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Dynamical Systems - Science topic

Studying the behavior of complex dynamical systems that are usually described by differential equations or difference equations.
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Publications related to Dynamical Systems (10,000)
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Fixed point theory has many applications in diverse fields ranging from different branches of mathematics to engineering, and from economics to biology. For example, optimization problems including minimization problems, variational inequality problems, equilibrium problems, and variational inclusion problems, among others, are known to be very use...
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Reachability analysis is a powerful tool for studying the safety of nonlinear systems, in which one of the key points is the computation of reachable tubes. As a common method in engineering, the Hamiltonian Jacobi technique often faces the “curse of dimensionality”. Its computational complexity grows exponentially with the dimensionality of the sy...
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For a class of state-constrained dynamical systems described by evolution variational inequalities, we study the time evolution of a probability measure which describes the distribution of the state over a set. In contrast to smooth ordinary differential equations, where the evolution of this probability measure is described by the Liouville equati...
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This paper discusses a possibility of using a new type of bifurcation diagrams that can be computed for time series being either solutions of nonlinear oscillatory dynamical systems or measured and recorded in a laboratory when no mathematical model is known. The obtained or measured noisy time-series in mechanical (and in other areas of engineerin...
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In this paper, we introduce and study the notions of hypercyclicity and transitivity for random dynamical systems and we establish the relation between them in a topological space. We also introduce the notions of mixing and weakly mixing for random dynamical systems and give some of their properties.
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Data assimilation (DA) is a key component of many forecasting models in science and engineering. DA allows one to estimate better initial conditions using an imperfect dynamical model of the system and noisy/sparse observations available from the system. Ensemble Kalman filter (EnKF) is a DA algorithm that is widely used in applications involving h...
Book
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This book addresses the problem of multi-agent systems, considering that it can be interpreted as a generalized multi-synchronization problem. From manufacturing tasks, through encryption and communication algorithms, to high-precision experiments, the simultaneous cooperation between multiple systems or agents is essential to successfully carrying...
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The advances in data science and machine learning have resulted in significant improvements regarding the modeling and simulation of nonlinear dynamical systems. It is nowadays possible to make accurate predictions of complex systems such as the weather, disease models or the stock market. Predictive methods are often advertised to be useful for co...
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We investigate the stationary flow of a colloidal gel under an inhomogeneous external shear force using adaptive Brownian dynamics simulations. The interparticle forces are derived from the Stillinger–Weber potential, where the three-body term is tuned to enable network formation and gelation in equilibrium. When subjected to the shear force field,...
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We calculate the hybrid entanglement entropy between coin and walker degrees of freedom in a class of two-step non-unitary quantum walks. The model possesses a joint parity and time-reversal symmetry or PT-symmetry and supports topological phases when this symmetry is unbroken by its eigenstates. An asymptotic analysis at long times reveals that th...
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Moisl [1, 2] proposed a model of how the brain implements intrinsic intentionality with respect to lexical and sentence meaning, where 'intrinsic' is understood as 'independent of interpretation by observers external to the cognitive agent'. The discussion in both was mainly philosophical and qualitative; the present paper gives a mathematical acco...
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This work develops a long-term conjunction toolkit by combining an efficient and accurate semi-analytical uncertainty propagation method, realistic dynamics, and an encounter model. The epoch Gaussian uncertainty distribution is split into smaller Gaussian Mixture Model components. These mixture components are propagated individually using State Tr...
Conference Paper
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About the Lecture Series: Dynamic equations have become very popular in many areas of science and engineering. Stability, bifurcations, and long-term behaviour of continuous dynamical systems and maps play a crucial role in modeling and interpreting real world problems. In this lecture series, the eminent speakers will talk about the stability of...
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The qualitative theory for planar dynamical systems is used to study the bifurcation of the wave solutions for the space-fractional nonlinear Schrödinger equation with multiplicative white noise. Employing the first integral, we introduce some new wave solutions, assorted into periodic, solitary, and kink wave solutions. The dependence of the solut...
