Deterministic Dynamical Systems Research Articles

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.

Back to basics: fundamental principles of system dynamics and queueing theory

Back to basics: fundamental principles of system dynamics and queueing theory

Macroscopic Dynamical Fluctuations in Kac Ring Model

We study dynamical fluctuations in the macroscopic paths around the most probable path of the Kac ring model, which is a simple deterministic and reversible dynamical system exhibiting the macroscopic irreversible relaxation. We derive the form of the generating function for macroscopic paths and show that the small deviations are described by a discrete-time Ornstein-Uhlenbeck process. We also argue that the microscopic reversibility leads to the fluctuation relation of the rate function, and prove it based on the form of the generating function.

Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures

This paper presents fast non-sampling based methods to assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents’ futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models to predict both agent positions and control inputs conditioned on the scene contexts. We show that the problem of risk assessment when Gaussian mixture models of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using nonlinear Chebyshev’s Inequality and sums-of-squares programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require higher order statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent control inputs as opposed to positions, we propagate the moments of uncertain control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planning horizon. To this end, we construct deterministic linear dynamical systems that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.

Relaxation of Some Confusions about Confounders

This work is about observational causal discovery for deterministic and stochastic dynamic systems. We explore what additional knowledge can be gained by the usage of standard conditional independence tests and if the interacting systems are located in a geodesic space.

Prioritization of the Target Population for Coronavirus Disease 2019 (COVID-19) Vaccination Program in Thailand.

Thailand was hit by the second wave of Coronavirus Disease 2019 (COVID-19) in a densely migrant-populated province (Samut Sakhon). COVID-19 vaccines were known to be effective; however, the supply was limited. Therefore, this study aimed to predict the effectiveness of Thailand’s COVID-19 vaccination strategy. We obtained most of the data from the Ministry of Public Health. Deterministic system dynamics and compartmental models were utilized. The reproduction number (R) between Thais and migrants was estimated at 1.25 and 2.5, respectively. Vaccine effectiveness (VE) to prevent infection was assumed at 50%. In Samut Sakhon, there were 500,000 resident Thais and 360,000 resident migrants. The contribution of migrants to the province’s gross domestic product was estimated at 20%. Different policy scenarios were analyzed. The migrant-centric vaccination policy scenario received the lowest incremental cost per one case or one death averted compared with the other scenarios. The Thai-centric policy scenario yielded an incremental cost of 27,191 Baht per one life saved, while the migrant-centric policy scenario produced a comparable incremental cost of 3782 Baht. Sensitivity analysis also demonstrated that the migrant-centric scenario presented the most cost-effective outcome even when VE diminished to 20%. A migrant-centric policy yielded the smallest volume of cumulative infections and deaths and was the most cost-effective scenario, independent of R and VE values. Further studies should address political feasibility and social acceptability of migrant vaccine prioritization.

From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy

Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are "as random as a coin-toss". Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power $e_p(U)$ of the basic two-particle unitary building block, $U$, of the circuit, that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally we show, both analytically and numerically, how local-averaging over random realizations of the single-particle unitaries, $u_i$ and $v_i$ such that the building block is $U^\prime = (u_1 \otimes u_2 ) U (v_1 \otimes v_2 )$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power $e_p(U)$. Finally we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.

Complexity-based permutation entropies: From deterministic time series to white noise

This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this “ordinal” methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of allowed permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of allowed permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes.

Recognition Methods of Geometrical Images of Automata Models of Systems in Control Problem

The laws of functioning of discrete deterministic dynamical systems are investigated, presented in the form of automata models defined by geometric images. Due to the use of the apparatus of geometric images of automata, developed by V.A. Tverdokhlebov, the analysis of automata models is carried out on the basis of the analysis of mathematical structures represented by geometric curves and numerical sequences. The purpose of present research is to further develop the mathematical apparatus of geometric images of automaton models of systems, including the development of new methods for recognizing automata by their geometric images, given both geometric curves and numerical sequences.

