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Liquid–liquid phase separation in biology has recently been shown to play a major role in the spatial control of biomolecular components within the cell. However, as they are phase transitions, these processes also display nontrivial dynamics. A model phase-separating system of DNA nanostars provides unique access to nucleation physics in a biomolecular context, as phase separation is driven near room temperature by highly thermo-responsive DNA hybridization and at modest DNA concentrations. By measuring the delay time for phase-separated droplets to appear, we demonstrate that the dynamics of DNA nanostar phase separation reflect that of a metastable binary mixture of patchy particles. For sufficiently deep temperature quenches, droplets undergo spinodal decomposition and grow spontaneously, driven by Brownian motion and coalescence of phase-separated droplets, as confirmed by comparing experimental measurements to particle-based simulations. Near the coexistence boundary, droplet growth slows substantially, indicative of a nucleation process. The temperature dependence of droplet appearance times can be predicted by a classical nucleation picture with mean field exponents and demonstrates that a theory previously used to predict equilibrium phase diagrams can also distinguish spinodal and nucleation dynamical regimes. These dynamical principles are relevant to behaviors associated with liquid–liquid phase separating systems, such as their spatial patterning, reaction coupling, and biological function. [ABSTRACT FROM AUTHOR]

We present comprehensive numerical results from a study of model H, which describes phase separation kinetics in binary fluid mixtures. We study the pattern dynamics of both density and velocity fields in d = 2, 3. The density length scales show three distinct regimes, in accordance with analytical arguments. The velocity length scale shows a diffusive behavior. We also study the scaling behavior of the morphologies for density and velocity fields and observe dynamical scaling in the relevant correlation functions and structure factors. Finally, we study the effect of quenched random field disorder on spinodal decomposition in model H. [ABSTRACT FROM AUTHOR]

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. D. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape and can, in simple cases, be interpreted in terms of the barriers of the staircase landscape. [ABSTRACT FROM AUTHOR]

Memory effects are ubiquitous in a wide variety of complex physical phenomena, ranging from glassy dynamics and metamaterials to climate models. The Generalized Langevin Equation (GLE) provides a rigorous way to describe memory effects via the so-called memory kernel in an integro-differential equation. However, the memory kernel is often unknown, and accurately predicting or measuring it via, e.g., a numerical inverse Laplace transform remains a herculean task. Here, we describe a novel method using deep neural networks (DNNs) to measure memory kernels from dynamical data. As a proof-of-principle, we focus on the notoriously long-lived memory effects of glass-forming systems, which have proved a major challenge to existing methods. In particular, we learn the operator mapping dynamics to memory kernels from a training set generated with the Mode-Coupling Theory (MCT) of hard spheres. Our DNNs are remarkably robust against noise, in contrast to conventional techniques. Furthermore, we demonstrate that a network trained on data generated from analytic theory (hard-sphere MCT) generalizes well to data from simulations of a different system (Brownian Weeks–Chandler–Andersen particles). Finally, we train a network on a set of phenomenological kernels and demonstrate its effectiveness in generalizing to both unseen phenomenological examples and supercooled hard-sphere MCT data. We provide a general pipeline, KernelLearner, for training networks to extract memory kernels from any non-Markovian system described by a GLE. The success of our DNN method applied to noisy glassy systems suggests that deep learning can play an important role in the study of dynamical systems with memory. [ABSTRACT FROM AUTHOR]

When dielectric-elastomer (DE) resonators are operated under large alternating voltage excitations, strong geometric and material nonlinearities can significantly affect the dynamic performance. Based on the virtual work principle, a dynamic model of a circular DE resonator is established in this work. Static analysis is conducted to study the impacts of the control voltage on the strain and natural frequency of the DE resonator at equilibrium. A modified Incremental Harmonic Balance and arc-length methods are used to obtain dynamic responses of the system excited by alternating voltages, and stability analysis is conducted using the Floquet theory. The effects of the mechanical parameters are studied, and period-doubling bifurcations are observed, which are produced by the parametric excitation of the voltage. The bifurcation phenomenon is comprehensively investigated through analysis of the voltage, stress, and damping coefficient, which provides design guidance for DE resonators. [ABSTRACT FROM AUTHOR]

