Linear Dynamics Research Articles

Modeling of Static Stress Identification Using Electromechanical Impedance of Embedded Piezoelectric Plate.

Working stress is an important indicator reflecting the health status of structures. Passive-monitoring technology using the piezoelectric effect can effectively monitor the dynamic stress of structures. However, under static loads, the charge generated by the piezoelectric devices can only be preserved when the external circuit impedance is infinitely large, which means passive-monitoring techniques are unable to monitor static and quasi-static stress caused by slow-changing actions. In current studies, experimental observations have shown that the impedance characteristics of piezoelectric devices are affected by external static loads, yet the underlying mechanisms remain inadequately explained. This is because the impedance characteristics of piezoelectric devices are actually dynamic characteristics under alternating voltage. Most existing impedance analysis models are based on linear elastic dynamics. Within this framework, the impact of static stress on dynamic characteristics, including impedance characteristics, cannot be addressed. Accounting for static stress in impedance modeling is a challenging problem. In this study, the static stress applied on an embedded piezoelectric plate is abstracted as the initial stress of the piezoelectric plate. Based on nonlinear elastic dynamic governing equations, using the displacement method, an impedance analysis model of an embedded piezoelectric plate considering initial stress is established and verified through a fundamental experiment and a finite element analysis. Based on this, the explicit analytical relation between initial stress and impedance characterizations is provided, the mechanism of the effect of initial stress on the impedance characterizations is revealed, and procedures to identify static stress using impedance characterizations is proposed. Moreover, the sensitivities of the impedance characterizations in response to the initial stress are thoroughly discussed. This study mainly provides a theoretical basis for monitoring static stress using the electromechanical impedance of an embedded piezoelectric plate. And the results of the present study can help with the performance prediction and design optimization of piezoelectric-based static stress sensors.

One- and two-photon excitation dynamics using semiclassical electron force field model.

We have extended the semiclassical-based electron force-field simulation by introducing field-electron interaction to enable us to describe linear and nonlinear electronic excitation dynamics of a condensed matter system with low computational cost. To verify the simulation method, as a first step, numerical examples of interaction dynamics of simple systems (H atom, SiH4 molecule, and Si crystalline solid) with applied short electric field pulse as well as the obtained absorbed energies by the one- and two-photon excitations have been reported along with comparison with quantum dynamics calculations as reference.

Linear and nonlinear dynamics of self-consistent collisionless tearing modes in toroidal gyrokinetic simulations

We investigate tearing modes (TM) driven by current density gradient in collisionless tokamak plasmas by using the electromagnetic gyrokinetic simulation code ORB5. We elucidate the TM width by simulations for flat profiles, as the absence of background diamagnetic flows implies a small rotation speed, while finite gradients are included to investigate the TM rotation. For flat profiles, the initial saturation width of nonlinearly driven magnetic islands is related to the TM linear growth rate; however, large islands in the initial saturation phase are prone to current density redistribution that reduces the island width in the following evolution. Island-induced E×B and diamagnetic sheared flows develop at the separatrix, able to destabilize the Kelvin–Helmholtz instability (KHI). The KHI turbulence enhances a strong quadrupole vortex flow that reinforces the island decay, resulting in a strong reduction of the island width in an eventual steady state. This process is enhanced by trapped electrons. For finite gradients profile, the TM usually rotates in the electron diamagnetic direction but can change direction when the ion temperature gradient dominates the other gradients. The reduced growth of the TM by diamagnetic effects results in a moderate island size, which remains almost unchanged after the initial saturation. At steady state, strong zonal flows are nonlinearly excited and dominate the island rotation, as expected from previous theoretical and numerical studies. When β is increased, the TM mode is suppressed and a mode with the same helicity but with twisting parity, coupled with the neighboring poloidal harmonics, is destabilized, similar to the kinetic ballooning mode.

Spontaneous symmetry breaking in generative diffusion models*

Abstract Generative diffusion models have recently emerged as a leading approach for generating high-dimensional data. In this paper, we show that the dynamics of these models exhibit a spontaneous symmetry breaking phenomenon that divides the generative dynamics into two distinct phases: (1) a linear steady-state dynamics around a central fixed-point and, (2) an attractor dynamics directed towards the data manifold. These two ‘phases’ are separated by the change in stability of the central fixed-point, with the resulting window of instability being responsible for the diversity of the generated samples. Using both theoretical and empirical evidence, we show that an accurate simulation of the early dynamics does not significantly contribute to the final generation, since early fluctuations are reverted to the central fixed point. To leverage this insight, we propose a Gaussian late initialization scheme, which significantly improves model performance, achieving up to 3× Fréchet inception distance improvements on fast samplers, while also increasing sample diversity (e.g. racial composition of generated CelebA images). Our work offers a new way to understand the generative dynamics of diffusion models that has the potential to bring about higher performance and less biased fast-samplers.

