Continuous-time Differential Equations Research Articles

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter.

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

A Novel Mathematical Characterization for Switched Linear Systems Based on Automata and Its Stabilizability Analysis

Mathematical characterization of switched systems based on automata is one of the most challenging problems in the context of hybrid dynamical systems. This paper investigates the mathematical representation and stabilization of a class of slowly and fastly switched linear systems with automata in the hybrid coupling framework. To address such an issue, by using the semi-tensor product of matrices, a so-called “automaton-dependent switched linear system” is established to generalize the traditional switched system, which is characterized mathematically in the new form of a hybrid dynamic system consisting of a continuous-time differential equation and a discrete-time difference equation. Moreover, a novel automaton-dependent constrained switching scheme is developed for the stabilization of the proposed model with unstable subsystems. Next, by designing slow mode-dependent average dwell time (S-MDADT), fast mode-dependent average dwell time (F-MDADT), slow cycle-dependent cycle dwell time (S-CDCDT) and fast cycle-dependent cycle dwell time (F-CDCDT) switching laws, a generalized stabilization lemma for automaton-dependent switched nonlinear systems with both stable and unstable modes inside and outside the cycle is derived in the term of digital transformations and multiple Lyapunov functions. Furthermore, based on the stabilizability criteria of nonlinear case, by setting digital transformations and linear matrix inequalities, a new stabilization theorem for automaton-dependent switched linear systems with unstable subsystems is obtained. It should be noted that the proposed scheme provides a more precise estimation of the upper and lower bounds for the dwell time of the system with unstable modes. Finally, a practical converter circuit example and a numerical example are given to show the validity of the designed techniques.

Suboptimal Filtering Over Sensor Networks With Random Communication

The problem of filter design is considered for continuous-time linear stochastic systems using distributed sensors. Each sensor unit, represented by a node in an undirected and connected graph, collects some information about the state and communicates its own estimate with the neighbors. It is stipulated that this communication between sensor nodes connected by an edge is time-sampled randomly and for each edge, the sampling process is an independent Poisson counter. Our proposed filtering algorithm for each sensor node is a stochastic hybrid system: It comprises a continuous-time differential equation, and at random time instants when communication takes place, each sensor node updates its state estimate based on the information received by its neighbors. In this setting, we compute the expectation of the error covariance matrix for each unit which is governed by a matrix differential equation. To study the asymptotic behavior of these covariance matrices, we show that if the gain matrices are appropriately chosen and the mean sampling rate is large enough, then the error covariances practically converge to a constant matrix.

Asymptotic error in Euler’s method with a constant step size

In the gradient descent method, one often focus on the convergence of the sequence generated by the algorithm, but less often on the deviation of these points from the solutions of the original continuous-time differential equation (gradient flow). This also happens when discretizing other ordinary differential equations. In the case of a discretization by explicit Euler’s method with a constant step h, we provide here sufficient conditions, in terms of strong monotonicity and co-coercivity, for the deviation between discrete and continuous solutions to tend asymptotically towards zero. This analysis could shed new light on some applications of the gradient descent algorithm.

Lyapunov equation in open quantum systems and non-Hermitian physics

The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

The equivalence of continuous-/discrete-time autoregressive-moving average (ARMA) systems is considered in this paper. For the integer-order cases, the interrelations between systems defined by continuous-time (CT) differential and discrete-time (DT) difference equations are found, leading to formulae relating partial fractions of the continuous and discrete transfer functions. Simple transformations are presented to allow interconversions between both systems, recovering formulae obtained with the impulse invariant method. These transformations are also used to formulate a covariance equivalence. The spectral correspondence implied by the bilinear (Tustin) transformation is used to study the equivalence between the two types of systems. The general fractional CT/DT ARMA systems are also studied by considering two DT differential fractional autoregressive-moving average (FARMA) systems based on the nabla/delta and bilinear derivatives. The interrelations CT/DT are also considered, paying special attention to the systems defined by the bilinear derivatives.