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We study the stability of regular pullback attractors for non-autonomous dynamical systems generated by partial differential equations with non-autonomous forcing term. We first introduce the concept of the backward compact regular pullback attractor. We then establish the existence theorem of the backward compact regular pullback attractor. Finall...
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When a Hamiltonian system undergoes a stochastic, time-dependent anharmonic perturbation, the values of its adiabatic invariants as a function of time follow a distribution whose shape obeys a Fokker–Planck equation. The effective dynamics of the body’s centre-of-mass during human walking is expected to represent such a stochastically perturbed dyn...
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The interdependency between interest rates, investment demands and inflation rates in a given economy has a continuous dynamics. We propose a four-dimensional model which describes these interactions by imposing a control law on the interest rate. By a qualitative analysis based on tools from dynamical systems theory, we obtain in the new model tha...
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This paper provides the theoretical foundation for the approximation of the regions of attraction in hyperbolic and polynomial systems based on the eigenfunctions deduced from the data-driven approximation of the Koopman operator. In addition, it shows that the same method is suitable for analyzing higher-dimensional systems in which the state spac...
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Given a dynamical system with constrained outputs, the maximal admissible set (MAS) is defined as the set of all initial conditions such that the output constraints are satisfied for all time. It has been previously shown that for discrete-time, linear, time-invariant, stable, observable systems with polytopic constraints, this set is a polytope de...
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In this paper, we introduce some new classes of exponentially variational inclusions. Several important special cases are obtained as applications. Using the resolvent operator, it is shown that the exponentially variational inclusions are equivalent to the fixed point problem. This alternative formulation is used to suggest and investigate a wide...
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This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo periodic, pseudo recurrent and Poisson stable) motions of monotone sub-linear non-autonomous dynamical systems. The main results we establish in the framework of general non-autono...
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Recently, the research community has been exploring fractional calculus to address problems related to cosmology; in this approach, the gravitational action integral is altered, leading to a modified Friedmann equation, then the resulting theory is compared against observational data. In this context, dynamical systems can be used along with an ana...
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Modelling the evolution of a system using stochastic dynamics typically implies a greater subjective uncertainty in the adopted system coordinates as time progresses, and stochastic entropy production has been developed as a measure of this change. In some situations the evolution of stochastic entropy production can be described using an It\^o pro...
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Using the concept of information distance derived from Kolmogorov randomness, we study damage spreading for elementary cellular automata acting on a one-dimensional lattice. In contrast to previous definitions of the Lyapunov exponent based on Hamming distance, the new magnitude allows a better clustering of chaotic rules. The combined use of the L...
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Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work we study a dynamical systems where the edge conductivities of a network are regulated by time-varying mass loads injected on nodes....
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A mathematical correspondence between nonlinear dynamical systems modelling of seismic activation and statistical physics modelling of seismic activation is presented. Starting from earthquake fault dynamic equations, it is explained how the real time increase in mean rupture area of earthquakes occurring during seismic activation can be correspond...
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In a Hilbert framework, we introduce a new class of fast continuous dissipative dynamical systems for approximating zeroes of an arbitrary maximally monotone operator. This system originates from some change of variable operated in a continuous Nesterov-like model that is driven by the Yosida regularization of the operator and that involves an asym...
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We classify the possible closures of leaves of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus g≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...
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This paper adopts a new terminology, "delta q−Mittag-Leffler stability," for studying the stability of nonlinear q−fractional dynamical systems on the time scale. In fact, the idea of delta q−Mittag-Leffler stability is inspired by the idea of Mittag-Leffler stability, which is designed to investigate the stability of fractional dynamical systems....
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This paper focuses on the analysis of two particular models, from deterministic and random perspective respectively, for spreading processes. With a proper encoding of propagation patterns, the spread rate of each pattern is discussed for both models by virtue of the substitution dynamical systems and branching process. In view of this, we are empo...
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The present research objective was to analyze the offensive phase from Complex I in high-level male volleyball teams in a macro- and micro-level view, through the inter e intra-team variability analysis of eight best teams of the 2018 Men's Volleyball World Championship over the social network analysis and eigenvector centrality. The sample consist...