Heart rate variability and insomnia in depressed patients with breast cancer

ObjectivesDepression is associated with unhealthy autonomic regulation. However, whether patients with breast cancer (BC) with different degrees of depression can be identified from linear and non-linear dynamics in the autonomic...

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document}, for t∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1)$$\\end{document}, are investigated when they act on the disc algebra A(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A({\\mathbb {D}})$$\\end{document} and on the Hardy spaces Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document}, for 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}. We study the continuity, compactness, spectrum and point spectrum of Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} as well as their linear dynamics and mean ergodicity on these spaces.

Reexamining the economic globalization-welfare state nexus: a Bayesian mixed approach to linear and non-linear dynamics

Reexamining the economic globalization-welfare state nexus: a Bayesian mixed approach to linear and non-linear dynamics

Numerical analysis of the fungibility of disclination dipole by dislocation array or dislocation monopole

Disclinations have been discovered and applied to a wide variety of materials, underscoring the importance of their analysis. Previous studies have demonstrated that a disclination dipole can be represented by a dislocation array in a deformation schematic and by a dislocation monopole in a stress state. Based on these observations, we propose that the elastic field of a disclination dipole can be substituted with that of a dislocation array or dislocation monopole, allowing the development of the disclination theory grounded in dislocation theory. In this study, we conducted a quantitative analysis of the elastic strain–energy of disclination dipoles, dislocation arrays, and dislocation monopoles using linear elasticity theory and molecular dynamics (MD) simulation. While linear elasticity theory facilitates a quantitative evaluation of the elastic field, it fails to account for out-of-plane deformation of the material. Therefore, MD analysis, which considers out-of-plane deformation, was employed to obtain versatile results applicable to a broader range of materials. Our findings from both theoretical analyses indicate that, in terms of strain–energy, a disclination dipole with a large separation can be represented by a dislocation array, whereas a dipole with a small separation can be represented by a dislocation monopole. This study newly performs strain–energy analysis of dislocations and disclinations using these two theoretical approaches.

Linear RNNs Provably Learn Linear Dynamical Systems

Linear RNNs Provably Learn Linear Dynamical Systems

Nonlinear Dynamics for the Ising Model

We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann’s model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely spin systems at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} and O(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log n)$$\\end{document} respectively, where n is the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erdős-Rényi random graphs, and non-traditional branching processes augmented by a “fragmentation” process. Our results extend immediately to any spin system with a finite number of spins and bounded interactions.

Weak and strong solutions for a fluid‐poroelastic‐structure interaction via a semigroup approach

A filtration system comprising a Biot poroelastic solid coupled to an incompressible Stokes free‐flow is considered in 3D. Across the flat 2D interface, the Beavers‐Joseph‐Saffman coupling conditions are taken. In the inertial, linear, and non‐degenerate case, the hyperbolic‐parabolic coupled problem is posed through a dynamics operator on a chosen energy space, adapted from Stokes‐Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. The generation of a strongly continuous semigroup for the dynamics operator is obtained via a non‐standard maximality argument. The latter employs a mixed‐variational formulation in order to invoke the Babuška‐Brezzi theorem. The Lumer‐Philips theorem then yields semigroup generation, and thereby, strong and generalized solutions are obtained. For the linear dynamics, density obtains the existence of weak solutions; we extend to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial linear Biot‐Stokes filtration system, we provide a clear elucidation of strong solutions and a construction of weak solutions, as well as their regularity through associated estimates.

Linear Dynamics of Multiplication and Composition Operators on $$\textrm{Hol}(\mathbb {D})$$

Linear Dynamics of Multiplication and Composition Operators on $$\textrm{Hol}(\mathbb {D})$$

Operational Modal Analysis of Self-excited Vibrations in Milling Considering Periodic Dynamics

Abstract A new method is presented to identify the dynamics of regenerative chatter from measured process vibrations in milling. This method combines the synchronous once-per-revolution sampling of process vibrations with Operational Modal Analysis to estimate the Floquet multipliers of the delayed linear time-periodic dynamics in milling, all from the natural process vibrations without external excitation. The identified multipliers quantify vibration stability, enabling chatter prediction before it occurs. In addition to this, they can be used to calibrate physics-based chatter models based on vibration measurements solely within the stable region. The method's accuracy in identifying Floquet multipliers is validated through extensive numerical simulations and two experimental case studies. The results show that chatter due to both Hopf and period-doubling bifurcations can be predicted from the process vibrations during stable cuts. Moreover, the experimental case studies demonstrate a vibration measurement system for implementing the presented method in standard milling operations and confirm its effectiveness in practice.

