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The rich non‐linear dynamics of the coupled oscillators (under second harmonic injection) can be leveraged to solve computationally hard problems in combinatorial optimization such as finding the ground state of the Ising Hamiltonian. While prior work on the stability of the so‐called Oscillator Ising Machines (OIMs) has used the linearization method, in this letter, the authors present a complementary method to analyze stability using the second‐order derivative test of the energy/cost function. The authors establish the equivalence between the two methods, thus augmenting the tool kit for the design and implementation of OIMs. While prior work has focused on the use of linearization methods to analyze stability of oscillator Ising machines (OIMs), here, the authors introduce an alternative approach to analyze the stability of the fixed points using the second‐order derivative test of the energy/cost function. The authors’ work is uniquely enabled by a novel theoretical relationship between the eigenvalues of the Jacobian matrix and the eigenvalues of the second‐order Hessian in OIMs, elucidated in this work. Moreover, the authors’ approach is applicable to a broader class of gradient descent systems.

Willems' fundamental lemma (WFL) is widely used in the data‐driven model predictive controllers (MPCs) to model the plant and predict its response to the given input sequence. The advantage of this model is that it can be incorporated as a linear matrix equality constraint into the optimization problem of MPC. However, the response of a dynamical system depends not only on the input applied to it but also on its initial condition. The WFL does not directly state anything about how the initial condition of the system should be defined and incorporated into the data‐driven model equations. The conventional method for incorporating the initial condition of the system in a data‐driven model is to insert a number of the most recent input and output samples of the system into the same data‐driven model used for predicting the system's response, and this equation is then added as a new constraint to the optimization problem. This paper shows that the conventional method of incorporating the system's initial condition into the data‐driven model is inaccurate and leads to errors in predicting the system's output. Moreover, the correct method for doing this has been presented for linear time‐invariant systems, and the results have also been extended to the non‐linear case. The proposed method has been used to design a data‐driven MPC for a lab‐scale wind turbine and experimental results have been presented. A new data‐driven model predictive controller for multi‐input multi‐output non‐linear dynamical systems are proposed. The main novelty of the manuscript is in the method developed for predicting the response of the non‐linear dynamical system to the given input sequence with high accuracy.

The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using Lagrange’s equations, which is then solved analytically up to the third approximation employing the approach of multiple scales (AMS). The emerging cases of resonance are categorized according to the solvability requirements wherein the modulation equations (ME) have been found. The stability areas and the instability ones are examined utilizing the Routh–Hurwitz criteria (RHC) and analyzed in line with the solutions at the steady state. The obtained results, resonance responses, and stability regions are addressed and graphically depicted to explore the positive influence of the various inputs of the physical parameters on the rheological behavior of the inspected system. The significance of the present work stems from its numerous applications in theoretical physics and engineering.

We review recent developments of sequential data assimilation techniques used in oceanography to integrate spatio-temporal observations into numerical models describing physical and ecological dynamics. Theoretical aspects from the simple case of linear dynamics to the general case of nonlinear dynamics are described from a geostatistical point-of-view. Current methods derived from the Kalman filter are presented from the least complex to the most general and perspectives for nonlinear estimation by sequential importance resampling filters are discussed. Furthermore an extension of the ensemble Kalman filter to transformed Gaussian variables is presented and illustrated using a simplified ecological model. The described methods are designed for predicting over geographical regions using a high spatial resolution under the practical constraint of keeping computing time sufficiently low to obtain the prediction before the fact. Therefore the paper focuses on widely used and computationally efficient methods. /// Nous recensons quelques développements récents de techniques d'assimilation séquentielle utilisées en océanographie, qui intègrent des observations spatio-temporelles dans des modèles numériques décrivant des dynamiques physiques et écologiques. Les aspects théoriques allant du cas simple d'une dynamique linéaire au cas général d'une dynamique non-linéaire sont examinées du point de vue géostatistique. Des méthodes usuelles dérivées du filtre de Kalman sont présentées en partant du cas le moins complexe au cas le plus général et des perspectives pour une estimation non-linéaire sont discutées. Nous présentons en outre une extension du filtre de Kalman d'ensemble au cas de variables ayant subi une transformation gaussienne et nous l'illustrons en utilisant un modèle écologique simplifié. Les méthodes exposées sont conçues pour prédire dans une région géographique avec une haute résolution spatiale sous la contrainte pratique que les temps de calcul soient suffisamment courts pour obtenir une prédiction avant l'heure. Ainsi l'article se concentre sur des méthodes couramment utilisées et de grande efficacité calculatoire.

This comprehensive book provides the first unified framework for stability and dissipativity analysis and control design for nonnegative and compartmental dynamical systems, which play a key role in a wide range of fields, including engineering, thermal sciences, biology, ecology, economics, genetics, chemistry, medicine, and sociology. Using the highest standards of exposition and rigor, the authors explain these systems and advance the state of the art in their analysis and active control design. Nonnegative and Compartmental Dynamical Systemspresents the most complete treatment available of system solution properties, Lyapunov stability analysis, dissipativity theory, and optimal and adaptive control for these systems, addressing continuous-time, discrete-time, and hybrid nonnegative system theory. This book is an indispensable resource for applied mathematicians, dynamical systems theorists, control theorists, and engineers, as well as for researchers and graduate students who want to understand the behavior of nonnegative and compartmental dynamical systems that arise in areas such as biomedicine, demographics, epidemiology, pharmacology, telecommunications, transportation, thermodynamics, networks, heat transfer, and power systems.

