Arbitrary Growth Rates Research Articles

Consensus of networked nonlinear systems based on the distributed stochastic approximation algorithm

Consensus of networked nonlinear nonparametric time-varying systems is studied, and the nonlinearity at input may possess arbitrary growth rate. In addition, the communication noise is also under consideration. The purely distributed control algorithm is designed on the basis of distributed stochastic approximation algorithm with expanding truncations (DSAAWET). The truncation mechanism neutralizes the divergent tendency caused by unknown nonlinearities and noises, which makes the algorithm well tackle more general nonlinearities with high growth rate and more complicated structure noises. We first prove the average consensus over fixed digraph, and then extend the results to averagely connected time-varying digraphs. Finally, the validity of the algorithm is justified by numerical simulations.

Mean attractors of stochastic delay lattice [formula omitted]-Laplacian equations driven by superlinear noise in high-order product Bochner spaces

This paper is concerned with the existence and uniqueness of mean random attractors of a class non-autonomous stochastic delay p-Laplacian lattice systems defined on a high-dimensional integer set Zd driven by a family infinite-dimensional superlinear noise. We first establish the global-in-time existence and uniqueness of the solutions in C([τ,∞),L2k(Ω,ℓ2(Zd)))∩Lq(Ω,Llocq((τ,∞),ℓq(Zd))) for any k⩾1 when the draft term has an arbitrary polynomial growth rate q>2 and the coefficient of the noise admits a superlinear growth order q̃∈[2,q). We then show that the mean random dynamical system generated by the solution operators has a unique weakly compact and weakly attracting mean random attractor in the high-order product Bochner space L2k(Ω,F;ℓ2(Zd))×L2k(Ω,F;L2k((−ϱ,0),ℓ2(Zd))), where ϱ is the time delay parameter. The dissipative property of the draft term is employed to carefully controlling the superlinear growth diffusion term. When k=1, our results are new even in the product Hilbert space L2(Ω,F;ℓ2(Zd))×L2(Ω,F;L2((−ϱ,0),ℓ2(Zd))). This work can be regard as a further study of mean attractors of stochastic p-Laplacian lattice systems in the works of Wang and Wang (2020) and Chen et al. (2023).

Tracking Control for Nonlinear Nonparametric Systems Based on the Stochastic Approximation Algorithm

In this article, we study the tracking control problem for nonlinear nonparametric systems, and additive random observation noise is also taken into account. The dynamical function is allowed to be time varying and possess an arbitrary growth rate at control input. The control algorithm is designed on the basis of the stochastic approximation algorithm with expanding truncations, and it is found that there is a tradeoff between the growth rate of the dynamical function at control input and that of the truncation bound sequence to be chosen. We prove that the average tracking or strong tracking is asymptotically achieved for a class of reference state sequences, which can be strongly averaged. Finally, numerical simulations given in this article justify the theoretical assertions.

Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

On the theory of crystal growth in metastable systems with biomedical applications: protein and insulin crystallization.

A generalized theory of nucleation and growth of crystals in a metastable (supercooled or supersaturated) liquid is developed taking into account two principal effects: the diffusion mechanism of the particle-size distribution function in the space of particle radii and the unsteady-state growth rates of individual crystals induced by fluctuations in external temperature or concentration field. A system of the Fokker-Planck and balance integro-differential equations is formulated and analytically solved in a parametric form for arbitrary nucleation kinetics and arbitrary growth rates of individual crystals. The particle-size distribution function and system metastability are found in an explicit form. The Weber-Volmer-Frenkel-Zel'dovich and Meirs kinetic mechanisms, as well as the unsteady-state growth rates of nuclei (Alexandrov & Alexandrova 2019 Phil. Trans. R. Soc. A 377, 20180209 ( doi:10.1098/rsta.2018.0209 )), are considered as special cases. Some potential biomedical applications of the present theory for crystal growth from supersaturated solutions are discussed. The theory is compared with experimental data on protein and insulin crystallization (growth dynamics of the proteins lysozyme and canavalin as well as of bovine and porcine insulin is considered). The hat-shaped particle-size distribution functions for lysozyme and canavalin crystals as well as for bovine and porcine insulin are found. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Analytic form of the size distribution in irreversible growth of nanoparticles.