In Born–Oppenheimer molecular dynamics (BOMD) simulations based on the density functional theory (DFT), the potential energy and the interatomic forces are calculated from an electronic ground state density that is determined by an iterative self-consistent field optimization procedure, which, in practice, never is fully converged. The calculated energies and forces are, therefore, only approximate, which may lead to an unphysical energy drift and instabilities. Here, we discuss an alternative shadow BOMD approach that is based on backward error analysis. Instead of calculating approximate solutions for an underlying exact regular Born–Oppenheimer potential, we do the opposite. Instead, we calculate the exact electron density, energies, and forces, but for an underlying approximate shadow Born–Oppenheimer potential energy surface. In this way, the calculated forces are conservative with respect to the approximate shadow potential and generate accurate molecular trajectories with long-term energy stabilities. We show how such shadow Born–Oppenheimer potentials can be constructed at different levels of accuracy as a function of the integration time step, δt, from the constrained minimization of a sequence of systematically improvable, but approximate, shadow energy density functionals. For each energy functional, there is a corresponding ground state Born–Oppenheimer potential. These pairs of shadow energy functionals and potentials are higher-level generalizations of the original "zeroth-level" shadow energy functionals and potentials used in extended Lagrangian BOMD [Niklasson, Eur. Phys. J. B 94, 164 (2021)]. The proposed shadow energy functionals and potentials are useful only within this extended dynamical framework, where also the electronic degrees of freedom are propagated as dynamical field variables together with the atomic positions and velocities. The theory is quite general and can be applied to MD simulations using approximate DFT, Hartree–Fock, or semi-empirical methods, as well as to coarse-grained flexible charge models. [ABSTRACT FROM AUTHOR]

As a nonlinear dynamic system, the dynamic equation of triboelectric nanogenerator system is not only difficult to accurately establish, but also difficult to solve. The output signals of triboelectric nanogenerator system depends on the initial conditions of the system, which reflects the dynamic behavior of the system. Currently, there is little research on using the output signals to study the dynamic behavior of triboelectric nanogenerator system. Therefore, a dynamic simulation model of a freestanding triboelectric nanogenerator (F-TENG) system was established using COMSOL software. The mathematical model of the F-TENG system was fitted. Using chaos theory and its phase-space reconstruction technology, the phase-space trajectories of the F-TENG system were reconstructed through voltage signals under external factors, and the dynamic behavior of the F-TENG system was discussed and studied. Finally, validation experiments were carried out. The research shows that with the variation of internal factors and operating frequency, the mathematical model of the F-TENG system remains basically unchanged, and the variation law of output characteristics also remains basically unchanged. The F-TENG system is in periodic motion; with the variation of external factors such as resistance and friction layer spacing, the mathematical model of the F-TENG system will evolve in a complex direction, and the output characteristics will gradually distort. The F-TENG system will evolve from periodic motion to chaotic motion. The research results not only reveal the dynamic behavior and enrich the nonlinear dynamic theory of triboelectric nanogenerator system, but also provide theoretical guidance for increasing the working bandwidth and stability of triboelectric nanogenerator system. [ABSTRACT FROM AUTHOR]

In this paper, two dynamic models, recently proposed to describe the adaptive repeated choices of a boundedly rational consumer, are joined together. One considers a consumer adjusting the consumption level of a given good over time according to the observed discrepancy between expected and realized utility gain and modifies the utility function according to past consumption experience, also including saturation effects when past consumption is excessive. The other one considers the same adjustment mechanism with constant preferences but with a behavioral effect that introduces a tendency (or bias) to imitate a reference group of consumers. Merging these two models, a two-dimensional nonlinear dynamical system is obtained which describes consumers that decide their next period consumption of a given good by following two different (sometimes contrasting) criteria: their own utility maximization on the one side and imitation of a reference group of consumers on the other side. This leads to a greater uncertainty with respect to the model without the behavioral bias. Such uncertainty is studied through a numerical exploration of the long-run dynamics, guided by some global dynamical features of the nonlinear model, such as the folding action of the critical curves that characterize the behavior of the iterated noninvertible map and the singularities related to the presence of a vanishing denominator, namely focal points and prefocal curves. So, the aim of the paper is twofold: on the one side, it tries to contribute to the literature on the economic theory of boundedly rational consumers represented by evolutionary and behavioral approaches; On the other side, it tries to contribute to the recent literature about the global analysis of discrete dynamical systems characterized by contact bifurcations leading to the creation of complex topological structures of the attractors and their basins of attraction. [ABSTRACT FROM AUTHOR]

We present a model in which capital and environmental quality co-evolve over time. To improve the environmental quality, the government intervenes by means of a limitation of the capital use and awareness campaigns. In case of severe degradation of the environment, a restriction on capital use is introduced that is proportional to the damage caused by human activity; at the same time, awareness campaigns are used to increase the public concern about sustainability. By means of a discrete-time dynamical system and considering homogeneous agents, we found that multiple equilibria may exist and that awareness campaigns are a useful tool to push an economy toward sustainable levels of production. The limitation in the use of available capital, however, might be useless or even harmful, deteriorating the level of capital disposable for those countries that are trapped in an equilibrium in which the environmental quality is low. [ABSTRACT FROM AUTHOR]