A bridge between invariant dynamical structures and uncertainty quantification

We develop a new quantifier for forward time uncertainty for trajectories that are solutions of models generated from data sets. Our uncertainty quantifier is defined on the phase space in which the trajectories evolve and we show that it has a rich structure that is directly related to phase space structures from dynamical systems theory, such as hyperbolic trajectories and their stable and unstable manifolds. We apply our approach to an ocean data set, as well as standard benchmark models from deterministic dynamical systems theory. A significant application of our results, is that they allow a quantitative comparison of the transport performance described from different ocean data sets. This is particularly interesting nowadays when a wide variety of sources are available since our methodology provides avenues for assessing the effective use of these data sets in a variety of situations.

A General Metric for the Similarity of Both Stochastic and Deterministic System Dynamics.

Many problems in the study of dynamical systems—including identification of effective order, detection of nonlinearity or chaos, and change detection—can be reframed in terms of assessing the similarity between dynamical systems or between a given dynamical system and a reference. We introduce a general metric of dynamical similarity that is well posed for both stochastic and deterministic systems and is informative of the aforementioned dynamical features even when only partial information about the system is available. We describe methods for estimating this metric in a range of scenarios that differ in respect to contol over the systems under study, the deterministic or stochastic nature of the underlying dynamics, and whether or not a fully informative set of variables is available. Through numerical simulation, we demonstrate the sensitivity of the proposed metric to a range of dynamical properties, its utility in mapping the dynamical properties of parameter space for a given model, and its power for detecting structural changes through time series data.

Handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics.

We introduce a general formulation of the fluctuation-dissipation relations (FDRs) holding also in far-from-equilibrium stochastic dynamics. A great advantage of this version of the FDR is that it does not require explicit knowledge of the stationary probability density function. Our formula applies to Markov stochastic systems with generic noise distributions: When the noise is additive and Gaussian, the relation reduces to those known in the literature; for multiplicative and non-Gaussian distributions (e.g., Cauchy noise) it provides exact results in agreement with numerical simulations. Our formula allows us to reproduce, in a suitable small-noise limit, the response functions of deterministic, strongly nonlinear dynamical models, even in the presence of chaotic behavior: This could have important practical applications in several contexts, including geophysics and climate. As a case of study, we consider the Lorenz '63 model, which is paradigmatic for the chaotic properties of deterministic dynamical systems.

AVALANCHES IN A SHORT-MEMORY EXCITABLE NETWORK.

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

Predicted Impact of Vaccination and Active Case Finding Measures to Control Epidemic of Coronavirus Disease 2019 in a Migrant-Populated Area in Thailand.

BackgroundThailand experienced the first wave of Coronavirus Disease 2019 (COVID-19) during March–May 2020 and has been facing the second wave since December 2020. The area facing the greatest impact was Samut Sakhon, a main migrant-receiving province in the country. The Department of Disease Control (DDC) of the Thai Ministry of Public Health (MOPH) considered initiating a vaccination strategy in combination with active case finding (ACF) in the epidemic area. The DDC commissioned a research team to predict the impact of various vaccination and ACF policy scenarios in terms of case reduction and deaths averted, which is the objective of this study.MethodsThe design of this study was a secondary analysis of quantitative data. Most of the data were obtained from the DDC, MOPH. Deterministic system dynamics and compartmental models were exercised. A basic reproductive number (R0) was estimated at 3 from the beginning. Vaccine efficacy against disease transmission was assumed to be 50%. A total of 10,000 people were estimated as an initial population size.ResultsThe findings showed that the greater the vaccination coverage, the smaller the size of incident and cumulative cases. Compared with a no-vaccination and no-ACF scenario, the 90%-vaccination coverage combined with 90%-ACF coverage contributed to a reduction of cumulative cases by 33%. The case reduction benefit would be greater when R0 was smaller (~53% and ~51% when R0 equated 2 and 1.5, respectively).ConclusionThis study reaffirmed the idea that a combination of vaccination and ACF measures contributed to favourable results in reducing the number of COVID-19 cases and deaths, relative to the implementation of only a single measure. The greater the vaccination and ACF coverage, the greater the volume of cases saved. Though we demonstrated the benefit of vaccination strategies in this setting, actual implementation should consider many more policy angles, such as social acceptability, cost-effectiveness and operational feasibility. Further studies that address these topics based on empirical evidence are of great value.

