Continuous-time Dynamical Systems Research Articles

On dissipative symplectic integration with applications to gradient-based optimization

Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in this line of work is how to discretize the system in such a way that its stability and rates of convergence are preserved. In this paper we propose a geometric framework in which such discretizations can be realized systematically, enabling the derivation of ‘rate-matching’ algorithms without the need for a discrete convergence analysis. More specifically, we show that a generalization of symplectic integrators to non-conservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error. Moreover, such methods preserve a shadow Hamiltonian despite the absence of a conservation law, extending key results of symplectic integrators to non-conservative cases. Our arguments rely on a combination of backward error analysis with fundamental results from symplectic geometry. We stress that although the original motivation for this work was the application to optimization, where dissipative systems play a natural role, they are fully general and not only provide a differential geometric framework for dissipative Hamiltonian systems but also substantially extend the theory of structure-preserving integration.

Physical Deep Learning Based on Optimal Control of Dynamical Systems

Deep learning is the backbone of artificial intelligence technologies, and it can be regarded as a kind of multilayer feedforward neural network. An essence of deep learning is information propagation through layers. This suggests that there is a connection between deep neural networks and dynamical systems in the sense that information propagation is explicitly modeled by the time-evolution of dynamical systems. In this study, we perform pattern recognition based on the optimal control of continuous-time dynamical systems, which is suitable for physical hardware implementation. The learning is based on the adjoint method to optimally control dynamical systems, and the deep (virtual) network structures based on the time evolution of the systems are used for processing input information. As a key example, we apply the dynamics-based recognition approach to an optoelectronic delay system and demonstrate that the use of the delay system allows for image recognition and nonlinear classifications using only a few control signals. This is in contrast to conventional multilayer neural networks, which require a large number of weight parameters to be trained. The proposed approach provides insight into the mechanisms of deep network processing in the framework of an optimal control problem and presents a pathway for realizing physical computing hardware.

Applications of the Poincaré–Hopf Theorem: Epidemic Models and Lotka–Volterra Systems

This article focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré–Hopf theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria and ensures that the system is in fact locally exponentially stable. We apply this result to the susceptible–infected–susceptible (SIS) networked epidemic model, and a generalized Lotka–Volterra system. We use the result further to extend the SIS model via the introduction of decentralized feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincaré–Hopf approach, we identify a necessary and sufficient condition, under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady state), and monotone systems theory is used to show that this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented.

Exponential stabilization of continuous-time dynamical systems via time and event triggered aperiodic intermittent control

This paper studies the exponential stabilization of continuous-time dynamical systems via aperiodic intermittent controls (APIC). Time-triggered and event-triggered APIC schemes are designed respectively for continuous-time dynamical systems. Under time-triggered APIC (T-APIC), by proposing the notions of (minimum) average control width and using the method of Lyapunov-like function, the exponential stabilization criteria expressed by converse average dwell-time conditions are derived. Based on indices of threshold-value and check-period, the event-triggered APIC (E-APIC) is proposed. E-APIC is state dependent and the algorithm for E-APIC is given to derive criteria of exponential stabilization. Moreover, by introducing control-free index into APIC, an E-APIC with state-based control widths (SCW) is proposed to improve the constraints on control widths in E-APIC and T-APIC. In order to verify performances of the intermittent control schemes including periodic intermittent control (PIC), some discussions are given by using notions of number of control, rate of control, and cost of control. Finally, one example with numerical simulations is given for illustrations. It is shown that both PIC and T-APIC are stronger control schemes than E-APIC and E-APIC with SCW. E-APIC and E-APIC with SCW have better performances than PIC and T-APIC with lower values on rate of control, number of control, and cost of control. And E-APIC with SCW has lowest rate of control and can achieve least total control time.