In the field of molecular nonequilibrium transports, physical mechanisms of multiple reaction dynamics of these systems are the core of deep understanding complex reactions and transport mechanisms. In order to explore related mechanisms, establishing multiple systems coupled with tremendous exit dynamics and studying their exit dynamics properties are quite vital. Beyond previous researches, new stochastic transport processes are emphasized here. Multiple new exit dynamic systems are established, which are motivated by the multiplicity of paths and products of real biochemical processes in organisms. In order to ensure research universality, core system modeling factors are fully considered. Countable parallel orbits, uniform connection with external sources, countable parallel orbits as subsystems in middle lattices and influences of all lattices on transport trajectories on dynamic properties are analyzed. Dynamic properties of different particles located in orbits are explored by deeply studying average exit time and time scale. Quantitative spatiotemporal impacts are extensively studied. The rationality of average exit time as a time scale in the universal exit dynamic system is proved. Main findings and fruitful results can not only serve as theoretical bases for broadening reaction path modeling, but also be helpful to support understanding nonequilibrium transport mechanisms, especially stochastic biochemical processes. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model of fluid flows in a convective thermal system is developed, and a five-dimensional dynamical system is developed for the investigation of the convective fluid dynamics. The analytical solutions of periodic motions to chaos of the convective fluid flows are developed for steady-state vortex flows, and the corresponding stability and bifurcations of periodic motions in the five-dimensional dynamical system are studied. The harmonic frequency-amplitude characteristics for periodic flows are obtained, which provide energy distribution in the parameter space. Analytical homoclinic orbits for the convective fluid flow systems are developed for the asymptotic convection through the infinite-many homoclinic orbits in the five-dimensional dynamical system. The dynamics of fluid flows in the convective thermal systems are revealed, and one can use such methodology to predict atmospheric and oceanic phenomena through thermal convections. [ABSTRACT FROM AUTHOR]

We present a simple method for estimating fundamental frequencies from orbit's time series of triaxial dynamical systems based on the zero-crossing technique. The simplicity and high performance of the method make it a viable alternative to more advanced, but rather complex and computationally expensive techniques based on Fourier spectral analysis and iterative torus-mapping with action-angle variables. More importantly, the method provides a simple and quick way to identify irregular orbits and quantify their level of chaoticity, allowing to detect even a subtle onset of chaos. To validate the proposed algorithm, we performed comparison tests with synthetic quasi-periodic signals and with other methods using the triaxial Dehnen potential. We found noticeable differences in frequency maps constructed with our method and methods based on Fourier spectral analysis that we attributed to the way we assign static properties to chaotic orbits. At the same time, our frequency maps turn out to be in very good agreement with the maps obtained using the action-based technique. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative approach for assessing the stability of linear time-invariant (LTI) systems. The key innovations in our methodology include the calculation of Markov parameters, the extension to nonlinear continuous-time systems, and significantly improved computational efficiency. Our novel approach leverages Markov parameters derived from the characteristic polynomial of the system. Unlike traditional methods, we employ the Leverrier-Takeno algorithm to efficiently determine the coefficients of the characteristic polynomial for state-space representation. The systematic calculation of Markov parameters using a specialized Maclaurin series expansion enhances our method's precision. One of our major achievements is the rigorous verification of stability through Hankel matrices, ensuring positive definiteness using the Cholesky decomposition algorithm. In addition, we introduce a second stability criterion based on iterated polynomial long divisions, although we acknowledge its limitations in handling high-order systems due to computational inefficiency. Our pioneering contributions extend beyond LTI systems. We adapt our innovative techniques to analyze the local stability of nonlinear continuous-time systems by introducing the concept of Jacobian matrices and employing the indirect Lyapunov method. This expansion broadens the applicability of our approach to a wider range of real-world systems. Notably, our approach exhibits remarkable computational efficiency. According to CPU-time analysis, our first stability criterion outperforms the classical Routh–Hurwitz method by more than 12 times for dynamic systems with orders ranging from 2 to 180. This significant reduction in computation time positions our method as a valuable option for real-time stability analysis applications. [ABSTRACT FROM AUTHOR]

Data assimilation (DA) integrates experimental measurements into computational models to enable high-fidelity predictions of dynamical systems. However, the cost associated with solving this inverse problem, from measurements to the state, can be prohibitive for complex systems such as transitional hypersonic flows. We introduce an accurate and efficient deep-learning approach that alleviates this computational burden, and that enables approximately three orders of magnitude computational acceleration relative to variational techniques. Our method pivots on the deployment of a deep operator network (DeepONet) as an accurate, parsimonious and efficient meta-model of the compressible Navier–Stokes equations. The approach involves two main steps, each addressing specific challenges. Firstly, we reduce the computational load by minimizing the number of costly direct numerical simulations to construct a comprehensive dataset for effective supervised learning. This is achieved by optimally sampling the space of possible solutions. Secondly, we expedite the computation of high-dimensional assimilated solutions by deploying the DeepONet. This entails efficiently navigating the DeepONet's approximation of the cost landscape using a gradient-free technique. We demonstrate the successful application of this method for data assimilation of wind-tunnel measurements of a Mach 6, transitional, boundary-layer flow over a 7-degree half-angle cone. [ABSTRACT FROM AUTHOR]