Nested sequences of period-adding stability phases in a CO2 laser map proxy

• Nested sequences of period-adding bifurcations found in Henon map. • Period-adding sequences exist in the Hamiltonian limits of the map. • Adding cascade crisscross the whole control parameter space. We report the discovery of wide nested sequences of period-adding complexification routes spawning the whole control parameter space of the Hénon map, a discrete-time proxy of single-mode loss-modulated CO 2 lasers. For a realistic map, the new adding routes found reproduce analogous phenomena previously observed in continuous-time differential equations describing, for instance, oscillatory phases in an enzyme reaction model. In contrast to differential equations, the map is analytically tractable to a large extent, and provides a prototypic framework to investigate intricate features of generic dissipative flows.

Dynamics, stability and sustainable optimal control in wolf-moose systems

A wolf-moose predator-prey system is defined as a nonlinear continuous time differential equation system. The parameters are estimated via a discretized approximation and empirical data from Isle Royale, USA, representing a 61 year period. All parameter estimates confirm the hypotheses and all P-values are below 5%. Possible dynamic equilibria are determined as explicit general functions of the parameters and as numerical values based on the empirical data. General dynamic properties of the system are determined via phase -plane analysis and nonlinear simulation. The nonlinear system is also linearized close to the single equilibrium with two strictly positive populations. The explicit equations of the time path of the linearized system are compared to the nonlinear simulation. Both methods give almost identical solutions close to the equilibrium. Far from the equilibrium, the time paths of the two methods deviate considerably. The solution is a stable converging spiral (center) (unstable diverging spiral) in case the system equilibrium prey population is located at a higher (the same) (lower) level than the population level that gives maximum sustainable net production. Based on the empirically determined expected parameter values, the system is stable but converges very slowly, as a spiral, to the two-species equilibrium. The estimated standard deviations of the parameters can be used to determine the probability that the system solution is a center or an unstable diverging spiral. The optimal management of the wolf-moose system is also determined via sustainable optimal control. Moose hunting and adjustments of the wolf population based on different prices and values of sustainable population levels are derived. Optimal hunting and stock levels are determined and reported as explicit functions of all parameters and prices.

Hyperspace fermions, Möbius transformations, Krein space, fermion doubling, dark matter

We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter T and equations of motion converted into difference equations. These equations are solved and the physical limit T → 0 then taken. In principle, this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to Möbius transformations in projective geometry. Quantization via Schwinger’s action principle recovers standard particle-antiparticle modes for bosons but in the case of fermions, Hilbert space has to be replaced by Krein space. We discuss possible links with the fermion doubling problem and with dark matter.

Macroeconomic Impacts of the Public Health Response to COVID-19

The economic impact of public health measures to contain the COVID-19 novel coronavirus is a matter of contentious debate. Given the high uncertainties, there is a need for combined epidemiological-macroeconomic scenarios. We present a model in this paper for developing such scenarios. The epidemiological sub-model is a discrete-time matrix implementation of an SEIR model. This approach avoids known problems with the more usual set of continuous-time differential equations. The post-Keynesian macroeconomic sub-model is a stylized representation of the United States economy with three sectors: core, social (most impacted by social distancing), and hospital, which may experience excessive demand. Simulations with the model show the clear superiority of a rigorous testing and contact tracing regime in which infected individuals, symptomatic or not, are isolated. Social distancing leads to an abrupt and deep recession. With expanded unemployment benefits, the drop is shallower. When testing and contact tracing is introduced, social spending can be scaled back and the economy recovers quickly. Ending social distancing without a testing and tracing regime leads to a high death toll and severe economic impacts. Results suggest that social distancing and fiscal stimulus have had their desired effects of reducing the health and economic impacts of the disease.

First Order and Second Order Learning Algorithms on the Special Orthogonal Group to Compute the SVD of Data Matrices

The present paper deals with neural algorithms to learn the singular value decomposition (SVD) of data matrices. The neural algorithms utilized in the present research endeavor were developed by Helmke and Moore (HM) and appear under the form of two continuous-time differential equations over the special orthogonal group of matrices. The purpose of the present paper is to develop and compare different numerical schemes, under the form of two alternating learning rules, to learn the singular value decomposition of large matrices on the basis of the HM learning paradigm. The numerical schemes developed here are both first-order (Euler-like) and second-order (Runge-like). Moreover, a reduced Euler scheme is presented that consists of a single learning rule for one of the factors involved in the SVD. Numerical experiments performed to estimate the optical-flow (which is a component of modern IoT technologies) in real-world video sequences illustrate the features of the novel learning schemes.