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis

Interval approaches for the reachability analysis of initial value problems for sets of classical ordinary differential equations have been investigated and implemented by many researchers during the last decades. However, there exist numerous applications in computational science and engineering, where continuous-time system dynamics cannot be described adequately by integer-order differential equations. Especially in cases in which long-term memory effects are observed, fractional-order system representations are promising to describe the dynamics, on the one hand, with sufficient accuracy and, on the other hand, to limit the number of required state variables and parameters to a reasonable amount. Real-life applications for such fractional-order models can, among others, be found in the field of electrochemistry, where methods for impedance spectroscopy are typically used to identify fractional-order models for the charging/discharging behavior of batteries or for the dynamic relation between voltage and current in fuel cell systems if operated in a non-stationary state. This paper aims at presenting an iterative method for reachability analysis of fractional-order systems that is based on an interval arithmetic extension of Mittag-Leffler functions. An illustrating example, inspired by a low-order model of battery systems concludes this contribution.

Kernel-based identification of asymptotically stable continuous-time linear dynamical systems

ABSTRACT In many engineering applications, continuous-time models are preferred to discrete-time ones, in that they provide good physical insight and can be derived also from non-uniformly sampled data. However, for such models, model selection is a hard task if no prior physical knowledge is given. In this paper, we propose a non-parametric approach to infer a continuous-time linear model from data, by automatically selecting a proper structure of the transfer function and guaranteeing to preserve the system stability properties. By means of benchmark simulation examples, the proposed approach is shown to outperform state-of-the-art continuous-time methods, also in the critical case when short sequences of canonical input signals, like impulses or steps, are used for model learning.

Booms and Busts in the Oil Market: Identifying Speculative Bubbles Using a Continuous-Time Dynamic System

The sharp changes in oil prices since 2004 featured a nonlinear data-generating mechanism which displayed bubble-like behavior. A popular view is that such a salient pattern cannot be explained by shifts in economic fundamentals, but was driven by speculative bubbles as a consequence of the increased financialization of oil future markets. Testing this hypothesis, however, is challenging since the fundamental component of the oil price is unobservable. This paper attempts to isolate the contribution of speculative bubbles and fundamentals to the evolution of oil prices by providing a stylized model of commodity pricing. Motivated by our theoretical model, we adopt a continuous-time model with a random and time-varying persistence parameter to empirically investigate the presence of speculative bubbles in daily oil future prices over the period April 1983 to June 2020. We do not find any evidence in favor of speculative bubbles, although we indeed find that oil prices exhibit episodes of unstable behavior after 2004.

Lyapunov exponent algorithm based on perturbation theory for discontinuous systems

Lyapunov exponent is a significant symbol to identify the nonlinear dynamic characteristics of the system. However, most of algorithms are not universal enough and complex. According to the classic Lyapunov exponent algorithm and perturbation theory, in this paper we propose a new algorithm which can be used to compute Lyapunov exponents for discontinuous systems. Firstly, the initial value of the system state parameter and the disturbance of each basic vector along the phase space are taken as initial conditions to determine the phase trajectory. Secondly, the method of difference quotient approximate derivative is adopted to obtain the Jacobi matrix. Thirdly, the eigenvalues of the Jacobi matrix are calculated to obtain the Lyapunov exponent spectrum of the system. Finally, the algorithm in a two-degree-of-freedom system with impacts and friction is used, showing its effectiveness and correctness by comparing its results with the counterparts from the synchronization method. The algorithm can not only be used for discrete systems and continuous-time dynamic systems, but also quickly calculate the Lyapunov exponent of complex discontinuous systems, which provides a new idea for determining the dynamic behavior of complex discontinuous systems.

Constrained reference learning for continuous-time model predictive tracking control of autonomous systems

Abstract Often, systems need to adapt their behavior to other systems in their surroundings while obeying constraints to achieve good performance or due to safety reasons. We consider repetitive applications, where the reference for the controller stems from noisy sensor data. Including preview information of the reference, e.g. extrapolating from previous cycles or similar instances, can significantly improve the overall tracking performance and ensure constraint satisfaction. We propose a learning-supported predictive controller that uses Gaussian processes as reference generators for its control task. A Gaussian process is used to learn, filter, and predict the references. It learns references tailored to model predictive control, taking into account continuous-time system dynamics and constraints via constrained hyperparameter optimization. We illustrate the benefits concerning approximation and control performance of the informed Gaussian-process training by a cooperative, mobile robot example.