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We study the existence and uniqueness of solutions to the inverse quasi-variational inequality problem. Motivated by the dynamical approach to solving optimization problems such as variational inequality, monotone inclusion, and inverse variational problems, we consider a dynamical system associated with the inverse quasi-variational inequality pro...
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Simulations of large-scale dynamical systems require expensive computations. Low-dimensional parametrization of high-dimensional states such as Proper Orthogonal Decomposition (POD) can be a solution to lessen the burdens by providing a certain compromise between accuracy and model complexity. However, for really low-dimensional parametrizations (f...
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Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax theory. It consists of a couple of $$\sqrt{T\overline{T}}$$ T T ¯ -deformed oscillators that nevertheless pr...
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The dynamics of the 1.5-degree-of-freedom model of towed wheel is investigated. Dry friction at the king pin is considered, leading to a non-smooth dynamical system. Beyond analytical and numerical linear stability analysis, the nonlinear vibrations are investigated by numerical bifurcation analysis with smoothing and by numerical simulations with...
Article
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A folded chain hanged by its two ends in U-shape at the same level with an opening, distance, between its two tips is known as U-folded chain, as studied in Markou et al. (2023). When one of the tips is fixed and the other one is released, the free tip of the U-folded falling chain accelerates faster than gravity, due to momentum conservation. This...
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We exploit analogies between first-order algorithms for constrained optimization and non-smooth dynamical systems to design a new class of accelerated first-order algorithms for constrained optimization. Unlike Frank-Wolfe or projected gradients, these algorithms avoid optimization over the entire feasible set at each iteration. We prove convergenc...
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These last years, scientific research focuses on the dynamical aspects of psychiatric disorders and their clinical significance. In this article, we proposed a theoretical framework formalized as a generic mathematical model capturing the heterogeneous individual evolutions of psychiatric symptoms. The first goal of this computational model based o...
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Augmenting mechanistic ordinary differential equation (ODE) models with machine-learnable structures is a novel approach to create highly accurate, low-dimensional models of engineering systems incorporating both expert knowledge and reality through measurement data. Our exploratory study focuses on training universal differential equation (UDE) mo...
Article
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This article deals with the construction of high-gain observers for autonomous polynomial dynamical systems. In contrast to the usual approach, the system’s state is embedded into a higher dimensional Euclidean space. The observer state will be contained in said Euclidean space, which has usually higher dimension than the system’s state space. Due...
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Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit...
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Although the problem of locked-in deep stall is well documented over many years, there currently exists no consistent procedure that can guarantee recovery. Past studies have suggested that it might be possible to rock the aircraft in pitch to destabilise the statically-stable deep stall trim point, thereby gaining enough momentum to push the nose...
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The use of a digital twin to update a feedback controller is considered, and this is illustrated using simulations of a position-controlled dynamical system with a time-varying nonlinear element. The feedback control system consists of a dc motor driving the displacement of a three degree of freedom structure through a lead screw that is subject to...
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We consider a special class of bipartite graphs, called chain graphs, defined as {C_3, C_5, 2K_2}-free graphs, that have no repeated Laplacian eigenvalues. Our results include structure theorems, degree constraints and examinations of the corresponding eigenspaces. For example , it occurs that such chain graphs do not contain a triplet of vertices...
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To better understand the orbital dynamics of exoplanets around close binary stars, i.e., circumbinary planets (CBPs), we applied techniques from dynamical systems theory to a physically motivated set of solutions in the Circular Restricted Three-Body Problem (CR3BP). We applied Floquet theory to characterize the linear dynamical behavior -- static,...
Article
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The problem of detecting linear attacks on industrial systems is presented in this paper. The object is attacked by linear attack is the wireless communication process from sensors to controller with simulated mathematical model (stochastic dynamical systems and random noises). The attack matrices are calculated to ensure that Kullback-Leiber (K-L)...
