General Dynamical Systems Research Articles

Internal Noise Interference to Warnings of Tipping Points in Generic Multidimensional Dynamical Systems

Internal Noise Interference to Warnings of Tipping Points in Generic Multidimensional Dynamical Systems

Decomposition and equivalence of general nonlinear dynamical control systems

AbstractProjection and equivalence concepts have been widely studied in the literature. In the present paper two nonlinear systems with the same number of inputs, but not the same number of state variables, are considered. Under mild assumptions, necessary and sufficient conditions are given for the existence of a submersion such that the higher dimensional system projects locally onto the other one. The solution to this problem has relevant applications, for instance in robotics. It includes as a special case the equivalence of nonlinear systems with no particular structure.

Identifying the bottleneck and its shifts for the nexus across water supply, power generation and environment systems: Hehuang, China

Identifying the bottleneck is the key to alerting and preventing unexpected failures of the nexus across water supply, power generation, and environment (WPE) systems. However, the bottleneck identification is challenged by the unclear concept of bottleneck as well as the difficulty of simulating complex emergent behaviors. Here, we develop a generic framework for identifying the bottleneck for the WPE nexus, which is defined as the subsystem that has the highest risk of leading to the failure of WPE nexus. First, the evolution of uncertainties of the state variables of WPE nexus is simulated by a generic stochastic dynamical system. Next, the propagation matrix, accumulation ratio (AR), and risk contribution ratio are defined as effective metrics and tools to identify the system bottleneck. Finally, the effects of human decisions and input stochasticity on the shifts of bottleneck are analyzed using sensitivity and scenario analysis. The WPE system in Hehuang Region, China, is selected as a case study. Results indicate that: (a) The effects of human decisions on ARs can be superimposed, suggesting a “superimposition method” for policy-making of Hehuang Region; (b) The subsystems can be classified as “endogenous” or “exogenous”, depending on the sources of accumulated uncertainties that are from local inputs, or inputs from the other subsystems; (c) There exist multiple bottlenecks at the same time that could shift from one subsystem to another, which complicates the prevention of the failures of WPE nexus. For the Hehuang Region, water and energy subsystems are more likely to be bottlenecks at the beginning stage, while society and environment subsystems are more likely to be bottlenecks at the steady stage. This study provides the first attempt to identify the bottleneck within the WPE nexus, which helps to provide realistic insights for preventing unexpected failures.

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Mean dimension of induced hyperspace dynamical systems

In this paper, we prove that the mean dimension of the hyperspace dynamical system induced by a dynamical system with positive mean dimension is infinite. Specifically, for the full shift (KZ,σ) over a finite-dimensional compact metric space K, we find that mdim(2KZ,2σ)=+∞ if and only if mdim(KZ,σ)>0. However, for general dynamical systems, we cannot assume that this result holds universally. In this paper, we present a counterexample where the mean dimension of the dynamical system is 0 while the mean dimension of the induced hyperspace dynamical system is infinite.

A General, Noise-Driven Mechanism for the 1/f-Like Behavior of Neural Field Spectra.

Consistent observations across recording modalities, experiments, and neural systems find neural field spectra with 1/f-like scaling, eliciting many alternative theories to explain this universal phenomenon. We show that a general dynamical system with stochastic drive and minimal assumptions generates 1/f-like spectra consistent with the range of values observed in vivo without requiring a specific biological mechanism or collective critical behavior.

Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonance.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.

Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.

The potential impact of the new ‘Right to Repair’ rules on electrical and electronic equipment waste: A case study of the UK

Every year an estimated two million tonnes of waste electrical and electronic equipment (WEEE) are discarded by householders and companies in the United Kingdom (UK). While the UK has left the European Union (EU), its waste-related policies still mirror those of the EU, including the WEEE-related policies. Motivated by the recent introduction the so-called ‘Right to Repair’ policy for electrical and electronic equipment (EEE) across the EU and UK, this paper aims to demonstrate that, depending on the commitment and behavioural changes by the consumers and the government, the future of the WEEE management of the UK will vary. To this end, focusing on landfilled WEEE reduction we develop a generic system dynamics model and apply it to eleven WEEE categories. They depict the flow of EEE and WEEE representing the interaction among the stakeholders (e.g., consumers and producers of EEE) and relevant government regulations of the UK. Our four scenario analyses find that longer use of EEE and better WEEE collection seem to be effective in reducing landfilled WEEE, while more reuse and more recycling and recovery have negligible impacts, despite excluding the additional generation of landfilled WEEE as a result of recycling and recovery. Comparing with the business-as-usual scenario, one year longer EEE use and 10% more of WEEE collection could at maximum reduce landfilled WEEE by 14.05% of monitoring and control instruments and 93.93% of display equipment respectively. Backcasting scenario analyses reveal that significant efforts are required to reduce the targeted amounts.

Nonlinear model reduction to temporally aperiodic spectral submanifolds.

