Discrete-time Dynamical Systems Research Articles

Propagating uncertainty through system dynamics in reproducing kernel Hilbert space

We present a data-driven approach to propagate uncertainty in initial conditions through the dynamics of an unknown system in a reproducing kernel Hilbert space (RKHS). The uncertainty in initial conditions is represented through its kernel mean embedding (KME) in the RKHS. For a discrete-time Markovian dynamical system, we utilize the conditional mean embedding (CME) operator to encode the underlying dynamics. Learning in RKHS often incurs prohibitive data storage requirements. To circumvent said limitation, we propose an algorithm to propagate uncertainty via a learned sparse CME operator, and provide theoretical guarantees on the approximation error for the embedded distribution with time. We empirically study our algorithm over illustrative dynamical systems and power systems.

State Dependent Delay Maps: Numerical Algorithms and Dynamics of Projections

This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a function of the state variable. A delay map with constant delays can be rewritten explicitly as a discrete time dynamical system on an appropriate function space, and a delay map with small state dependent terms can be viewed as a “non-autonomous” perturbation. We develop a fixed point formulation for the Cauchy problem of such perturbations, and under appropriate assumptions obtain the existence of forward iterates of the map. The proof is constructive and leads to numerical procedures which we implement for illustrative examples, including the cubic Ikeda and Mackey-Glass systems with constant and state-dependent delays. After proving a local convergence result for the method, we study more qualitative/global convergence issues using data analytic tools for time series analysis (dimension and topological measures derived from persistent homology). Using these tools we quantify the convergence of the dynamics in the finite dimensional projections to the dynamics of the infinite dimensional system.

Contract composition for dynamical control systems: Definition and verification using linear programming

Designing large-scale control systems to satisfy complex specifications is hard in practice, as most formal methods are limited to systems of modest size. Contract theory has been proposed as a modular alternative, in which specifications are defined by assumptions on the input to a component and guarantees on its output. However, current contract-based methods for control systems either prescribe guarantees on the state of the system, going against the spirit of contract theory, or are not supported by efficient computational tools. In this paper, we present a contract-based modular framework for discrete-time dynamical control systems. We extend the definition of contracts by allowing the assumption on the input at a time k to depend on outputs up to time k−1, which is essential when considering feedback systems. We also define contract composition for arbitrary interconnection topologies, and prove that this notion supports modular design, analysis and verification. This is done using graph theory methods, and specifically using the notions of topological ordering and backward-reachable nodes. Lastly, we present an algorithm for verifying vertical contracts, which are claims of the form “the conjunction of given component-level contracts implies given contract on the integrated system”. These algorithms are based on linear programming, and scale linearly with the number of components in the interconnected network. A numerical example is provided to demonstrate the scalability of the presented approach, as well as the modularity achieved by using it.

PAC-Bayes Generalisation Bounds for Dynamical Systems including Stable RNNs

In this paper, we derive a PAC-Bayes bound on the generalisation gap, in a supervised time-series setting for a special class of discrete-time non-linear dynamical systems. This class includes stable recurrent neural networks (RNN), and the motivation for this work was its application to RNNs. In order to achieve the results, we impose some stability constraints, on the allowed models. Here, stability is understood in the sense of dynamical systems. For RNNs, these stability conditions can be expressed in terms of conditions on the weights. We assume the processes involved are essentially bounded and the loss functions are Lipschitz. The proposed bound on the generalisation gap depends on the mixing coefficient of the data distribution, and the essential supremum of the data. Furthermore, the bound converges to zero as the dataset size increases. In this paper, we 1) formalize the learning problem, 2) derive a PAC-Bayesian error bound for such systems, 3) discuss various consequences of this error bound, and 4) show an illustrative example, with discussions on computing the proposed bound. Unlike other available bounds the derived bound holds for non i.i.d. data (time-series) and it does not grow with the number of steps of the RNN.