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A nonlinear feedback stabilizing controller technique for a hybrid excitation synchronous machine (HESM) system is developed in this study. The desired control law is designed using the immersion and invariance (I&I) technique. To ensure that the equilibrium point is asymptotically stable and that all the overall closed-loop system trajectories are...
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Repeatedly solving nonlinear partial differential equations with varying parameters is often an essential requirement to characterise the parametric dependences of dynamical systems. Reduced-order modelling (ROM) provides an economical way to construct low-dimensional parametric surrogates for rapid predictions of high-dimensional physical fields....
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Recent findings suggesting the potential transdiagnostic efficacy of psychedelic-assisted therapy have fostered the need to deepen our understanding of psychedelic brain action. Functional neuroimaging investigations have found that psychedelics reduce the functional segregation of large-scale brain networks. However, beyond this general trend, fin...
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The sufficient settings of the space generated by absolute type-weighted gamma matrices of rank p in the Nakano complex functions of the formal power series, as well as its associated prequasioperators’ ideal equipped with definite functions, are defined and explained in this paper. Assorted prequasinorms are shown to have the Fatou characteristic....
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We clarify the geometrical background of integrable systems, lattice models and two-body interaction models in particle physics according to their topological two degrees of freedom structure. The basic concept of global 2DF geometry of equations of motion as the actual geometry or topology of integrable dynamical systems is introduced which replac...
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In particle-laden turbulence, the Fourier Lagrangian spectrum of each phase is regularly computed, and analytically derived response functions relate the Lagrangian spectrum of the fluid- and the particle phase. However, due to the periodic nature of the Fourier basis, the analysis is restricted to statistically stationary flows. In the present wor...
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The ability to predict and characterize bifurcations from the onset of unsteadiness to the transition to turbulence is of critical importance in both academic and industrial applications. Numerous tools from dynamical system theory can be employed for that purpose. In this review, we focus on the practical computation and stability analyses of stea...
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The background numerical noise $\varepsilon_{0} $ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $\varepsilon(t)$ grows exponentially, say, $\varepsilon(t) = \varepsilon_{0} \exp(\kappa\,t)$, where $\kappa>0$ is the so-called noise-growing exponent. This is the reason why one can not...
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We address the following open problem, implicit in the 1990 article "Automorphisms of one-sided subshifts of finite type" of Boyle, Franks and Kitchens (BFK): "Does there exists an element $\psi$ in the group of automorphisms of the one-sided shift $\operatorname{Aut}(\{0,1,\ldots,n-1\}^{\mathbb{N}}, \sigma_{n})$ so that all points of $\{0,1,\ldots...
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En las primeras partes de este trabajo, se mostró la realización electrónica con circuitos analógicos de algunos sistemas caóticos continuos cuadráticos y lineales por tramos, mediante circuitos con amplificadores operacionales y otros componentes, así como la equivalencia de sus variables electrónicas con los modelos matemáticos establecidos. En e...
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Modeling spatiotemporal dynamical systems is a fundamental challenge in machine learning. Transformer models have been very successful in NLP and computer vision where they provide interpretable representations of data. However, a limitation of transformers in modeling continuous dynamical systems is that they are fundamentally discrete time and sp...
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Verification of discrete time or continuous time dynamical systems over the reals is known to be undecidable. It is however known that undecidability does not hold for various classes of systems: if robustness is defined as the fact that reachability relation is stable under infinitesimal perturbation, then their reachability relation is decidable....
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Extreme events are unusual and rare large-amplitude fluctuations that occur can unexpectedly in nonlinear dynamical systems. Events above the extreme event threshold of the probability distribution of a nonlinear process characterize extreme events. Different mechanisms for the generation of extreme events and their prediction measures have been re...
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Plant litter decomposition stands at the intersection between carbon (C) loss and sequestration in terrestrial ecosystems. Organic matter during this process experiences chemical and physical transformations that affect decomposition rates of distinct components with different transformation fates. However, most decomposition studies only fit one-p...