We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions, and control problems. The time-dependent SSMs we construct under these assumptions are normally hyperbolic and hence will persist for larger forcing and faster time dependence that are beyond the reach of our precise existence theory. For this reason, we also derive formal asymptotic expansions that, under explicitly verifiable nonresonance conditions, approximate SSMs and their aperiodic anchor trajectories accurately for stronger, faster, or even temporally discontinuous forcing. Reducing the dynamical system to these persisting SSMs provides a mathematically justified model- reduction technique for non-autonomous physical systems whose time dependence is moderate either in magnitude or speed. We illustrate the existence, persistence, and computation of temporally aperiodic SSMs in mechanical examples under chaotic forcing.

100 years of extremum seeking: A survey

Extremum seeking (ES) is a powerful approach to the optimization and stabilization of unknown dynamic systems and is an active field of research in control theory. This paper surveys recent ES algorithm developments. ES has evolved from its origins as a simple but brilliant engineering invention in the 1922 paper of Leblanc to the breakthrough work of Krstić and Wang in 2000 in which the first general stability analysis was given, to the most recent and numerous analytical results which prove convergence for a wide range of general dynamic systems. This survey provides a thorough overview of the dynamic systems for which ES stabilization and optimization results have been provided, the various analytical methods for proving convergence, and an overview of many real world applications for which ES has been utilized in industry and at scientific laboratories.

Nonequilibrium Schwinger-Keldysh formalism for density matrix states: Analytic properties and implications in cosmology

Motivated by cosmological Hartle-Hawking and microcanonical density matrix prescriptions for the quantum state of the Universe we develop a Schwinger-Keldysh in-in formalism for generic nonequilibrium dynamical systems with the initial density matrix. We build the generating functional of in-in Green’s functions and expectation values for a generic density matrix of the Gaussian type and show that the requirement of particle interpretation selects a distinguished set of positive/negative frequency basis functions of the wave operator of the theory, which is determined by the density matrix parameters. Then we consider a special case of the density matrix determined by the Euclidean path integral of the theory, which in the cosmological context can be considered as a generalization of the no-boundary pure state to the case of the microcanonical ensemble, and show that in view of a special reflection symmetry its Wightman Green’s functions satisfy Kubo-Martin-Schwinger periodicity conditions which hold despite the nonequilibrium and nonstationary nature of the physical setup. Rich analyticity structure in the complex plane of the time variable reveals the combined Euclidean-Lorentzian evolution of the theory, which depending on the properties of the initial density matrix can be interpreted as a decay of a classically forbidden quantum state. Published by the American Physical Society 2024

Adaptive exchange protocol for multi-agent communication in augmented reality system

Augmented Reality (AR) is one of the modern ways of providing users with different types of information. In applications based on this technology, the main influence on the subjective assessment of product quality is the speed of interaction between the system and the user and between users of the system. The main purpose of the work series are creation and modification for algorithms to assessing and improving the quality of AR services. This paper describes and justifies an adaptive communication protocol for multi-agent interaction. We consider a general nonlinear dynamic system with introducing feedback control which is based on measurements under almost arbitrary noise. In practice, this control is realized as a superposition of neural networks with the estimation of the result of mutual additional learning based on the system identification method (M.K.Campi and E.Weyer’s LSCR is used). The prototype is built for a system with a distributed server architecture and multiagent behavior of both clients and servers in the system.

Exact Response Theory for Time-Dependent and Stochastic Perturbations.

The exact, non perturbative, response theory developed within the field of non-equilibrium molecular dynamics, also known as TTCF (transient time correlation function), applies to quite general dynamical systems. Its key element is called the dissipation function because it represents the power dissipated by external fields acting on the particle system of interest, whose coupling with the environment is given by deterministic thermostats. This theory has been initially developed for time-independent external perturbations, and then it has been extended to time-dependent perturbations. It has also been applied to dynamical systems of different nature, and to oscillator models undergoing phase transitions, which cannot be treated with, e.g., linear response theory. The present work includes time-dependent stochastic perturbations in the theory using the Karhunen-Loève theorem. This leads to three different investigations of a given process. In the first, a single realization of the stochastic coefficients is fixed, and averages are taken only over the initial conditions, as in a deterministic process. In the second, the initial condition is fixed, and averages are taken with respect to the distribution of stochastic coefficients. In the last investigation, one averages over both initial conditions and stochastic coefficients. We conclude by illustrating the applicability of the resulting exact response theory with simple examples.

A Study on Some Properties for Weak Stability of Non-Autonomous Discrete Dynamical Systems

The interest for non-autonomous discrete dynamical systems has been increasing in recent years, because they are adequate to tell of real activities. For examples, when the mapping is disturbed in each iteration because of external factors, or model of some phenomena in physics, biology and economy, in specific, the population of human and weather forecasting, plus to solve problems generated in mathematics. In mathematics, stability theory addresses the stability of solutions of trajectories of dynamical systems under small perturbations of initial conditions. Besides that, the topological dynamics is the main method used in this paper, since we studied the non-autonomous discrete dynamical systems on a topological space. Next, we present a conception of weak stability (stableness) of non-autonomous discrete dynamical systems (NADDS). We evaluate a collection of weak stable points, and survey the connection among weak stableness and shadowing property. We also consider the connection among weak stableness of a non-autonomous discrete dynamical system and its induced system. In conclusion, the results of this paper will provide some helps in the modelling of a general dynamical system.