Conformal symplectic and relativistic optimization

Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov’s accelerated gradient and Polyaks’s heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyse the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost.

Finite-Time Control of Continuous-Time Networked Dynamical Systems

Finite-time stability (FTS) analysis and control problems are investigated in this paper for continuous-time networked dynamical systems constituted by many subsystems. Interactions among these subsystems are arbitrary and time-varying, and every subsystem has different and time-varying dynamics. Several differential linear matrix inequality (LMI) formed methods are presented for the FTS analysis and control, which avoid the inverse computation of matrices with large dimensions and efficiently utilize the block diagonal characteristic of system parameters and the sparseness of the subsystem connection matrix. Sufficient conditions only depending on the parameters of each individual subsystem are then derived. Furthermore, LMI-based conditions are also provided for the FTS analysis and control of linear time-invariant networked systems. Numerical simulations show that they are computationally effective to analyze and synthesize a large-scale networked dynamical systems.

Incremental stability and contraction via impulsive control for continuous-time dynamical systems

This paper investigates the incremental stability and contraction via impulsive control for continuous-time dynamical systems (CDS). By using the methods of Lyapunov-like function and average dwell-time (ADT), criteria are established for incremental stability including incremental global asymptotic stability (δGAS) and incremental global exponential stability (δGES). The notion of rate of control is proposed and the ADT condition is shown to be equivalent to the condition via the inferior limit of rate of control. And it is shown that the δGES via impulsive control is robust with respect to the time-delays and data dropout. The results are then applied to time-varying Lipschitz-type CDS which can achieve δGES via a linear impulsive control. Moreover, the notion of global contracting via impulsive control is proposed for CDS. The global contracting via impulsive control is proved to be equivalent to the δGES via impulsive control. Some less conservative criteria are derived for global contracting/δGES via impulsive control. And the equivalence is used to derive a byproduct of δGES and global contracting for CDS with destabilizing impulses. Finally, three examples with numerical simulations are presented for illustrations.

On the continuous time limit of the ensemble Kalman filter

We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by Burgers, van Leeuwen, and Evensen [Monthly Weather Review 126 (1998), pp. 1719–1724] as well as a recent variant of de Wiljes, Reich, and Stannat [SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, pp. 1152–1181] to the respective discretizations and show that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Our analysis also allows for the derivation of convergence rates with respect to the stepsize. An application of our analysis is the rigorous derivation of continuous-time ensemble filtering algorithms consistent with discrete-time approximation schemes. Conversely, the continuous time limit allows for a better qualitative and quantitative analysis of the discrete-time counterparts using the rich theory of dynamical systems in continuous time.

PAC Model Checking of Black-Box Continuous-Time Dynamical Systems

In this article, we present a novel model checking approach to finite-time safety verification of black-box continuous-time dynamical systems within the framework of probably approximately correct (PAC) learning. The black-box dynamical systems are the ones, for which no model is given but whose states changing continuously through time within a finite-time interval can be observed at some discrete-time instants for a given input. The new model checking approach is termed as the PAC model checking due to the incorporation of learned models with correctness guarantees expressed using the terms error probability and confidence. Based on the error probability and confidence level, our approach provides statistically formal guarantees that the time-evolving trajectories of the black-box dynamical system over finite-time horizons fall within the range of the learned model plus a bounded interval, contributing to insights on the reachability of the black-box system and thus on the satisfiability of its safety requirements. The learned model together with the bounded interval is obtained by scenario optimization, which boils down to a linear programming problem. Three examples demonstrate the performance of our approach.

Bayesian inference for diffusion processes: using higher-order approximations for transition densities.