We study theoretically the size distributions of nanoparticles (surface islands, droplets, molecular chains, and semiconductor nanowires) which grow without decay and with arbitrary size and time-dependent growth rates. Using a special transformation of variables, the analytic Green's function is obtained in the form of a Gaussian the variance of which is determined by the size dependence of the growth rate k(s). In the case of the power-law growth rates k(s)=(a+s)^{α}, the explicit formulas for the expectation and variance are given that contain earlier results in the limiting regimes. In the case of heterogeneous nucleation in a closed system, by convoluting Green's function with the exponential nucleation rate, we find an analytic size distribution which takes into account a delay in forming the smallest dimer and shows how it affects the distribution shapes. The recently discovered sub-Poissonian narrowing of the size distribution by nucleation antibunching is also included in the treatment. We briefly consider the length distribution of vapor-liquid-solid nanowires in the context of the obtained results. Overall, simple analytic size distributions obtained here under rather general assumptions may be useful for understanding and modeling statistical properties of different growth systems.

One-sided dichotomies versus two-sided dichotomies: arbitrary growth rates

We obtain necessary and sufficient conditions for the existence of two-sided nonuniform exponential dichotomies with arbitrary growth rates in terms of the existence of one-sided nonuniform exponential dichotomies on the past and on the future. We consider both linear nonautonomous dynamics with discrete and continuous time, on an arbitrary Banach space.

Strong nonuniform spectrum for arbitrary growth rates

We consider the notion of strong nonuniform spectrum for a nonautonomous dynamics with discrete time obtained from a sequence of matrices, which is defined in terms of the existence of strong nonuniform exponential dichotomies with an arbitrarily small nonuniform part. The latter exponential dichotomies are ubiquitous in the context of ergodic theory and correspond to have both lower and upper bounds along the stable and unstable directions, besides possibly a nonuniform conditional stability although with an arbitrarily small exponential dependence on the initial time. Moreover, we consider arbitrary growth rates instead of only the usual exponential rates. We give a complete characterization of the possible strong nonuniform spectra and for a Lyapunov regular trajectory, we show that the spectrum is the set of Lyapunov exponents. In addition, we provide explicit examples of nonautonomous dynamics for all possible strong nonuniform spectra. A remarkable consequence of our results is that for a sequence of matrices [Formula: see text], either [Formula: see text] does not admit a strong exponential dichotomy for any [Formula: see text], or if [Formula: see text] admits an exponential dichotomy for some [Formula: see text], then it also admits a strong exponential dichotomy for that [Formula: see text]. We emphasize that this result is not in the literature even in the special case of uniform exponential dichotomies.

Local discrepancies in the problem of fractional parts distribution of a linear function

In this paper we consider a problem of distribution of fractional parts of the sequence obtained as multiples of some irrational number with bounded partial quotients of its continued fraction expansion. Local discrepancies are the remainder terms of asymptotic formulas for the number of points of the sequence lying in given intervals. Earlier only intervals with bounded and logarithmic local discrepancies were known. We prove that there exists an infinite set of intervals with arbitrary small growth rate of local discrepancies. The proof is based on the connection of considered problem with some of those from Diophantine approximations.

NON‐UNIFORM TRICHOTOMIES AND ARBITRARY GROWTH RATES

For a non-autonomous dynamics defined by a sequence of matrices, we consider the notion of a non-uniform exponential trichotomy for an arbitrary growth rate (this means that there may exist contracting, expanding and neutral directions with an arbitrary fixed growth rate). The purpose of our work is two-fold: to use a regularity coefficient in order to show that these trichotomies occur naturally and to provide several alternative characterizations of those for which the non-uniform part is arbitrarily small. This includes characterizations in terms of the growth rate of volumes and of the Lyapunov exponents of the dynamics and its adjoint. We also obtain sharp lower and upper bounds for the regularity coefficient.

Nonautonomous Dynamics with Discrete Time and Topological Equivalence

For evolution families with discrete time, we show that any exponential dichotomy is topologically equivalent to a certain normal form, in which the exponential behavior along the stable and unstable directions are multiples of the identity. We consider the general case of a generalized exponential dichotomy in which the usual exponential behavior is replaced by an arbitrary growth rate. In addition, we show that the topological equivalence between two evolution families with generalized exponential dichotomies can be completely characterized in terms of a notion of equivalence between the growth rates.

