm MSOMA in optimal open-loop control problems of switching systems

The article discusses the application of the metaheuristic algorithm of global constrained optimization for solving the problem of finding the optimal open-loop control for nonlinear switching deterministic dynamical systems. The quality of control is assessed by the value of the functional defined on individual trajectories. The optimal control problem is reduced to a parametric optimization problem, which is solved using theMSOMA algorithm, which belongs to the evolutionary group. The MSOMA algorithm is a new algorithm based on the SOMA self-organizing migration algorithm. The modification consists in identifying three leaders among the individuals forming the current population. For each member of the population, two clones are generated with the same position. Thus, in fact, three populations are generated, each of which then realizes a migration cycle (evolutionary process) relative to one of the three selected leaders. A step-by-step algorithm for piecewise-constant, piecewiselinear, quadratic spline and cubic spline methods of control laws approximation is proposed. The effectiveness of the proposed method is demonstrated by the example of solving optimal control problems for switching systems with two and three subsystems. The influence of the parameters of the MSOMA algorithm on the quality of the obtained result is investigated. Comparison of the operation of the algorithm with the known solution is carried out.

Safe Model-Based Reinforcement Learning for Systems With Parametric Uncertainties.

Reinforcement learning has been established over the past decade as an effective tool to find optimal control policies for dynamical systems, with recent focus on approaches that guarantee safety during the learning and/or execution phases. In general, safety guarantees are critical in reinforcement learning when the system is safety-critical and/or task restarts are not practically feasible. In optimal control theory, safety requirements are often expressed in terms of state and/or control constraints. In recent years, reinforcement learning approaches that rely on persistent excitation have been combined with a barrier transformation to learn the optimal control policies under state constraints. To soften the excitation requirements, model-based reinforcement learning methods that rely on exact model knowledge have also been integrated with the barrier transformation framework. The objective of this paper is to develop safe reinforcement learning method for deterministic nonlinear systems, with parametric uncertainties in the model, to learn approximate constrained optimal policies without relying on stringent excitation conditions. To that end, a model-based reinforcement learning technique that utilizes a novel filtered concurrent learning method, along with a barrier transformation, is developed in this paper to realize simultaneous learning of unknown model parameters and approximate optimal state-constrained control policies for safety-critical systems.

Application of Mini-Batch Metaheuristic Algorithms in Problems of Optimization of Deterministic Systems with Incomplete Information about the State Vector

In this paper, we consider the application of the zero-order mini-batch optimization method in the problem of finding optimal control of a pencil of trajectories of nonlinear deterministic systems in the case of incomplete information about the state vector. The pencil of trajectories originates from a given set of initial states. To solve the problem, the structure of a feedback system is proposed, which contains models of the plant, measuring system, nonlinear state observer and control law of the fixed structure with unknown coefficients. The objective function proposed considers the quality of pencil of trajectories control, which is estimated by the average value of the Bolz functional over the given set of initial states. Unknown control laws of a plant and an observer are found in the form of expansions in terms of orthonormal systems of basis functions, which are specified on the set of possible states of a dynamical system. The original pencil of trajectories control problem is reduced to a global optimization problem, which is solved using the well-proven zero-order method, which uses a modified mini-batch approach in a random search procedure with adaptation. An algorithm for solving the problem is proposed. The satellite stabilization problem with incomplete information is solved.

Application of the modified self-organizing migration algorithm MSOMA in optimal open-loop control problems

The article discusses the application of the metaheuristic algorithm of global constrained optimization for solving the problem of finding the optimal open-loop control for nonlinear continuous deterministic dynamical systems. The quality of control is assessed by the value of the functional defined on individual trajectories. The optimal control problem is reduced to a parametric optimization problem, which is solved using the MSOMA algorithm, which belongs to the evolutionary group. The MSOMA algorithm is a new algorithm based on the SOMA self-organizing migration algorithm. A step-by-step algorithm for piecewise-constant, piecewise-linear, quadratic spline and cubic spline methods of control laws approximation is proposed. The effectiveness of proposed method is demonstrated by the example of solving the problem of optimal control of a chemical process in a mixing reactor and a singular problem of optimal control. The influence of the parameters of the MSOMA algorithm on the quality of the obtained result is investigated. Comparison of the operation of the algorithm with a known solution, as well as with a solution using the original SOMA method is carried out.

Application of the mini-batch adaptive method of random search (MAMRS) in problems of optimal in mean control of the trajectory pencils

The article discusses the application of one of the new methods of constrained optimization to solve the problem of finding optimal control of a pencil of trajectories of nonlinear deterministic systems emanating from a given set of initial states. The structure of a feedback system is proposed, which contains a model of a control object, a measurement system model, a state observer that generates an estimate of the state vector from incoming measurements, and a regulator. The quality of control is assessed by the value of the average value of the functional determined on individual trajectories. Unknown control laws for the object model and the state observer are found in the form of expansions in terms of orthonormal systems of basic functions defined on the set of admissible states of the dynamic system. The problem of controlling a pencil of trajectories is reduced to the problem of parametric optimization, which is solved using a mini-batch adaptive random search method. A step-by-step algorithm for solving the problem is proposed, which is demonstrated by solving the problem of tracking various coordinates of a dynamical system according to the measurement results. The influence of the mini-batch size on the achieved tracking accuracy is investigated.

Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit‐time optimal control Part I: Theory

AbstractThis work studies the problem of constructing control Lyapunov functions (CLFs) and feedback stabilization strategies for deterministic nonlinear control systems described by ordinary differential equations. Many numerical methods for solving the Hamilton–Jacobi–Bellman partial differential equations specifying CLFs typically require dense state space discretizations and consequently suffer from the curse of dimensionality. A relevant direction of attenuating the curse of dimensionality concerns reducing the computation of the values of CLFs and associated feedbacks at any selected states to finite‐dimensional nonlinear programming problems. We propose to use exit‐time optimal control for that purpose. This article is the first part of a two‐part work. First, we state an exit‐time optimal control problem with respect to a sublevel set of an appropriate local CLF and establish that, under a number of reasonable conditions, the concatenation of the corresponding value function and the local CLF is a global CLF in the whole domain of asymptotic null‐controllability. We also investigate the formulated optimal control problem. A modification of these constructions for the case when one does not find a suitable local CLF is provided as well. Our developments serve as a theoretical basis for a curse‐of‐dimensionality‐free approach to feedback stabilization, that is presented in the second part Yegorov et al. (2021) of this work together with supporting numerical simulation results.