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In fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadiness to the transition to turbulence, is of critical importance for both academic and industrial applications. Different tools from dynamical systems theory can be used for this purpose. In this review, we present a concise theoretical and numerical framework f...
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We introduce the concept of ``covariant Lyapunov field'', which assigns all the components of covariant Lyapunov vectors at almost all points of the phase space of a dynamical system. We show that in ergodic systems such fields can be characterized as the global solutions of a partial differential equation on the phase space. Due to the arbitrarine...
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In this paper, we discuss the time-fractional mKdV–ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV–ZK equation to an integer-order ordinary differential equation utili...
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Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for per...
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The features of the main models of spiking neurons are discussed in this review. We focus on the dynamical behaviors of five paradigmatic spiking neuron models and present recent literature studies on the topic, classifying the contributions based on the most-studied items. The aim of this review is to provide the reader with fundamental details re...
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Sparse model identification enables nonlinear dynamical system discovery from data. However, the control of false discoveries for sparse model identification is challenging, especially in the low-data and high-noise limit. In this paper, we perform a theoretical study on ensemble sparse model discovery, which shows empirical success in terms of acc...
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We develop the first end-to-end sample complexity of model-free policy gradient (PG) methods in discrete-time infinite-horizon Kalman filtering. Specifically, we introduce the receding-horizon policy gradient (RHPG-KF) framework and demonstrate $\tilde{\mathcal{O}}(\epsilon^{-2})$ sample complexity for RHPG-KF in learning a stabilizing filter that...
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We study the problem of performance optimization of closed-loop control systems with unmodeled dynamics. Bayesian optimization (BO) has been demonstrated to be effective for improving closed-loop performance by automatically tuning controller gains or reference setpoints in a model-free manner. However, BO methods have rarely been tested on dynamic...
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Nanoscale anisotropic dynamical systems are tedious to solve in a classical domain without any approximation techniques. In such cases, the interactive space and force are significant in defining their motions. We solve and simulate the phase transition motion of water in the interface of 3 μm × 3 μm confined water domain. We demonstrate the presen...
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The phenomenologically emergent dark energy(PEDE) model is a varying dark energy model with no extra degrees of freedom proposed initially to alleviate the Hubble tension by Li and Shafieloo\cite{Li_2019}. The statistical consistency of the model have been discussed by many authors. Since the model presents a phantom dark energy which is an increas...
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In this paper, we analyze a speed restarting scheme for the dynamical system given by $$ \ddot{x}(t) + \dfrac{\alpha}{t}\dot{x}(t) + \nabla \phi(x(t)) + \beta \nabla^2 \phi(x(t))\dot{x}(t)=0, $$ where $\alpha$ and $\beta$ are positive parameters, and $\phi:\mathbb{R}^n \to \mathbb{R}$ is a smooth convex function. If $\phi$ has quadratic growth, we...
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This is a text written for the Ennio De Giorgi Colloquio volume. It covers analogies between algebraic number theory and knot theory, analogies between analytic number theory and certain dynamical systems, and a report on our construction of dynamical systems for arithmetic schemes which realize some of these analogies.
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We introduce a new concept of Yosida distance between two (unbounded) linear operators $A$ and $B$ in a Banach space $\mathbb{X}$ defined as $d_Y(A,B):=\limsup_{\mu\to +\infty} \| A_\mu-B_\mu\|$, where $A_\mu$ and $B_\mu$ are the Yosida approximations of $A$ and $B$, respectively, and then study the persistence of evolution equations under small Yo...
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In a dynamical systems description of spatiotemporally chaotic PDEs including those describing turbulence, chaos is viewed as a trajectory evolving within a network of non-chaotic, dynamically unstable, time-invariant solutions embedded in the chaotic attractor of the system. While equilibria, periodic orbits and invariant tori can be constructed u...
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This paper introduces a computational framework to reconstruct and forecast a partially observed state that evolves according to an unknown or expensive-to-simulate dynamical system. Our reduced-order autodifferentiable ensemble Kalman filters (ROAD-EnKFs) learn a latent low-dimensional surrogate model for the dynamics and a decoder that maps from...