Learning controllers from data for stabilizing dynamical systems typically follows a two-step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) n , then there always exist n states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model. [ABSTRACT FROM AUTHOR]

Introduction: The occurrence of pulmonary consolidation in children with Mycoplasma pneumoniae pneumonia (MPP) can lead to exacerbation of the disease. Therefore, early identification of children with MPP in combination with pulmonary consolidation is critical. The purpose of this study was to develop a straightforward, easy-to-use online dynamic nomogram for the identification of children with MPP who are at high risk of developing pulmonary consolidation. Methods: 491 MPP patients were chosen and divided randomly into a training cohort and an internal validation cohort at a 4:1 ratio. Multi-factor logistic regression was used to identify the risk variables for mixed pulmonary consolidation in children with Mycoplasma pneumoniae (MP). The selected variables were utilized to build the nomograms and validated using the C-index, decision curve analysis, calibration curves, and receiver operating characteristic (ROC) curves. Results: Seven variables were included in the Nomogram model: age, fever duration, lymphocyte count, C-reactive protein (CRP), ferritin, T8 lymphocyte percentage, and T4 lymphocyte percentage. We created a dynamic nomogram that is accessible online (https://ertong.shinyapps.io/DynNomapp/). The C-index was 0.90. The nomogram calibration curves in the training and validation cohorts were highly comparable to the standard curves. The area under the curve (AUC) of the prediction model was, respectively, 0.902 and 0.883 in the training cohort and validation cohort. The decision curve analysis (DCA) curve shows that the model has a significant clinical benefit. Conclusions: We developed a dynamic online nomogram for predicting combined pulmonary consolidation in children with MP based on 7 variables for the first time. The predictive value and clinical benefit of the nomogram model were acceptable. [ABSTRACT FROM AUTHOR]

This study introduces a simple memristor cellular neural network structure, a minimalist configuration with only two cells, designed to concurrently address two logic problems. The unique attribute of this system lies in its adaptability, where the nature of the implemented logic gate, be it AND, OR, and XOR, is determined exclusively by the initial states of the memristors. The memristors' state, alterable through current flow, allows for dynamic manipulation, enabling the setting of initial conditions and consequently, a change in the circuit's functionality. To optimize the parameters of this dynamic system, contemporary machine learning techniques are employed, specifically gradient descent optimization. Through a case study, the potential of leveraging intricate circuit dynamics is exemplified to expand the spectrum of problems solvable with a defined number of neurons. This work not only underscores the significance of adaptability in logical circuits but also demonstrates the efficacy of memristive elements in enhancing problem‐solving capabilities. [ABSTRACT FROM AUTHOR]

In unreinforced masonry structures, among the most dangerous events that can occur during earthquakes are the out-of-plane mechanisms. This type of response significantly changes if the wall is restrained by a horizontal element. The collapse, in this case, could take place for slipping/failure of the diaphragm connection or for overturning of the wall, following the formation of a crack at an intermediate height between the base and the top. A specific analytical model is used to capture the complex dynamic behavior of the wall, formed by two stacked rigid bodies (free to rock) with the top one connected to a flexible diaphragm. The model is calibrated using experimental data available in the literature and it is then used to carry out a dynamic parametric analysis. The variation range of relevant parameters refers to the building features surveyed in Emilia-Romagna region, Italy, and their effect on the global response of the system is investigated. Further, the influence of ground motion is considered, using different ground accelerations. The results of the analysis highlight that, for the considered area and return period, the maximum rotations of the system are significant only for large slenderness values. Further, the investigation shows that the diaphragm plays a crucial role in the dynamic response of the system. The stiffness of the diaphragm can significantly reduce the rotations and consequently the risk of overturning. Additionally, the study on the effect of the wall size pointed out how a top spring causes a reverse scale effect. [ABSTRACT FROM AUTHOR]

A driven‐dissipative system is considered, consisting of an atomic Bose‐Einstein condensates loaded into a 2D Hubbard lattice and coupled to a single mode of an optical cavity. Due to the interplay between strong, repulsive atomic interaction and the atom‐cavity coupling, the system exhibits several phases of atoms and photons including the atomic superfluid (SF) and supersolid (SS). The dynamical behavior of the system, where dissipation is included by means of Lindblad master equation formalism. Due to the discontinuous nature of the Dicke transition for strong atomic repulsion, extended co‐existence region of different phases are found. The resulting switching dynamics are investigated, particularly between the coexisting SF and SS phases, which eventually becomes damped by the dissipation. [ABSTRACT FROM AUTHOR]