Rotational stabilization of the resistive wall modes in tokamaks with a ferritic wall

The dynamics of the rotating resistive wall modes (RWMs) is analyzed in the presence of a uniform ferromagnetic resistive wall with μ̂≡μ/μ0≤4 (μ is the wall magnetic permeability, and μ0 is the vacuum one). This mimics a possible arrangement in ITER with ferromagnetic steel in test blanket modules or in future experiments in JT-60SA tokamak [Y. Kamada, P. Barabaschi, S. Ishida, the JT-60SA Team, and JT-60SA Research Plan Contributors, Nucl. Fusion 53, 104010 (2013)]. The earlier studies predict that such a wall must provide a destabilizing influence on the plasma by reducing the beta limit and increasing the growth rates, compared to the reference case with μ̂=1. This is true for the locked modes, but the presented results show that the mode rotation changes the tendency to the opposite. At μ̂>1, the rotational stabilization related to the energy sink in the wall becomes even stronger than at μ̂=1, and this “external” effect develops at lower rotation frequency, estimated as several kHz at realistic conditions. The study is based on the cylindrical dispersion relation valid for arbitrary growth rates and frequencies. This relation is solved numerically, and the solutions are compared with analytical dependences obtained for slow (s/dw≫1) and fast (s/dw≪1) “ferromagnetic” rotating RWMs, where s is the skin depth and dw is the wall thickness. It is found that the standard thin-wall modeling becomes progressively less reliable at larger μ̂, and the wall should be treated as magnetically thick. The analysis is performed assuming only a linear plasma response to external perturbations without constraints on the plasma current and pressure profiles.

An exact analytical solution of the Fokker-Planck type equation in the presence of arbitrary growth rates of nuclei

An exact analytical solution of the Fokker-Planck type equation in the presence of arbitrary growth rates of nuclei

On qualitative properties of solutions of quasilinear elliptic equations with strong dependence on the gradient

We are interested in the regularity of positive, spherically symmetric solutions of a class of quasilinear elliptic equations involving the p-Laplace operator, with an arbitrary positive growth rate e0 on the gradient on the right-hand side.We study the regularity of a class of strong and weak solutions at the origin.Furthemore, we find some conditions under which strong solutions are classical

Nucleation and growth of crystals at the intermediate stage of phase transformations in binary melts

An exact analytical solution of integro-differential equations, which describe the phase transitions in supercooled binary melts, is constructed with allowance for fluctuating rates of crystal growth. A complete solution of the corresponding Fokker–Planck and balance equations is found for arbitrary nucleation kinetics and growth rates of crystals. In limiting cases, the obtained solutions transform to earlier known solutions for a special form of diffusivity in the space of crystal sizes and purely kinetic and “diffusion-kinetic” growth rates of nuclei.

On the theory of transient nucleation at the intermediate stage of phase transitions

Abstract The evolution of a system of growing aggregates in a macroscopically homogeneous medium with account of both the reduction in metastability and the continuing initiation of new nuclei is studied. The corresponding integro-differential model describing the intermediate stage of phase transitions is solved analytically for arbitrary nucleation kinetics and growth rates of nuclei. An exact solution of the Fokker–Planck equation is found with allowance for the diffusivity along the axis of nucleus radii. In limiting cases of purely kinetic and mixed kinetic-diffusion rates of crystal growth for a special form of diffusivity, the obtained solutions transform to earlier known expressions.

Quasilinear elliptic equations with positive exponent on the gradient

We study the existence and nonexistence of positive, spherically symmetric solutions of a quasilinear elliptic equation (1.1) involving p-Laplace operator, with an arbitrary positive growth rate $e_0$ on the gradient on the right-hand side. We show that $e_0 = p − 1$ is the critical exponent: for $e_0 p − 1$ we have existence-nonexistence splitting of the coefficients $\tilde f_0$ and $\tilde g_0$. The elliptic problem is studied by relating it to the corresponding singular ODE of the first order. We give sufficient conditions for a strong radial solution to be the weak solution. We also examined when -solutions of (1.1), defined in Definition 2.3, are weak solutions. We found conditions under which strong solutions are weak solutions in the critical case of $e_0 = p − 1$.

Nonautonomous equations with arbitrary growth rates: A Perron-type theorem

For a linear equation x′=A(t)x, we show that the asymptotic behavior of its solutions is reproduced by the solutions of the nonlinear equation x′=A(t)x+f(t,x) for any sufficiently small perturbation f. More precisely, we show that if the Lyapunov exponents of the linear equation are limits, even for general exponential rates ecρ(t) for an arbitrary function ρ, then the same happens with the Lyapunov exponents of the solutions of the nonlinear equations, without introducing new values. Our approach is based on Lyapunov’s theory of regularity.

Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.

Conjugacies between general contractions

The main aim of this note is to construct topological conjugacies between linear nonautonomous contractions with discrete time. As a consequence, we obtain topological conjugacies between linear and nonlinear nonautonomous contractions. We consider the general case of linear contractions with respect to arbitrary growth rates e-cρ(m). This includes the usual exponential contractions with ρ(m)=m as a very special case. We also consider the case of contractions that are not uniformly stable. In addition, we show that all conjugacies are locally Hölder outside the origin, and locally Hölder everywhere for a large class of growth rates ρ(m).