Modelling random dynamical systems in continuous time, diffusion processes are a powerful tool in many areas of science. Model parameters can be estimated from time-discretely observed processes using Markov chain Monte Carlo (MCMC) methods that introduce auxiliary data. These methods typically approximate the transition densities of the process numerically, both for calculating the posterior densities and proposing auxiliary data. Here, the Euler–Maruyama scheme is the standard approximation technique. However, the MCMC method is computationally expensive. Using higher-order approximations may accelerate it, but the specific implementation and benefit remain unclear. Hence, we investigate the utilization and usefulness of higher-order approximations in the example of the Milstein scheme. Our study demonstrates that the MCMC methods based on the Milstein approximation yield good estimation results. However, they are computationally more expensive and can be applied to multidimensional processes only with impractical restrictions. Moreover, the combination of the Milstein approximation and the well-known modified bridge proposal introduces additional numerical challenges.

Intensity-Modulated Radiation Therapy Optimization for Acceptable and Remaining-One Unacceptable Dose-Volume and Mean-Dose Constraint Planning.

We give a novel approach for obtaining an intensity-modulated radiation therapy (IMRT) optimization solution based on the idea of continuous dynamical methods. The proposed method, which is an iterative algorithm derived from the discretization of a continuous-time dynamical system, can handle not only dose-volume but also mean-dose constraints directly in IMRT treatment planning. A theoretical proof for the convergence to an equilibrium corresponding to the desired IMRT planning is given by using the Lyapunov stability theorem. By introducing the concept of “acceptable,” which means the existence of a nonempty set of beam weights satisfying the given dose-volume and mean-dose constraints, and by using the proposed method for an acceptable IMRT planning, one can resolve the issue that the objective and evaluation are different in the conventional planning process. Moreover, in the case where the target planning is totally unacceptable and partly acceptable except for one group of dose constraints, we give a procedure that enables us to obtain a nearly optimal solution close to the desired solution for unacceptable planning. The performance of the proposed approach for an acceptable or unacceptable planning is confirmed through numerical experiments simulating a clinical setup.

Interval-Based Simulation of Zélus IVPs using DynIbex

Modeling continuous-time dynamical systems is a complex task. Fortunately some dedicated programming languages exist to ease this work. Zélus is one such language that generates a simulation executable which can be used to study the behavior of the modeled system. However, such simulations cannot handle uncertainties on some parameters of the system. This makes it necessary to run multiple simulations to check that the system fulfills particular requirements (safety for instance) for all the values in the uncertainty ranges. Interval-based guaranteed integration methods provide a solution to this problem. The DynIbex library provides such methods but it requires a manual encoding of the system in a general purpose programming language (C++). This article presents an extension of the Zélus compiler to generate interval-based guaranteed simulations of IVPs using DynIbex. This extension is conservative since it does not break the existing compilation workflow.

Noise-induced chaotic-attractor escape route

In the present article, a combination of numerical and experimental studies is undertaken to comprehend the influence of noise on the responses of continuous-time dynamical systems. In particular, the influence of white Gaussian noise on the chaotic and periodic responses of bistable, Duffing oscillators is the focus of this work. The noteworthy result of the conducted studies concerns the presence of a pair of attractors, one being periodic and the other being chaotic: the chaotic attractor response can be controlled and terminated with an appropriate noise level. For trajectories in the basin of the chaotic attractor, white Gaussian noise is added at a barely sufficient level to allow trajectories to eventually leave (within some specified time). The authors report that trajectories leave via a special escape route: the unstable manifold of a fixed point saddle on the basin boundary between the two basins of attraction. Striking similarities and differences between experimental and numerical investigation are discussed in the work.

On the dynamics of a modified Lorenz–Stenflo system

The Lorenz–Stenflo system is a four-parameter four-dimensional autonomous nonlinear continuous-time dynamical system, derived to model the time evolution of finite amplitude acoustic gravity waves in a rotating atmosphere. In this paper, we propose a modified Lorenz–Stenflo system, where the variable [Formula: see text] in the fourth equation of the original Lorenz–Stenflo system was replaced by [Formula: see text]. We investigate cross-sections of the parameter-space of this new system, characterizing regions of different dynamical behaviors. We show that the aforementioned replacement may promote the emergence of organized periodic structures in places of these cross-sections, where they did not exist before modification.