Continuous-time Dynamical Systems Research Articles

A Continuous Time Dynamical Turing Machine.

Continuous time recurrent neural networks (CTRNNs) are systems of coupled ordinary differential equations (ODEs) inspired by the structure of neural networks in the brain. CTRNNs are known to be universal dynamical approximators: given a large enough system, the parameters of a CTRNN can be tuned to produce output that is arbitrarily close to that of any other dynamical system. However, in practice, both designing systems of CTRNN to have a certain output, and the reverse-understanding the dynamics of a given system of CTRNN-can be nontrivial. In this article, we describe a method for embedding any specified Turing machine in its entirety into a CTRNN. As such, we describe in detail a continuous time dynamical system that performs arbitrary discrete-state computations. We suggest that in acting as both a continuous time dynamical system and as a computer, the study of such systems can help refine and advance the debate concerning the Computational Hypothesis that cognition is a form of computation and the Dynamical Hypothesis that cognitive systems are dynamical systems.

Identification of Continuous-Time Dynamic Systems With Uncertainties Measured by Fuzzy Sets Subject to Model Structure Errors

Identification of Continuous-Time Dynamic Systems With Uncertainties Measured by Fuzzy Sets Subject to Model Structure Errors

Controllability and observability of conformable fractional finite dimensional linear systems

In this research paper, we investigate the controllability and observability of continuous-time fractional dynamical systems. The characterisation of all fractional continuous one-parameter semi groups on a finite dimensional vector space as fractional matrix-valued exponential functions is given. This characterisation is used to express the solution of continuous-time fractional dynamical system. Furthermore, we analyse controllability and observability of linear conformable fractional systems in the mathematical context of linear operators on a finite dimensional vector space. By defining the controllability and observability maps, we establish necessary and sufficient conditions for controllability and observability of this type of systems. We also present the Hautus test for controllability and observability, and introduce controllability and observability Gramians, both for finite and infinite time interval. Finally, we present several illustrative applications and examples to validate all the obtained results.

A Relaxed Inertial Method for Solving Monotone Inclusion Problems with Applications

We study a relaxed inertial forward–backward–half-forward splitting approach with variable step size to solve a monotone inclusion problem involving a maximal monotone operator, a cocoercive operator, and a monotone Lipschitz operator. The convergence of the sequence of iterations generated by the discretisations of a continuous-time dynamical system is established under suitable conditions. Given the challenges associated with computing the resolvent of the composite operator, the proposed method is employed to tackle the composite monotone inclusion problem. Additionally, a convergence analysis is conducted under certain conditions. To demonstrate the effectiveness of the algorithm, numerical experiments are performed on the image deblurring problem.

Oscillatory Neural Network-Based Ising Machine Using 2D Memristors.

Neural networks are increasingly used to solve optimization problems in various fields, including operations research, design automation, and gene sequencing. However, these networks face challenges due to the nondeterministic polynomial time (NP)-hard issue, which results in exponentially increasing computational complexity as the problem size grows. Conventional digital hardware struggles with the von Neumann bottleneck, the slowdown of Moore's law, and the complexity arising from heterogeneous system design. Two-dimensional (2D) memristors offer a potential solution to these hardware challenges, with their in-memory computing, decent scalability, and rich dynamic behaviors. In this study, we explore the use of nonvolatile 2D memristors to emulate synapses in a discrete-time Hopfield neural network, enabling the network to solve continuous optimization problems, like finding the minimum value of a quadratic polynomial, and tackle combinatorial optimization problems like Max-Cut. Additionally, we coupled volatile memristor-based oscillators with nonvolatile memristor synapses to create an oscillatory neural network-based Ising machine, a continuous-time analog dynamic system capable of solving combinatorial optimization problems including Max-Cut and map coloring through phase synchronization. Our findings demonstrate that 2D memristors have the potential to significantly enhance the efficiency, compactness, and homogeneity of integrated Ising machines, which is useful for future advances in neural networks for optimization problems.

Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.

Data-Driven Stochastic Optimal Control With Safety Constraints Using Linear Transfer Operators

We provide a data-driven framework for optimal control of a continuous-time control-affine stochastic dynamical system. The proposed framework relies on the linear operator theory involving Perron-Frobenius (P-F) and Koopman operators. Our first result involving the P-F operator provides a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the linear operators. We provide rigorous error bounds and convergence rate for the data-driven approximation of the linear operators and comment on the convergence rate of the optimal control w.r.t. data size. Our second results demonstrate using the Koopman operator, dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving the P-F operator and the Koopman-based formulation using the HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of the P-F and Koopman operators. Finally, we present examples to demonstrate the efficacy of the developed framework and verify the convergence rates for the operator and optimal control numerically.