Fixed time stability of discrete-time stochastic dynamical systems

In this paper, we address fixed time stability in probability of discrete-time stochastic dynamical systems. Unlike finite time stability in probability, wherein the finite time almost sure convergence behavior of the dynamical system depends on the system initial conditions, fixed time stability in probability involves finite time stability in probability for which the stochastic settling-time is guaranteed to be independent of the system initial conditions. More specifically, we develop Lyapunov theorems for fixed time stability in probability for Itô-type stationary nonlinear stochastic difference equations including a Lyapunov theorem that involves a Lyapunov difference satisfying an exponential inequality of the Lyapunov function that gives rise to a minimum bound on the average stochastic settling-time characterized by the primary and secondary branches of the Lambert W function.

Discrete-time systems properties defined on state-space regions

In this paper various properties of discrete-time dynamic control systems trajectories with respect to state-space corner regions are considered. The known notion of state-space invariance serves as a basis for derivation of the whole family of dynamics behaviors for which both necessary and sufficient conditions are derived in a general nonlinear case as well as in the linear time-invariant case, shortly LTI-case. Specific examples are given for every case considered, and the proposed notions are analyzed with both theoretical and practical usefulness in mind. For the general nonlinear case a geometric approach is used, which provides a direct insight into the nature of trajectories’ behavior. In the LTI-case a geometric approach is used as well, but it is also translated into the purely algebraic set of conditions allowing for a direct analysis of system matrices. The presented family of control systems expands on the classical state-space invariance (and positive systems) analysis, thus potentially opening new research venues in this branch of control theory and system dynamics in general.

Capital exploitation and environmental awareness: How they affect the economy and the environment in a dynamic framework

AbstractWe present a model in which capital and environmental quality co-evolve over time. To improve the environmental quality, the government intervenes by means of a limitation of the capital use and awareness campaigns. In case of severe degradation of the environment, a restriction on capital use is introduced that is proportional to the damage caused by human activity; at the same time, awareness campaigns are used to increase the public concern about sustainability. By means of a discrete-time dynamical system and considering homogeneous agents, we found that multiple equilibria may exist and that awareness campaigns are a useful tool to push an economy toward sustainable levels of production. The limitation in the use of available capital, however, might be useless or even harmful, deteriorating the level of capital disposable for those countries that are trapped in an equilibrium in which the environmental quality is low.

Construction of contraction metrics for discrete-time dynamical systems using meshfree collocation

Construction of contraction metrics for discrete-time dynamical systems using meshfree collocation

Fractional Cumulative Residual Inaccuracy Information Measure and Its Extensions with Application to Chaotic Maps

The purpose of this work is to introduce fractional cumulative residual inaccuracy (FCRI) information, Jensen-cumulative residual inaccuracy (JCRI), and Jensen-fractional cumulative residual inaccuracy (JFCRI) information measure. Further, we study the FCRI information for some well-known models used in reliability, economics and survival analysis. The associated results reveal some interesting connections between the FCRI information measure and cumulative residual entropy and Gini mean difference measures. Applications to two chaotic discrete-time dynamical systems (Chebyshev and Logistic) are presented to illustrate the behavior of the proposed information measures. FCRI and JFCRI measures allow to determine regions of discrepancy between systems, depending on their respective fractional and chaotic map parameters.

Structurally Unstable Synchronization and Border-Collision Bifurcations in the Two-Coupled Izhikevich Neuron Model

This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton’s method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the “strike-slip fault” observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system.

A new study for global dynamics and numerical simulation of a discrete-time computer virus propagation model

Truong Ha Hai, Pham Hoai Thu, Hoang Manh Tuan Abstract— This work is devoted to conducting a new study for global dynamics and numerical simulation of a discrete-time computer virus propagation model, which was constructed in our recent work. By utilizing well-known results on asymptotic stability of discrete-time dynamical systems, we establish the global asymptotic stability of a unique viral equilibrium point, whereas only its local asymptotic stability was previously analyzed. After that, we investigate convergence and provide an error bound for the discrete-time model. Next, the step doubling strategy is applied to control errors. The result is that the accuracy of approximations generated by the discrete-time model is enhanced. The obtained results not only improve the ones constructed in the benchmark work, but also can be useful to study reliable numerical methods for mathematical models of malware. Finally, we present two numerical experiments that support and illustrate the theoretical findings of this study.