Minimal motifs for habituating systems

Habituation-a phenomenon in which a dynamical system exhibits a diminishing response to repeated stimulations that eventually recovers when the stimulus is withheld-is universally observed in living systems from animals to unicellular organisms. Despite its prevalence, generic mechanisms for this fundamental form of learning remain poorly defined. Drawing inspiration from prior work on systems that respond adaptively to step inputs, we study habituation from a nonlinear dynamics perspective. This approach enables us to formalize classical hallmarks of habituation that have been experimentally identified in diverse organisms and stimulus scenarios. We use this framework to investigate distinct dynamical circuits capable of habituation. In particular, we show that driven linear dynamics of a memory variable with static nonlinearities acting at the input and output can implement numerous hallmarks in a mathematically interpretable manner. This work establishes a foundation for understanding the dynamical substrates of this primitive learning behavior and offers a blueprint for the identification of habituating circuits in biological systems.

Exploring nonlinear dynamics in brain functionality through phase portraits and fuzzy recurrence plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Exploring Nonlinear Dynamics In Brain Functionality Through Phase Portraits And Fuzzy Recurrence Plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Detecting stochasticity in population time series using a non‐parametric test of intrinsic predictability

Abstract Many ecological systems dominated by stochastic dynamics can produce complex time series that inherently limit forecast accuracy. The ‘intrinsic predictability’ of these systems can be approximated by a time series complexity metric called weighted permutation entropy (WPE). While WPE is a useful metric to gauge forecast performance prior to model building, it is sensitive to noise and may be biased depending on the length of the time series. Here, we introduce a simple randomized permutation test (rWPE) to assess whether a time series is intrinsically more predictable than white noise. We apply rWPE to both simulated and empirical data to assess its performance and usefulness. To do this, we simulate population dynamics under various scenarios, including a linear trend, chaotic, periodic and equilibrium dynamics. We further test this approach with observed abundance time series for 932 species across four orders of animals from the Global Population Dynamics Database. Finally, using Adélie (Pygoscelis adeliae) and emperor penguin (Aptenodytes forsteri) time series as case studies, we demonstrate the application of rWPE to multiple populations for a single species. We show that rWPE can determine whether a system is significantly more predictable than white noise, even with time series as short as 10 years that show an apparent trend under biologically realistic stochasticity levels. Additionally, rWPE has statistical power close to 100% when time series are at least 30 time steps long and show chaotic or periodic dynamics. Power decreases to ~10% under equilibrium dynamics, irrespective of time series length. Among four classes of animal taxa, mammals have the highest relative frequency (28%) of time series that are both longer than 30 time steps and indistinguishable from white noise in terms of complexity, followed by insects (16%), birds (16%) and bony fishes (11%). rWPE is a straightforward and useful method widely applicable to any time series, including short ones. By informing forecasters of the inherent limitations to a system's predictability, it can guide a modeller's expectations for forecast performance.

River and drain: Ambiguity in the waterways of Accra

This article analyses a range of diverse and often conflicting practices in and around a waterway in Accra, Ghana, emphasising the ambiguity of their politics. Urban political ecology approaches have demonstrated how flows of (waste)water reveal power dynamics across scales. I extend such insights by developing an understanding of practices that are more ambiguous, complexifying otherwise linear dynamics of power. I do this by centring the narratives of those living and working in the waterway. Through providing needed attention to the everyday and micro-level interactions, this approach makes room for situated ambiguities within wider power dynamics. The argument is based on qualitative research along one primary waterway in Accra, along which people live, farm, dump waste and manage flood risk. These diverse practices suggest that, in addition to distributing risks in an often uneven fashion, the waterway also affords connection, solidarity and adaptability across urban divides. I argue that accounting for these practices does not deny the inequalities of the waterway's geography; rather, it encourages the development of a vocabulary to understand situated processes, even where they appear ambiguous. Embracing the ambiguities of the waterway allows us to fully consider the complex forms that power takes in different settings.

The fine spectra of a difference of weighted shift operators acting in some classical sequence spaces

In this paper we investigate the fine spectra, the ergodic properties and the linear dynamics of the operatorC(x)=(−nxn+n(n−1)xn−2)n∈N0,x=(xn)n∈N0∈CN0, and of its formal adjoint C′ acting in the classical sequence spaces s, s′, ω and φ. The operators C and C′ acting in s are conjugate to the one-dimensional Ornstein-Uhlenbeck differential operator Ou=u″−xu′ and to the differential operator O′u=u″+xu′+u acting in S(R), respectively.

Fractional conformal map, qubit dynamics and the Leggett–Garg inequality

Abstract A pure state of a qubit can be geometrically represented as a point on the extended complex plane through stereographic projection. By employing successive conformal maps on the extended complex plane, we can generate an effective discrete-time evolution of the pure states of the qubit. This work focuses on a subset of analytic maps known as fractional linear conformal maps. We show that these maps serve as a unifying framework for a diverse range of quantum-inspired conceivable dynamics, including (i) unitary dynamics,(ii) non-unitary but linear dynamics and (iii) \ non-unitary and non-linear dynamics where linearity (non-linearity) refers to the action of the discrete time evolution operator on the Hilbert space. We provide a characterization of these maps in terms of Leggett-Garg Inequality complemented with No-signaling in Time (NSIT) and Arrow of Time (AoT) conditions.