Exact Inference for Continuous-Time Gaussian Process Dynamics

Many physical systems can be described as a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. While this scheme is mathematically tempting, it can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. We tackle this task by leveraging higher-order numerical integrators. These integrators provide the necessary tools to discretize dynamical systems with arbitrary accuracy. However, most higher-order integrators require dynamics evaluations at intermediate time steps, making exact GP inference intractable. In previous work, this problem is often addressed by approximate inference techniques. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to enable direct inference, we propose to leverage multistep and Taylor integrators. We demonstrate how exact inference schemes can be derived for these types of integrators. Further, we derive tailored sampling schemes that allow one to draw consistent dynamics functions from the posterior. The learned model can thus be integrated with arbitrary integrators, just like a standard dynamical system. We show empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyse their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in Beyn, et al. [Angular values of nonautonomous and random linear dynamical systems: Part I – fundamentals, SIAM J. Appl. Dyn. Syst. 21(2) (2022), pp. 1245–1286; Angular values of nonautonomous linear dynamical systems: Part II – reduction theory and algorithm, SIAM J. Appl. Dyn. Syst. 22(1) (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

High-order sensory processing nanocircuit based on coupled VO2 oscillators

Conventional circuit elements are constrained by limitations in area and power efficiency at processing physical signals. Recently, researchers have delved into high-order dynamics and coupled oscillation dynamics utilizing Mott devices, revealing potent nonlinear computing capabilities. However, the intricate yet manageable population dynamics of multiple artificial sensory neurons with spatiotemporal coupling remain unexplored. Here, we present an experimental hardware demonstration featuring a capacitance-coupled VO2 phase-change oscillatory network. This network serves as a continuous-time dynamic system for sensory pre-processing and encodes information in phase differences. Besides, a decision-making module for special post-processing through software simulation is designed to complete a bio-inspired dynamic sensory system. Our experiments provide compelling evidence that this transistor-free coupling network excels in sensory processing tasks such as touch recognition and gesture recognition, achieving significant advantages of fewer devices and lower energy-delay-product compared to conventional methods. This work paves the way towards an efficient and compact neuromorphic sensory system based on nano-scale nonlinear dynamics.

Data-Driven Inverse Reinforcement Learning Control for Linear Multiplayer Games.

This article proposes a data-driven inverse reinforcement learning (RL) control algorithm for nonzero-sum multiplayer games in linear continuous-time differential dynamical systems. The inverse RL problem in the games is solved by a learner reconstructing the unknown expert players' cost functions from demonstrated expert's optimal state and control input trajectories. The learner, thus, obtains the same control feedback gains and trajectories as the expert, only using data along system trajectories without knowing system dynamics. This article first proposes a model-based inverse RL policy iteration framework that has: 1) policy evaluation step for reconstructing cost matrices using Lyapunov functions; 2) state-reward weight improvement step using inverse optimal control (IOC); and 3) policy improvement step using optimal control. Based on the model-based policy iteration algorithm, this article further develops an online data-driven off-policy inverse RL algorithm without knowing any knowledge of system dynamics or expert control gains. Rigorous convergence and stability analysis of the algorithms are provided. It shows that the off-policy inverse RL algorithm guarantees unbiased solutions while probing noises are added to satisfy the persistence of excitation (PE) condition. Finally, two different simulation examples validate the effectiveness of the proposed algorithms.

Online Corrections to Neural Policy Guidance for Pinpoint Powered Descent

This study presents incremental correction methods for refining neural network parameters or control functions entering into a continuous-time dynamic system to achieve improved solution accuracy in satisfying the interim point constraints placed on the performance output variables. The proposed approach is to linearize the dynamics around the baseline values of its arguments and then to solve for the corrective input required to transfer the perturbed trajectory to precisely known or desired values at specific time points, in other words, the interim points. Depending on the type of decision variables to adjust, parameter correction and control function correction methods are developed. These incremental correction methods can be used as a means to compensate for the prediction errors of pretrained neural networks in real-time applications where high accuracy of the prediction of dynamical systems at prescribed time points is imperative. In this regard, the online update approach can be useful for enhancing overall targeting accuracy of finite-horizon control subject to point constraints using a neural policy. A numerical example demonstrates the effectiveness of the proposed approach in an application to a powered descent problem on Mars.

Distributed order estimation for continuous-time stochastic systems

AbstractIn this paper, we investigate the distributed estimation problem of continuous-time stochastic dynamic systems over sensor networks when both the system order and parameters are unknown. We propose a local information criterion (LIC) based on the $$L_0$$ L 0 penalty term. By minimizing LIC at the diffusion time instant and utilizing the continuous-time diffusion least squares algorithm, we obtain a distributed estimation algorithm to simultaneously estimate the unknown order and the parameters of the system. By dealing with the effect of the system noises and the coupling relationship between estimation of system orders and parameters, we establish the almost sure convergence results of the proposed distributed estimation algorithm. Furthermore, we give a simulation example to verify the effectiveness of the distributed algorithm in estimating the system order and parameters.