Observability and Detectability of Stochastic Labeled Graphs

In this paper, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">observable stochastic graphs and detetectable stochastic graphs</i> are, respectively, defined with the detailed implementation for the observability and detectability of stochastic discrete-time and discrete-state dynamic systems. More specifically, they are generally two classes of vertex-colored and edge-labeled graphs rendering a walking agent therein to determine his initial and current positions, respectively, in probability one by measuring the color sequence of his traversed vertices. In the part of analysis, we follow the aforementioned two definitions and establish the corresponding polynomial-time verifying algorithms. Notably, the implicit formulas are also established to calculate the observability and detectability probability for any pairwise vertex pair. In the synthesis part, the minimal number of colors dyeing the vertices to make any considered graph stochastically observable and detectable is investigated, respectively. Our results indicate that the minimal coloring problem is NP-hard for observability in the stochastic situations but is solvable in polynomial time for detectability in the deterministic cases. The observability and detectability of stochastic finite-valued systems, assembling with the finite-cardinal state spaces, are validated as a compelling application of these two types of directed graphs, whereas the minimal sensors placement problems subject to observability and detectability problems are accordingly interpreted by the minimum set cover algorithm.

The [formula omitted]-compound of a difference–algebraic system

The multiplicative and additive compounds of a matrix have important applications in geometry, linear algebra, and the analysis of dynamical systems. In particular, the k-compounds allow to build a k-compound dynamical system that tracks the evolution of k-dimensional parallelotopes along the original dynamics. This has recently found many applications in the analysis of non-linear systems described by ODEs and difference equations. Here, we introduce the k-compound system corresponding to a difference–algebraic system, and describe several applications to the analysis of discrete-time dynamical systems.

Set-based state estimation for discrete-time constrained nonlinear systems: An approach based on constrained zonotopes and DC programming

This paper introduces a novel state estimator designed for discrete-time nonlinear dynamical systems that encompass unknown-but-bounded uncertainties, along with state linear inequality and nonlinear equality constraints. Our algorithm is based on constrained zonotopes (CZs) and employs a DC programming approach (where DC stands for difference of convex functions). Recently, techniques such as mean value extension and first-order Taylor extension have been adapted from zonotopes to facilitate the propagation of CZs across nonlinear mappings. While the resulting algorithms, known as CZMV and CZFO, achieve higher precision compared to the original zonotopic versions, they still exhibit sensitivity to the wrapping and dependency effects inherent in interval arithmetic. These interval-related challenges can be mitigated through the use of DC programming, as it allows for the determination of approximation error bounds by solving optimization problems. One direct advantage of this technique is the elimination of the dependency effect. Our set-membership filter, referred to as CZDC, provides an alternative solution to CZMV and CZFO. To showcase the effectiveness of our proposed approach, we conducted experiments using CZDC on two numerical examples.

A Generalized Stacked Reinforcement Learning Method for Sampled Systems

A common setting of reinforcement learning (RL) is a Markov decision process (MDP) in which the environment is astochastic discrete-time dynamical system. Whereas MDPs are suitable in such applications as video-games or puzzles, physical systems are time-continuous. A general variant of RL is of digital format, where updates of the value (or cost) and policy are performed at discrete moments in time. The agent-environment loop then amounts to a sampled system, whereby sample-and-hold is a specific case. In this paper, we propose and benchmark two RL methods suitable for sampled systems. Specifically, we hybridize model-predictive control (MPC) with critics learning the optimal Q- and value (or cost-to-go) function. Optimality is analyzed and performance comparison is done in an experimental case study with a mobile robot.