Data-driven model identification using forcing-induced limit cycles

This paper presents a data-driven model identification strategy that characterizes the behavior of a general dynamical system relative to a set of limit cycles that emerge in response to periodic forcing. Using time series data to infer the phase–amplitude dynamics associated with the underlying forced limit cycles, a low-order model can be obtained that accurately captures the dynamical behavior in response to arbitrary external inputs. The proposed strategy can be readily implemented in situations where full state measurements are unavailable and does not require any prior knowledge of the underlying model equations. This technique is applied to a model of coupled planar oscillators and to a model that considers the spike rate in a population of coupled conductance-based neurons where it outperforms two other commonly used data-driven model identification techniques.

PiP-X: Online feedback motion planning/replanning in dynamic environments using invariant funnels

Computing kinodynamically feasible motion plans and repairing them on-the-fly as the environment changes is a challenging, yet relevant problem in robot navigation. We propose an online single-query sampling-based motion re-planning algorithm using finite-time invariant sets, commonly referred to as “ funnels”. We combine concepts from nonlinear systems analysis, sampling-based techniques, and graph-search methods to create a single framework that enables feedback motion re-planning for any general nonlinear dynamical system in dynamic workspaces. A volumetric network of funnels is constructed in the configuration space using sampling-based methods and invariant set theory; and an optimal sequencing of funnels from robot configuration to a desired goal region is then determined by computing the shortest-path subgraph (tree) in the network. Analyzing and formally quantifying the stability of trajectories using Lyapunov level-sets ensures kinodynamic feasibility and guaranteed set-invariance of the solution paths. Though not required, our method is capable of using a pre-computed library of motion primitives to speedup online computation of controllable motion plans that are volumetric in nature. We introduce a novel directed-graph data structure to represent the funnel-network and its inter-sequencibility; helping us leverage discrete graph-based incremental search to quickly rewire feasible and controllable motion plans on-the-fly in response to changes in the environment. We validate our approach on a simulated cart-pole, car-like robot, and 6DOF quadrotor platform in a variety of scenarios within a maze and a random forest environment. Using Monte Carlo methods, we evaluate the performance in terms of algorithm success, length of traversed trajectory, and runtime.

High-dimensional multi-fidelity Bayesian optimization for quantum control

We present the first multi-fidelity Bayesian optimization (BO) approach for solving inverse problems in the quantum control of prototypical quantum systems. Our approach automatically constructs time-dependent control fields that enable transitions between initial and desired final quantum states. Most importantly, our BO approach gives impressive performance in constructing time-dependent control fields, even for cases that are difficult to converge with existing gradient-based approaches. We provide detailed descriptions of our machine learning methods as well as performance metrics for a variety of machine learning algorithms. Taken together, our results demonstrate that BO is a promising approach to efficiently and autonomously design control fields in general quantum dynamical systems.

Data-Driven Framework for Uncovering Hidden Control Strategies in Evolutionary Analysis

We devised a data-driven framework for uncovering hidden control strategies used by an evolutionary system described by an evolutionary probability distribution. This innovative framework enables deciphering of the concealed mechanisms that contribute to the progression or mitigation of such situations as the spread of COVID-19. Novel algorithms are used to estimate the optimal control in tandem with the parameters for evolution in general dynamical systems, thereby extending the concept of model predictive control. This marks a significant departure from conventional control methods, which require knowledge of the system to manipulate its evolution and of the controller’s strategy or parameters. We use a generalized additive model, supplemented by extensive statistical testing, to identify a set of predictor covariates closely linked to the control. Using real-world COVID-19 data, we delineate the descriptive behaviors of the COVID-19 epidemics in five prefectures in Japan and nine countries. We compare these nine countries and group them on the basis of shared profiles, providing valuable insights into their pandemic responses. Our findings underscore the potential of our framework as a powerful tool for understanding and managing complex evolutionary processes.

Optimal Control of Anti-Angiogenesis and Radiation Treatments for Cancerous Tumor: Hybrid Indirect Solver

This paper proposes a real-life volume reduction for cancer cells using optimal doses of radiation and an anti-angiogenic drug. A generalized dynamical system based on the diffusion-consumption equation along with stimulation and inhibition factors is proposed. To achieve continuous and low-dose therapy, the related problem is simulated by an optimal regulator problem mathematically. By combining steepest descent, conjugate gradient, and Armijo techniques, a novel hybrid indirect iterative solver is designed. For accuracy and execution speed, the current solver is compared with an interior-point optimizer and sequential quadratic Hamiltonian methods. Cancer therapy under two different treatment strategies and 24 various versions of the general dynamical system is considered numerically. A comprehensive analysis of the corresponding outcomes is presented. Numerical results and related diagrams are provided.