Scalable resetting algorithms for synchronization of pulse-coupled oscillators over rooted directed graphs

We study the problem of robust global synchronization of pulse-coupled oscillators (PCOs) over directed graphs. It is known that when the digraphs are strongly connected, global synchronization can be achieved by using a class of deterministic binary set-valued resetting controllers (Poveda and Teel, 2019). However, for large-scale networks, these algorithms are not scalable because some of their tuning parameters have upper bounds of the order O1N, where N is the number of agents. This paper resolves this scalability issue by presenting several new results about global synchronization of PCOs with more general network topologies using the frameworks of deterministic and stochastic hybrid dynamical systems. First, we establish that similar deterministic binary resetting algorithms can achieve robust global and fixed-time synchronization in any rooted acyclic digraph. Moreover, in this case, we show that the synchronization dynamics are now scalable as the tuning parameters of the algorithm are network independent, i.e., of order O(1). However, the algorithms cannot be further extended to all rooted digraphs. We establish this new impossibility result by introducing a counter-example with a particular rooted digraph for which global synchronization cannot be achieved, irrespective of the tuning parameters. Nevertheless, we show that if the binary resetting algorithms are modified by accommodating an Erdös–Renýi type random graph model, then the resulting stochastic resetting dynamics will guarantee global synchronization almost surely for all rooted digraphs and, moreover, the tuning parameters of the dynamics are network independent. Stability and robustness properties of the resetting algorithms are studied using the tools from set-valued hybrid dynamical systems. Numerical simulations are provided at the end of the paper for demonstration of the main results.

Dynamical systems as a level of cognitive analysis of multi-agent learning

A dynamical systems perspective on multi-agent learning, based on the link between evolutionary game theory and reinforcement learning, provides an improved, qualitative understanding of the emerging collective learning dynamics. However, confusion exists with respect to how this dynamical systems account of multi-agent learning should be interpreted. In this article, I propose to embed the dynamical systems description of multi-agent learning into different abstraction levels of cognitive analysis. The purpose of this work is to make the connections between these levels explicit in order to gain improved insight into multi-agent learning. I demonstrate the usefulness of this framework with the general and widespread class of temporal-difference reinforcement learning. I find that its deterministic dynamical systems description follows a minimum free-energy principle and unifies a boundedly rational account of game theory with decision-making under uncertainty. I then propose an on-line sample-batch temporal-difference algorithm which is characterized by the combination of applying a memory-batch and separated state-action value estimation. I find that this algorithm serves as a micro-foundation of the deterministic learning equations by showing that its learning trajectories approach the ones of the deterministic learning equations under large batch sizes. Ultimately, this framework of embedding a dynamical systems description into different abstraction levels gives guidance on how to unleash the full potential of the dynamical systems approach to multi-agent learning.

Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods

We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere. We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values $z$ in the Riemann sphere, for almost every sequence of maps $\gamma =(\gamma_{1},\gamma_{2},\ldots )$, the Lyapunov exponent of $\gamma $ at $z$ is negative. Also, we show that for a generic system, for every initial value $z$ in the Riemann sphere, the orbit of the Dirac measure at $z$ under the iteration of the dual map of the transition operator tends to a periodic cycle of measures in the space of probability measures on the Riemann sphere. Note that these are new phenomena in random complex dynamics which cannot hold in deterministic complex dynamical systems. We apply the above theory and results of random complex dynamical systems to finding roots of any polynomials by random relaxed Newton's methods and we show that for any polynomial $g$, for any initial value $z$ in the complex plane which is not a root of $g'$, the random orbit starting with $z$ tends to a root of $g$ almost surely, which is the virtue of the effect of randomness.

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.

Potentials of continuous Markov processes and random perturbations

With a scalar potential and a bivector potential, the vector field associated with the drift of a diffusion is decomposed into a generalized gradient field, a field perpendicular to the gradient, and a divergence-free field. We give such decomposition a probabilistic interpretation by introducing cycle velocity from a bivectorial formalism of nonequilibrium thermodynamics. New understandings on the mean rates of thermodynamic quantities are presented. Deterministic dynamical system is further proven to admit a generalized gradient form with the emerged potential as the Lyapunov function by the method of random perturbations.

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.