Application of Migrating Optimization Algorithms in Problems of Optimal Control of Discrete-Time Stochastic Dynamical Systems

The problem of finding the optimal open-loop control for discrete-time stochastic dynamical systems is considered. It is assumed that the initial conditions and external influences are random. The average value of the Bolza functional defined on individual trajectories is minimized. It is proposed to solve the problem by means of classical and modified migrating optimization algorithms. The modification of the migrating algorithm consists of cloning the members of the initial population and choosing different strategies of migratory behavior for the main population and for populations formed by clones. At the final stage of the search for an extremum, an intensively clarifying migration cycle is implemented with the participation of three leaders of the populations participating in the search process. Problems of optimal control of bundles of trajectories of deterministic discrete dynamical systems, as well as individual trajectories, are considered as special cases. Seven model examples illustrating the performance of the proposed approach are solved.

Synchronizability of Discrete Nonlinear Systems: A Master Stability Function Approach

In recent times, studies on discrete nonlinear systems received much attention among researchers because of their potential applications in real-world problems. In this study, we conducted an in-depth exploration into the stability of synchronization within discrete nonlinear systems, specifically focusing on the Hindmarsh–Rose map, the Chialvo neuron model, and the Lorenz map. Our methodology revolved around the utilization of the master stability function approach. We systematically examined all conceivable coupling configurations for each model to ascertain the stability of synchronization manifolds. The outcomes underscored that only distinct coupling schemes manifest stable synchronization manifolds, while others do not exhibit this trait. Furthermore, a comprehensive analysis of the master stability function’s behavior was performed across a diverse range of coupling strengths σ and system parameters. These findings greatly enhance our understanding of network dynamics, as discrete-time dynamical systems adeptly replicate the dynamics of continuous-time models, offering significant reductions in computational complexity.

Gonosomal algebras and associated discrete-time dynamical systems

In this paper we study the discrete-time dynamical systems associated with gonosomal algebras used as algebraic model in the sex-linked genes inheritance. We show that the class of gonosomal algebras is disjoint from the other non-associative algebras usually studied (Lie, alternative, Jordan, associative power). To each gonosomal algebra, with the mapping x↦12x2, an evolution operator W is associated that gives the state of the offspring population at the birth stage, then from W we define the operator V which gives the frequency distribution of genetic types. We study discrete-time dynamical systems generated by these two operators, in particular we show that the various stability notions of the equilibrium points are preserved by passing from W to V. Moreover, for the evolution operators associated with genetic disorders in the case of a diallelic gonosomal lethal gene we give complete analysis of fixed and limit points of the dynamical systems.

Finite Sample Analysis for Structured Discrete System Identification

We consider a discrete-time dynamical system over a discrete state-space, which evolves according to a structured Markov model called Bernoulli Autoregressive (BAR) model. Our goal is to obtain sample complexity bounds for the problem of estimating the parameters of this model using an indirect Maximum Likelihood Estimator. Our sample complexity bounds exploit the structure of the BAR model and are established using concentration inequalities for random matrices and Lipschitz functions.

Discrete-time nonlinear feedback linearization via physics-informed machine learning

We present a physics-informed machine learning (PIML) scheme for the feedback linearization of nonlinear discrete-time dynamical systems. The PIML finds the nonlinear transformation law, thus ensuring stability via pole placement, in one step. In order to facilitate convergence in the presence of steep gradients in the nonlinear transformation law, we address a greedy training procedure. We assess the performance of the proposed PIML approach via a benchmark nonlinear discrete map for which the feedback linearization transformation law can be derived analytically; the example is characterized by steep gradients, due to the presence of singularities, in the domain of interest. We show that the proposed PIML outperforms, in terms of numerical approximation accuracy, the traditional numerical implementation, which involves the construction –and the solution in terms of the coefficients of a power-series expansion–of a system of homological equations as well as the implementation of the PIML in the entire domain, thus highlighting the importance of continuation techniques in the training procedure of PIML schemes.