This paper identifies and explains particular differences and properties of adjoint-free iterative ensemble methods initially developed for parameter estimation in petroleum models. The aim is to demonstrate the methods' potential for sequential data assimilation in coupled and multiscale unstable dynamical systems. For this study, we have introduced a new nonlinear and coupled multiscale model based on two Kuramoto–Sivashinsky equations operating on different scales where a coupling term relaxes the two model variables toward each other. This model provides a convenient testbed for studying data assimilation in highly nonlinear and coupled multiscale systems. We show that the model coupling leads to cross covariance between the two models' variables, allowing for a combined update of both models. The measurements of one model's variable will also influence the other and contribute to a more consistent estimate. Second, the new model allows us to examine the properties of iterative ensemble smoothers and assimilation updates over finite-length assimilation windows. We discuss the impact of varying the assimilation windows' length relative to the model's predictability time scale. Furthermore, we show that iterative ensemble smoothers significantly improve the solution's accuracy compared to the standard ensemble Kalman filter update. Results and discussion provide an enhanced understanding of the ensemble methods' potential implementation and use in operational weather- and climate-prediction systems. [ABSTRACT FROM AUTHOR]

High-speed and high-precision rotating machinery is widely applied in industry, which promotes the rapid development of non-contact bearings represented by active magnetic bearings (AMBs). The rotor-AMBs system is usually installed on the base. Due to the base motion, it affects the dynamic characteristics of the rotor-AMBs system. In this paper, the dynamic analysis of the rotor-AMBs system with base motion is carried out. The dynamic model of the system with base motion is first derived, and the control model is then established. The influence of the inherent property of the base, the excitation mode of the base motion, and the control method for the rotor-AMBs system are analyzed with numerical simulation. In addition, the combination of feedforward control and PID control is employed to eliminate the vibration disturbance caused by the base motion and improve the suspension accuracy of the rotor. The experimental results show that the proposed hybrid control can effectively reduce the vibration of the rotor-AMBs system with base motion. [ABSTRACT FROM AUTHOR]

The authors have examined a Friedmann Robertson Walker cosmological model in Hořava‐Lifshitz gravity by using a dynamical system approach. A set of autonomous equations is derived and their solutions are calculated. The critical points from these equations and find the characteristics values with the analysis of the physical interpretation of the phase space for this system are assessed. Three stable critical points are found and the values of the physical parameters and the scale factor's expressions at each critical points are displayed in Tables 1, 2, and 3. A hybrid scale factor to develop the model, which results in a phase transition from deceleration to acceleration is used. The suitable values of the parameters are governed by applying the Monte Chain Monte Carlo method technique to the Hubble 46 and joint Hubble 46 and Baryon Acoustic Oscillations 15 datasets. In contrast to the negative behavior of pressure, the positive behavior of energy density and q=−1$q=-1$ illustrate the Universe's acceleration epoch and the ΛCDM$\Lambda{\rm CDM}$ model is represented by the EoS parameter ω=−1$\omega =-1$. The authors investigated that the energy conditions and their model violates the strong energy condition. Utilizing Om$Om$ diagnostic test, it is found that the model represents phantom behavior. The thermodynamical perspective for the model is also examined. The model accurately explained the Universe's propagation history and fits well with contemporary cosmic data. The authors have examined a Friedmann Robertson Walker cosmological model in Hoˇrava‐Lifshitz gravity by using a dynamical system approach. A set of autonomous equations is derived and their solutions are calculated. The critical points from these equations and find the characteristics values with the analysis of the physical interpretation of the phase space for this system are assessed. Three stable critical points are found and the values of the physical parameters and the scale factor's expressions at each critical points are displayed in Tables 1, 2, and 3. A hybrid scale factor to develop the model, which results in a phase transition from deceleration to acceleration is used. The suitable values of the parameters are governed by applying the Monte Chain Monte Carlo method technique to the Hubble 46 and joint Hubble 46 and Baryon Acoustic Oscillations 15 datasets. In contrast to the negative behavior of pressure, the positive behavior of energy density and q=−1$q=-1$ illustrate the Universe's acceleration epoch and the ΛCDM$\Lambda{\rm CDM}$ model is represented by the EoS parameter ω=−1$\omega =-1$. The authors investigated that the energy conditions and their model violates the strong energy condition. Utilizing Om$Om$ diagnostic test, it is found that the model represents phantom behavior. The thermodynamical perspective for the model is also examined. The model accurately explained the Universe's propagation history and fits well with contemporary cosmic data. [ABSTRACT FROM AUTHOR]

Recently, optimal time variable learning in deep neural networks was introduced in Antil et al. In this manuscript we extend the concept by introducing a regularization term that directly relates to the time horizon in discrete dynamical systems. Furthermore, we propose an adaptive pruning approach for Residual Neural Networks (ResNets), which reduces network complexity without compromising expressiveness, while simultaneously decreasing training time. The results are illustrated by applying the proposed concepts to classification tasks on the well known MNIST and Fashion MNIST data sets. Our PyTorch code is available on https://github.com/frederikkoehne/time_variable_learning. [ABSTRACT FROM AUTHOR]