Multiscale Stochastic Reaction\u2013Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations

Two multiscale algorithms for stochastic simulations of reaction–diffusion processes are analysed. They are applicable to systems which include regions with significantly different concentrations of molecules. In both methods, a domain of interest is divided into two subsets where continuous-time Markov chain models and stochastic partial differential equations (SPDEs) are used, respectively. In the first algorithm, Markov chain (compartment-based) models are coupled with reaction–diffusion SPDEs by considering a pseudo-compartment (also called an overlap or handshaking region) in the SPDE part of the computational domain right next to the interface. In the second algorithm, no overlap region is used. Further extensions of both schemes are presented, including the case of an adaptively chosen boundary between different modelling approaches.

Sex-structured wild and sterile mosquito population models with different release strategies.

In this paper, we propose sex-structured mathematical models in terms of continuous-time differential equations. We investigate the interactive dynamics of the sex-structured wild and sterile mosquitoes from several aspects including the existence of equilibria and their stability. We consider different strategies of releasing the sterile mosquitoes to control mosquitoes in an effective way. In addition, numerical simulations are provided to illustrate the dynamical features of the models.

Multi-objective particle swarm optimisation approach for production-inventory control systems

PurposeThis paper aims to optimise the dynamic performance of production–inventory control systems in terms of minimisation variance ratio between the order rate and the consumption, and minimisation the integral of absolute error between the actual and the target level of inventory by incorporating the Pareto optimality into particle swarm optimisation (PSO).Design/method/approachThe production–inventory control system is modelled and optimised via control theory and simulations. The dynamics of a production–inventory control system are modelled through continuous time differential equations and Laplace transformations. The simulation design is conducted by using the state–space model of the system. The results of multi-objective particle swarm optimisation (MOPSO) are compared with published results obtained from weighted genetic algorithm (WGA) optimisation.FindingsThe results obtained from the MOPSO optimisation process ensure that the performance is systematically better than the WGA in terms of reducing the order variability (bullwhip effect) and improving the inventory responsiveness (customer service level) under the same operational conditions.Research limitations/implicationsThis research is limited to optimising the dynamics of a single product, single-retailer single-manufacturer process with zero desired inventory level.Originality/valuePSO is widely used and popular in many industrial applications. This research shows a unique application of PSO in optimising the dynamic performance of production–inventory control systems.

Computation of Lyapunov Functions for Nonlinear Differential Equations via a Massera-Type Construction

An approach for computing Lyapunov functions for nonlinear continuous-time differential equations is developed via a new, Massera-type construction. This construction is enabled by imposing a finite-time criterion on the integrated function. By means of this approach, we relax the assumptions of exponential stability on the system's equilibrium, while still allowing integration over a finite time interval. The resulting Lyapunov function can be computed based on any $\mathcal {K}_{\infty}$ -function of the norm of the solution of the system. In addition, we show how the developed converse theorem can be used to construct an estimate of the domain of attraction. Finally, a range of examples from the literature and biological applications, such as the genetic toggle switch, the repressilator, and the HPA axis, are worked out to demonstrate the efficiency and improvement in computation of the proposed approach.

Low-rank approximate solutions to large-scale differential matrix Riccati equations

In the present paper, we consider large-scale continuous-time differential matrix Riccati equations having low rank right-hand sides. These equations are generally solved by Backward Differentiation Formula (BDF) or Rosenbrock methods leading to a large scale algebraic Riccati equation which has to be solved for each timestep. We propose a new approach, based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimentional differential algebraic Riccati equation. The latter matrix differential problem is then solved by Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.