Resting-state functional connectivity is a powerful tool for studying human functional brain networks. Temporal fluctuations in functional connectivity, i.e., dynamic functional connectivity (dFC), are thought to reflect dynamic changes in brain organization and non-stationary switching of discrete brain states. However, recent studies have suggested that dFC might be attributed to sampling variability of static FC. Despite this controversy, a detailed exposition of stationarity and statistical testing of dFC is lacking in the literature. This article seeks an in-depth exploration of these statistical issues at a level appealing to both neuroscientists and statisticians. We first review the statistical notion of stationarity, emphasizing its reliance on ensemble statistics. In contrast, all FC measures depend on sample statistics. An important consequence is that the space of stationary signals is much broader than expected, e.g., encompassing hidden markov models (HMM) widely used to extract discrete brain states. In other words, stationarity does not imply the absence of brain states. We then expound the assumptions underlying the statistical testing of dFC. It turns out that the two popular frameworks - phase randomization (PR) and autoregressive randomization (ARR) - generate stationary, linear, Gaussian null data. Therefore, statistical rejection can be due to non-stationarity, nonlinearity and/or non-Gaussianity. For example, the null hypothesis can be rejected for the stationary HMM due to nonlinearity and non-Gaussianity. Finally, we show that a common form of ARR (bivariate ARR) is susceptible to false positives compared with PR and an adapted version of ARR (multivariate ARR). Application of PR and multivariate ARR to Human Connectome Project data suggests that the stationary, linear, Gaussian null hypothesis cannot be rejected for most participants. However, failure to reject the null hypothesis does not imply that static FC can fully explain dFC. We find that first order AR models explain temporal FC fluctuations significantly better than static FC models. Since first order AR models encode both static FC and one-lag FC, this suggests the presence of dynamical information beyond static FC. Furthermore, even in subjects where the null hypothesis was rejected, AR models explain temporal FC fluctuations significantly better than a popular HMM, suggesting the lack of discrete states (as measured by resting-state fMRI). Overall, our results suggest that AR models are not only useful as a means for generating null data, but may be a powerful tool for exploring the dynamical properties of resting-state fMRI. Finally, we discuss how apparent contradictions in the growing dFC literature might be reconciled. •Space of stationary models bigger than expected; includes hidden Markov model (HMM).•Phase & autoregressive randomizations test for stationarity, linearity, Gaussianity.•Resting-state fMRI is mostly stationary, linear, and Gaussian.•1st order autoregressive (AR) model encodes static & one-lag FC.•1st order AR model explains sliding window correlations very well, better than HMM.

We study, for a continuous linear operator T on an F-space X, when the direct sum operator T ⊕ T is recurrent on the direct sum space X ⊕ X. In particular: we establish the analogous notion, for recurrence, to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each separable infinite-dimensional Banach space, solving the T ⊕ T-recurrence problem in the negative way.

This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics. The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth. This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.

Understanding both the epidemiological and evolutionary dynamics of antimicrobial resistance is a major public health concern. In this paper, we propose a nested model, explicitly linking the within- and between-host scales, in which the level of resistance of the bacterial population is viewed as a continuous quantitative trait. The within-host dynamics is based on integro-differential equations structured by the resistance level, while the between-host scale is additionally structured by the time since infection. This model simultaneously captures the dynamics of the bacteria population, the evolutionary transient dynamics which lead to the emergence of resistance, and the epidemic dynamics of the host population. Moreover, we precisely analyze the model proposed by particularly performing the uniform persistence and global asymptotic results. Finally, we discuss the impact of the treatment rate of the host population in controlling both the epidemic outbreak and the average level of resistance, either if the within-host scale therapy is a success or failure. We also explore how transitions between infected populations (treated and untreated) can impact the average level of resistance, particularly in a scenario where the treatment is successful at the within-host scale.

Human action recognition with a dual-stream architecture using linear dynamical systems (LDSs) approach is discussed in this paper. First, a slice process is established to extract original slices from video sequences. Two slicing methods are adopted to subtract or reserve the remaining frames in the video sequences. By applying background subtraction to adjacent frames of the original slices, difference slices are also expressed. To capture the spatial component of the background and difference expressed in each slice simultaneously, a framework based on pre-trained convolutional neural networks (CNNs) is introduced for dual-stream deep feature extraction. Subsequently, LDSs are established to model the timing relationship between adjacent slices and obtain the temporal component of the background and difference features, which are expressed as linear dynamical background feature (LD-BF) and linear dynamical difference feature (LD-DF). Practical experiments were conducted to demonstrate the effectiveness and robustness of the proposed approach using different datasets. Specifically, our experiments were conducted on the UCF50, UCF101, and hmdb51 datasets. The impact of retaining various principal component analysis (PCA) feature dimensions and distinct slicing methods in terms of detail recognition were evaluated. In particular, combining LD-BF with LD-DF under appropriate feature dimensions and slicing methods further improved the accuracy for the UCF50, UCF101, and hmdb51 datasets. In addition, the computational cost of the feature extraction process was evaluated to illustrate the efficiency of the proposed approach. The experimental results show that the proposed approach is competitive with state-of-the-art approaches in the three datasets.