Summary: The concerns of this article is to investigate the fixed time function combination synchronization (FCS) among three different chaotic systems with disturbances. First, a new stability theorem is presented by virtue of the integral inequality technique and a more accurate upper estimation of the convergence time is given. Next, the fixed time FCS problem is considered for the perturbed chaotic systems in which the scaling functions are generated by the bounded dynamical systems. Besides, the synchronization time is related to the control parameters rather than the initial values. The time upper bound can be calculated out by the simple formula, which is the function of control parameters. Finally, a simulation example is provided to illustrate the correctness of the derived criteria. [ABSTRACT FROM AUTHOR]

This article deals with an inverse problem of determining two space-time-dependent sources in an integerfractional mobile-immobile two-region solute transport system by additional Dirichlet-Neumann data. The unique existence of a solution to the forward problem is obtained by the method of Laplace transform, and a dynamical system connecting the known data with the unknown sources is established by variational method and boundary homogenization. The dynamical system is discretized to a linear system at a given time in a homogenous polynomial space, and the sources are reconstructed by alternative iterations and Tikhonov regularization. Numerical examples are presented to illustrate the validity of the inversion algorithm. [ABSTRACT FROM AUTHOR]

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k , then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ -almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T -invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space. [ABSTRACT FROM AUTHOR]

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 problem, and establish the existence and uniqueness of a solution to the proposed system. We prove that every trajectory of the proposed dynamical system converges to the unique solution of the inverse quasi-variational inequality problem and that the system is globally asymptotically stable at its equilibrium point. We also prove that if the function which governs the inverse quasi-variational inequality problem is strongly monotone and Lipschitz continuous, then the dynamical system is globally exponentially stable at its equilibrium point. We discretize the dynamical system and show that the sequence generated by the discretization of the system converges strongly to the unique solution of the inverse quasi-variational inequality problem under certain assumptions on the parameters involved. Finally, we provide numerical examples to support and illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Let $ (X, f) $ (X , f) be a dynamical system, i.e. X is a compact metric space and f is a continuous self-map on X and let $ K(X) $ K (X) , $ M(X) $ M (X) and $ \mathbb {F}(X) $ F (X) be the sets of all non-empty compact subsets of X, Borel probability measures on X and upper semi-continuous fuzzy sets on X, respectively. Then $ K(X) $ K (X) , $ M(X) $ M (X) and $ \mathbb {F}(X) $ F (X) are metric spaces under Hausdorff metric, prohorov metric and level-wise metric, respectively. Therefore, $ (X, f) $ (X , f) naturally induces three new systems $ (K(X), \bar {f}) $ (K (X) , f ¯) , $ (M(X), \hat {f}) $ (M (X) , f ^) and $ (\mathbb {F}(X), \tilde {f}) $ (F (X) , f ~). In this article, we investigate the connection of $ (r,s) $ (r , s) -sensitivity, $ (r,s) $ (r , s) -asymptotic sensitivity, $ (r,s) $ (r , s) -Li-Yorke sensitivity and Δ-transitivity of $ (X, f) $ (X , f) and its induced systems $ (K(X),\bar {f}) $ (K (X) , f ¯) , $ (M(X),\hat {f}) $ (M (X) , f ^) and $ (\mathbb {F}(X),\tilde {f}) $ (F (X) , f ~) and we obtain some desired results. For instance, we prove that $ (K(X),\bar {f}) $ (K (X) , f ¯) is $ (r,s) $ (r , s) -sensitive ⇔ $ (\mathbb {F}^{1}(X),\widetilde {f_{g}}) $ (F 1 (X) , f g ~) is $ (r,s) $ (r , s) -sensitive for each $ g\in D_{m}(I) $ g ∈ D m (I) satisfying $ g^{-1}(1)=\{1\} $ g − 1 (1) = { 1 } ; $ (X,f) $ (X , f) is Δ-transitive ⇔ $ (K(X),\bar {f}) $ (K (X) , f ¯) is Δ-transitive ⇔ $ (M(X),\hat {f}) $ (M (X) , f ^) is Δ-transitive $ \Leftrightarrow \,(\mathbb {F}^{1}(X),\tilde {f}) $ ⇔ (F 1 (X) , f ~) is Δ-transitive. [ABSTRACT FROM AUTHOR]

This study aims to solve the problem of small key space in image cryptosystems based on logistic mapping. First, a new one-dimensional (1D) chaotic system, with a wide continuous chaotic interval, a large Lyapunov exponent and obvious chaotic characteristics are presented. Subsequently, a novel image encryption algorithm based on the new 1D chaotic system and dynamic DNA encoding is designed. Compared with other DNA coding methods, the proposed image encryption algorithm encodes chaotic sequences and ensures that the sequence elements at different positions correspond to different DNA coding schemes. This will help to overcome the fixity of DNA coding and make the proposed dynamic DNA coding easy to operate and implement. Finally, the cipher image is obtained by scrambling and bit XOR operation based on the chaotic sequences. The fixed DNA coding method is compared with other chaotic image encryption schemes, and the experimental results indicate that the image encryption algorithm has higher security and can resist common attacks. [ABSTRACT FROM AUTHOR]

