Ergodic Properties of Weak Asymptotic Pseudotrajectories for Semiflows

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow and are interested in the limiting behavior of the empirical measures of X. The main results are as follows: (1) the weak* limit points of the empirical measures for X axe almost surely -invariant measures; (2) given any semiflow , there exists a weak asymptotic pseudotrajectory X of such that the set of weak* limit points of its empirical measures almost surely equal the set of all ergodic measures for ; and (3) if X is an asymptotic pseudotrajectory for a semiflow , then conditions on that ensure convergence of the empirical measures are derived.     

Hausdorff Dimension of Invariant Sets for Random Dynamical Systems

Suppose X() is a compact random set, invariant with respect to a continuously differentiable random dynamical system (RDS) on a separable Hilbert space. It is shown that the Hausdorff dimension dim _H (X()) is an invariant random variable, and it is bounded by d, provided the RDS contracts d-dimensional volumes exponentially fast. Both exponential decrease of d-volumes as well as the approximation of the RDS by its linearization are assumed to hold uniformly in . The results are applied to reaction diffusion equations with additive noise and to two-dimensional Navier–Stokes equations with bounded real noise.     

Asymptotic Properties of Global Solutions of Systems of Functional Differential Equations of Neutral Type with Nonlinear Deviations of an Argument

For a system of functional differential equations of the neutral type with nonlinear deviations of an argument dependent on an unknown function, we establish sufficient conditions for the existence of a solution continuously differentiable and bounded for t and study its properties.     

Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions

We show that any global nonnegative and bounded solution to the degenerate parabolic problemu_t-u^m+f(u)=0 qquad {\rm on} quad R^N,u|{}=0converges to a single stationary state as time goes to infinity. Here m>0, f is a restriction of a real analytic function defined on a sector containing the half-line [0, ), and f(u ^1/m ) is a continuously differentiable function of u.     

The Two-Dimensional Attractor of a Differential Equation with State-Dependent Delay

The delay differential equation

Convergence to equilibrium for delay-diffusion equations with small delay

It is shown that for scalar dissipative delay-diffusion equationsu _t–u=f(u(t),u(t–)) with a small delay, all solutions are asymptotic to the set of equilibria ast tends to infinity.     

Second Order Multiple Scales for Oscillators with Large Delay

We use the method of multiple scales (MMS) to study small perturbations, governed by a parameter , of a harmonic oscillator by a small term with a large delay. These systems differ significantly from others where small terms have delays; or an term has delay in a system near a Hopf bifurcation. Here, the slow flow in time t depends strongly on even at lowest order, and itself has an delay. The MMS has already been applied elsewhere for such systems, but only to first order and with attention restricted to periodic and quasiperiodic solutions. Here, we address transients as well as proceed to second order. The second order analysis holds unless a special resonance occurs (we assume it does not). Several numerical examples are presented. In each case, the slow flows are infinite-dimensional, show strong -dependence, require significantly less computation time than the full solutions, yet agree well with the same.     

Geometric Analysis of One Singular Cauchy Problem

We consider the Cauchy problem , x(0) = 0, where a ₀₀₀ = 0, a ₀₀₁ = 0, and a ₀₀₂ = 0, and prove the existence of continuously differentiable solutions x(0,] with required asymptotic properties.     

Symmetry groups of attractors

For maps equivariant under the action of a finite group on ⁿ, the possible symmetries of fixed points are known and correspond to the isotropy subgroups. This paper investigates the possible symmetries of arbitrary, possibly chaotic, attractors and finds that the necessary conditions of Melbourne, Dellnitz & Golubitsky [15] are sufficient, at least for continuous maps.This result shows that the reflection hyperplanes are important in determining those groups which are admissible; more precisely, a subgroup of is admissible as the symmetry group of an attractor if there exists a with / cyclic such that fixes a connected component of the complement of the set of reflection hyperplanes of reflections in but not in . For finite reflection groups this condition on reduces to the condition that is an isotropy subgroup. Our results are illustrated for finite subgroups of O(3).     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

A regularity theorem for minimizers of quasiconvex integrals

Summary We prove C ^1, partial regularity for minimizers of functionals with quasiconvex integrand f(x, u, Du) depending on vector-valued functions u. The integrand is required to be twice continuously differentiable in Du, and no assumption on the growth of the derivatives of f is made: a polynomial growth is required only on f itself.     

Square and Pulse Waves with Two Delays

This work is concerned with the effects on the dynamics of a differential difference equation with two delays as the delays become unbounded in a fixed direction. This leads to a singularly perturbed delay differential equation with singular parameter and delays (1, d). We study in detail d=2 for the case when =0 yields the Hénon map. In a neighborhood of a generic period doubling point for the Hénon map, we show that there can be either a stable square wave or an unstable pulse wave even though the period two point for the map is always stable.     

Existence and regularity for solutions of the Korteweg-de Vries equation

The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC³(), that U is piecewise of class C ⁴, and that U ^(j) decays at an algebraic rate for j4. The faster the decay of U ^(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than ¦x¦^–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u ₁ of the Korteweg-de Vries equation such that u ₁(·, 0)C() may fail to remain in class C for all time if u ₁(x, 0) does not decay fast enough as ¦x¦.This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Topological entropy of a dynamical system on the space of one-dimensional maps

We study properties of the topological entropy of the map F: f , C(I), generated by a fixed continuous map f C(I) of an interval of the straight line. In particular, we show that the topological entropy h(F) > 0 if and only if h(f) > 0.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 180–187, April–June, 2004.     

On the Structure of the Set of Solutions for One Class of Systems of Nonlinear Differential-Functional Equations of Neutral Type

For one class of nonlinear differential-functional equations, we study the structure of the set of its solutions continuously differentiable for t R ⁺ = [0, +).     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.