Generalized Invariance Principles for Stochastic Dynamical Systems and Their Applications

Investigating long-term behaviors of stochastic dynamical systems often requires to establish criteria that are able to describe delicate dynamics of the considered systems. In this article, we develop generalized invariance principles for continuous-time stochastic dynamical systems. Particularly, in a sense of probability one and by the developed semimartingale convergence theorem, we not only establish a local invariance principle, but also provide a generalized global invariance principle that allows the sign of the diffusion operator to be positive in some bounded region. We further provide an estimation for the time when a trajectory, initiating outside a particular bounded set, eventually enters it. Finally, we use several representative examples, including stochastic oscillating dynamics, to illustrate the practical usefulness of our analytical criteria in deciphering the stabilization or/and the synchronization dynamics of stochastic systems.

Distributed Optimization With Projection-Free Dynamics: A Frank-Wolfe Perspective.

We consider solving distributed constrained optimization in this article. To avoid projection operations due to constraints in the scenario with large-scale variable dimensions, we propose distributed projection-free dynamics by employing the Frank-Wolfe method, also known as the conditional gradient. Technically, we find a feasible descent direction by solving an alternative linear suboptimization. To make the approach available over multiagent networks with weight-balanced digraphs, we design dynamics to simultaneously achieve both the consensus of local decision variables and the global gradient tracking of auxiliary variables. Then, we present the rigorous convergence analysis of the continuous-time dynamical systems. Also, we derive its discrete-time scheme with an accordingly proved convergence rate of O(1/k) . Furthermore, to clarify the advantage of our proposed distributed projection-free dynamics, we make detailed discussions and comparisons with both existing distributed projection-based dynamics and other distributed Frank-Wolfe algorithms.

The Symplectic Adjoint Method: Memory-Efficient Backpropagation of Neural-Network-Based Differential Equations.

The combination of neural networks and numerical integration can provide highly accurate models of continuous-time dynamical systems and probabilistic distributions. However, if a neural network is used [Formula: see text] times during numerical integration, the whole computation graph can be considered as a network [Formula: see text] times deeper than the original. The backpropagation algorithm consumes memory in proportion to the number of uses times of the network size, causing practical difficulties. This is true even if a checkpointing scheme divides the computation graph into subgraphs. Alternatively, the adjoint method obtains a gradient by a numerical integration backward in time; although this method consumes memory only for single-network use, the computational cost of suppressing numerical errors is high. The symplectic adjoint method proposed in this study, an adjoint method solved by a symplectic integrator, obtains the exact gradient (up to rounding error) with memory proportional to the number of uses plus the network size. The theoretical analysis shows that it consumes much less memory than the naive backpropagation algorithm and checkpointing schemes. The experiments verify the theory, and they also demonstrate that the symplectic adjoint method is faster than the adjoint method and is more robust to rounding errors.

Kernel Regression-Based State-Space Estimation of Continuous-Time Dynamic Systems From Noisy Sampled State Data

Kernel Regression-Based State-Space Estimation of Continuous-Time Dynamic Systems From Noisy Sampled State Data

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.

On discrete-time dissipative port-Hamiltonian (descriptor) systems

AbstractPort-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman–Yakubovich–Popov inequalities is also studied.

A Compositional Dissipativity Approach for Data-Driven Safety Verification of Large-Scale Dynamical Systems

This work is concerned with a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">compositional data-driven approach</i> for formal safety verification of large-scale continuous-time dynamical systems with unknown models. The proposed framework enjoys the interconnection matrix and joint dissipativity-type properties of subsystems, described by the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic storage certificates</i> . In the first part of the paper, we cast the required conditions for constructing storage certificates as a robust optimization program (ROP). Since the proposed ROP is not tractable due to the unknown model appearing in one of its constraints, we propose a scenario optimization program (SOP) corresponding to the original ROP by collecting finite numbers of data from trajectories of each subsystem. By establishing a probabilistic relation between the optimal value of SOP and that of ROP, we construct a storage certificate for each unknown subsystem based on the number of data and a required level of confidence. We accordingly propose a compositional technique based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dissipativity reasoning</i> to construct <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic barrier certificates</i> of interconnected systems based on storage certificates of individual subsystems. By leveraging the acquired barrier certificate, we quantify a lower bound on the probability that an interconnected system never reaches a certain unsafe region in finite time horizons with an a-priori guaranteed confidence. We also propose an auxiliary compositional approach without requiring any compositionality condition but at the cost of providing a potentially conservative safety guarantee. In the second part of the paper, we propose our approaches for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic</i> continuous-time systems with unknown dynamics. We verify our results over an unknown room temperature network containing 100 rooms.