Summary: This paper addresses a trajectory tracking control problem for marine surface vessels (MSVs) in the presence of mismatched conditions and input saturation. Firstly, an MSV model under mismatched conditions is established to solve the problem of unreasonable modeling methods in the existing literature. Secondly, fixed‐time disturbance observers (FXDOs) are designed, where the designed FXDOs not only avoid selecting complicated parameters but also has simple structures, and the disturbance estimation errors can converge to the origin within a setting time. Thirdly, based on the FXDOs and an anti–saturation auxiliary dynamic system, a trajectory tracking control scheme in the presence of mismatched conditions and input saturation is constructed by virtue of fixed‐time integral sliding mode control (SMC), and the actual tracking error signals are uniformly fixed‐time bounded rather than finite‐time one. Fourthly, in order to decrease theoretical calculation values of the original setting time in fixed‐time control, the new setting time under three different parameter conditions is derived, and thus reducing the conservatism. Finally, the simulation results are provided to show the effectiveness and superiority of the designed tracking control scheme. [ABSTRACT FROM AUTHOR]

In this study, the asymptotical stabilization problem of fuzzy fractional dynamic systems (FFDSs) with the semilinear form under the granular Caputo fractional derivative for the case q ∈ (1 , 2) is investigated. To tackle this, we propose a linear feedback controller aimed at stabilizing the unstable states of FFDSs. Taking advantage of the generalized fractional Laplace-like transform (GFLT) and the Gronwall-Bellman inequality, we provide a simple method to evaluate the stability of fractional dynamical systems. Finally, we validate the effectiveness of our approach through examples and corresponding simulations. [ABSTRACT FROM AUTHOR]

This study examines the reachability of a time-varying fractional dynamical system with Riemann-Liouville fractional derivative. The state transition matrix is used to solve the time-varying systems. Using the reachability Grammian matrix, the reachability linear time-varying fractional dynamical system is discussed. The existence and uniqueness of a solution of a nonlinear time-varying fractional dynamical system is established, and sufficient conditions for the reachability of nonlinear time-varying fractional dynamical systems are obtained with the help of Banach fixed point theorem. The reachability results are proved for a time-varying integro-fractional dynamical system for a particular case. A successive approximation method is proposed to give numerical solutions to the reachability problems. [ABSTRACT FROM AUTHOR]

We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analyzed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n -torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered. [ABSTRACT FROM AUTHOR]

Dynamical systems subject to perturbations that decay over time are relevant in describing many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, and celestial mechanics. For this purpose, we consider time-dependent Hamiltonian vector fields that are the sum of two components. The first has an invariant torus supporting quasiperiodic solutions, and the second decays as time tends to infinity. The time decay is modelled by functions satisfying suitable conditions verified by a proper polynomial decay in time. We prove the existence of orbits converging as time tends to infinity to the quasiperiodic solutions associated with the unperturbed system. The proof of this result relies on a new strategy based on a refined analysis of the Banach spaces and the functionals involved in the resolution of suitable nonlinear invariant equations. This result is proved for finite differentiable and real-analytic Hamiltonians. Analogous statements for time-dependent vector fields on the torus are also obtained as a corollary. These results extend a previous work of Canadell and de la Llave, where only exponential decay in time is considered. The relaxation of the decay in time makes the results in the present paper suited for applications in many physical problems, such as celestial dynamics. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear (4 + 1)-dimensional Fokas model is studied and its dynamic character is predicted. This inquiry aims to accomplish two main objectives. First, the Lie symmetries are created and the associated transformation is applied to reduce the model to ordinary differential equations. Plots are used to demonstrate and establish invariant solutions. Second, several methods such as bifurcation, sensitivity analysis, and chaos are used to investigate the dynamic behavior of the model. Bifurcation theories are used to analyse a dynamical system's bifurcation at equilibrium points. The nonperturbed system introduces the perturbation period, which deviates from regular patterns. Different initial conditions are used to investigate the sensitivity analysis the model is found to be highly sensitive. The findings are unique, stimulating and mathematically helpful to comprehend the model. It is essential to comprehend the systems and processes that behave dynamically to investigate new technology, suggest interventions and forecast the future. [ABSTRACT FROM AUTHOR]