Dynamics analysis of a production-inventory control system with two pipelines feedback

PurposeThe purpose of this study is to propose a new dynamic model of a production-inventory control system. The objective of the new model is to maximise the flexibility of the system so that it can be used by decision makers to design inventory systems that adopt various strategies that provide a balance between reducing the bullwhip effect and improving the responsiveness of inventory performance.Design/methodology/approachThe proposed production-inventory control system is modelled and analysed via control theory and simulations. The production-inventory feedback control system is modelled through continuous time differential equations. The simulation experiments design is conducted by using the state-space model of the system. The Automatic Pipeline Inventory and Order-Based Production Control System (APIOBPCS) model is used as a benchmark production-inventory control system.FindingsThe results showed that the Two Automatic Pipelines, Inventory and Order-Based Production Control System (2APIOBPCS) model outperforms APIOBPCS in terms of reducing the bullwhip effect. However, the 2APIOBPCS model has a negative impact on Customer Service Level. Therefore, with careful parameter setting, it is possible to design control decisions to be suitably responsive while generating smooth order patterns and obtain the best trade-off of the two objectives.Research limitations/implicationsThis research is limited to the dynamics of single-echelon production-inventory control systems with zero desired inventory level.Originality/valueThis present model is an extension and improvement to Towill’s (1982) and John et al.’s (1994) work, since it presents a new dynamic model of a production-inventory control system which utilises an additional flow of information to improve the efficiency of order rate decisions.

Discrete stochastic analogs of Erlang epidemic models

ABSTRACTErlang differential equation models of epidemic processes provide more realistic disease-class transition dynamics from susceptible (S) to exposed (E) to infectious (I) and removed (R) categories than the ubiquitous SEIR model. The latter is itself is at one end of the spectrum of Erlang SEIR models with concatenated E compartments and concatenated I compartments. Discrete-time models, however, are computationally much simpler to simulate and fit to epidemic outbreak data than continuous-time differential equations, and are also much more readily extended to include demographic and other types of stochasticity. Here we formulate discrete-time deterministic analogs of the Erlang models, and their stochastic extension, based on a time-to-go distributional principle. Depending on which distributions are used (e.g. discretized Erlang, Gamma, Beta, or Uniform distributions), we demonstrate that our formulation represents both a discretization of Erlang epidemic models and generalizations thereof. We consider the challenges of fitting SEIR models and our discrete-time analog to data (the recent outbreak of Ebola in Liberia). We demonstrate that the latter performs much better than the former; although confining fits to strict SEIR formulations reduces the numerical challenges, but sacrifices best-fit likelihood scores by at least 7%.

Noise in Neuronal and Electronic Circuits: A General Modeling Framework and Non-Monte Carlo Simulation Techniques.

The brain is extremely energy efficient and remarkably robust in what it does despite the considerable variability and noise caused by the stochastic mechanisms in neurons and synapses. Computational modeling is a powerful tool that can help us gain insight into this important aspect of brain mechanism. A deep understanding and computational design tools can help develop robust neuromorphic electronic circuits and hybrid neuroelectronic systems. In this paper, we present a general modeling framework for biological neuronal circuits that systematically captures the nonstationary stochastic behavior of ion channels and synaptic processes. In this framework, fine-grained, discrete-state, continuous-time Markov chain models of both ion channels and synaptic processes are treated in a unified manner. Our modeling framework features a mechanism for the automatic generation of the corresponding coarse-grained, continuous-state, continuous-time stochastic differential equation models for neuronal variability and noise. Furthermore, we repurpose non-Monte Carlo noise analysis techniques, which were previously developed for analog electronic circuits, for the stochastic characterization of neuronal circuits both in time and frequency domain. We verify that the fast non-Monte Carlo analysis methods produce results with the same accuracy as computationally expensive Monte Carlo simulations. We have implemented the proposed techniques in a prototype simulator, where both biological neuronal and analog electronic circuits can be simulated together in a coupled manner.

Hardware Tailored Linear Algebra for Implicit Integrators in Embedded NMPC

Nonlinear Model Predictive Control (NMPC) requires the efficient treatment of the dynamic model in the form of a system of continuous-time differential equations. Newton-type optimization relies on a numerical simulation method in addition to the propagation of first or higher order derivatives. In the case of stiff or implicitly defined dynamics, implicit integration schemes are typically preferred. This paper proposes a tailored implementation of the necessary linear algebra routines (LU factorization and triangular solutions), in order to allow for a considerable computational speedup of such integrators. In particular, the open-source BLASFEO framework is presented as a library of efficient linear algebra routines for small to medium-scale embedded optimization applications. Its performance is illustrated on the nonlinear optimal control example of a chain of masses. The proposed library allows for considerable speedups and it is found to be overall competitive with both a code-generated solver and a high-performance BLAS implementation.