This study covers the traveling wave analysis of the two-mode Cahn–Allen (TMCA) equation. The TMCA model plays the role of transmitting information into two different locations and preserving its physical properties. A wave transformation converts the considered model into an ordinary differential equation (ODE). By applying the two analytical techniques, the extended tanh - coth technique and the extended G ′ G 2 -expansion scheme, we have obtained a new type of the wave structures like singular, kink and periodic. To present the significance of the acquired outcomes, we have shown the graphical behaviors of the results by taking some suitable values of the involved parameters. The acquired ODE can be transformed into a dynamical system by applying the Galilean transformation. To examine the sensitive behavior, the dynamical system is expressed in a non-autonomous form, and the effects are analyzed by considering various initial conditions. The considered model admits the two-dimensional Lie algebra. The nonlinear self-adjointness of the considered model is checked using a new conservation theorem. The results show that the considered model is not self-adjoint and is made self-adjoint by computing a new dependent variable. Finally, the conserved quantities corresponding to each symmetry generator are computed. [ABSTRACT FROM AUTHOR]

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least C 2 , we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class r = 2 , ... , ∞ , ω . The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders. [ABSTRACT FROM AUTHOR]

Suppose (f,\mathcal {X},\mu) is a measure preserving dynamical system and \phi \colon \mathcal {X}\to \mathbb {R} a measurable function. Consider the maximum process M_n≔\max \{X_1,\ldots,X_n\}, where X_i=\phi \circ f^{i-1} is a time series of observations on the system. Suppose that (u_n) is a non-decreasing sequence of real numbers, such that \mu (X_1>u_n)\to 0. For certain dynamical systems, we obtain a zero–one measure dichotomy for \mu (M_n\leq u_n\,\text {i.o.}) depending on the sequence u_n. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences u_n. Our results on the permitted sequences u_n are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory. [ABSTRACT FROM AUTHOR]

Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active and refractory neurons in the network's populations. We do so by describing the evolution of the states of individual neurons with a continuous-time Markov chain, from which we formally derive a low-dimensional dynamical system. This is done by solving a moment closure problem in a way that is compatible with the nonlinearity and boundedness of the activation function. Our dynamical system captures the behavior of the high-dimensional stochastic model even in cases where the mean-field approximation fails to do so. Taking into account the second-order moments modifies the solutions that would be obtained with the mean-field approximation and can lead to the appearance or disappearance of fixed points and limit cycles. We moreover perform numerical experiments where the mean-field approximation leads to periodically oscillating solutions, while the solutions of the second-order model can be interpreted as an average taken over many realizations of the stochastic model. Altogether, our results highlight the importance of including higher moments when studying stochastic networks and deepen our understanding of correlated neuronal activity. [ABSTRACT FROM AUTHOR]

This article investigates the impact of three harmonically external moments on the motion of a four-degrees-of-freedom (DOF) nonlinear dynamical system composed of a double rigid pendulum connected to a nonlinear spring with linear damping. In light of the system's generalized coordinates, the governing system (GS) of motion is derived using Lagrange's equations. With the use of the multiple-scales method (MSM), the approximate solutions (AS) of the equations of this system are obtained up to a higher order of approximation, maybe the third order. Within the framework of the absence of secular terms, the conditions of solvability are obtained. Accordingly, the different resonance cases are categorized, and three of them are investigated simultaneously. Thus, these conditions are updated in preparation for achieving the modulation equations (ME) for the examined system. The numerical solutions (NS) of the GS are achieved using the algorithms of fourth-order Runge–Kutta (4RK), which are compared with the AS, which displays their high degree of consistency and demonstrates the precision of the MSM. The motion's time history, steady-state solutions, and resonance curves are graphed to demonstrate the influence of various system physical parameters. All relevant fixed points at steady-state situations are identified and graphed in accordance with the Routh–Hurwitz criteria (RHC). Therefore, the zones of stability/instability of are checked and analyzed. Numerous real-world applications in disciplines like engineering and physics attest to the significance of the nonlinear dynamical system under study such as in shipbuilding, automotive devices, structure vibration, developing robots, and analysis of human walking. [ABSTRACT FROM AUTHOR]

This article studies the motion of a three degrees-of-freedom (DOF) dynamical system, which consists of two parts, a dynamic part, and an electromagnetic harvesting device. The composition of the system depends on the magnet oscillating in the coil of this device. An electric potential is exerted because of the electromotive force to obtain energy harvesting (EH). Lagrange's equations (LE) are used to derive the system's equations of motion (EOM), which are solved analytically up to a third-order of approximation using the multiple-scales method (MSM). These solutions are considered novel, in which the mentioned method is applied to a novel dynamical system. The gained are confirmed through comparison with the numerical solutions of the EOM to reveal a high degree of consistency. To establish the system's solvable conditions and modulation equations (ME), three scenarios of primary external resonance are investigated simultaneously. Certain diagrams of time histories for the amplitudes and phases of the examined dynamical system are plotted to show their behaviors at any given instance of time. The Routh–Hurwitz criteria (RHC) are used to establish the stability and instability regions in light of the graphs of resonance response curves. The presence of an energy harvester in dynamic devices helps us to convert vibrational motion into electrical energy that can be exploited in several applications in a variety of daily tasks, including structural and environmental monitoring, military applications, space exploration, and remote medical sensing. [ABSTRACT FROM AUTHOR]