Orbits Homoclinic to Exponentially Small Periodic Orbits for a Class of Reversible Systems. Application to Water Waves

In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields, the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues , . In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to periodic orbits of size less than μ^N for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based on analyticity of the vector field. One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying . (Accepted October 2, 1995)     

Intermittency as a codimension-three phenomenon

We analyze the interaction of three Hopf modes and show that locally a bifurcation gives rise to intermittency between three periodic solutions. This phenomenon can occur naturally in three-parameter families. Consider a vector fieldf with an equilibrium and suppose that the linearization off about this equilibrium has three rationally independent complex conjugate pairs of eigenvalues on the imaginary axis. As the parameters are varied, generically three branches of periodic solutions bifurcate from the steady-state solution. Using Birkhoff normal form, we can approximatef close to the bifurcation point by a vector field commuting with the symmetry group of the three-torus. The resulting system decouples into phase amplitude equations. The main part of the analysis concentrates on the amplitude equations in R³ that commute with an action ofZ ₂+Z ₂+Z ₂. Under certain conditions, there exists an asymptotically stable heteroclinic cycle. A similar example of such a phenomenon can be found in recent work by Guckenheimer and Holmes. The heteroclinic cycle connects three fixed points in the amplitude equations that correspond to three periodic orbits of the vector field in Birkhoff normal form. We can considerf as being an arbitrarily small perturbation of such a vector field. For this perturbation, the heteroclinic cycle disappears, but an invariant region where it was is still stable. Thus, we show that nearby solutions will still cycle around among the three periodic orbits.     

Dependence of Topological Conjugacies on Parameters

For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C ¹ outside the jumping times, we show that the invariant manifolds are also of class C ¹ outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.     

Hyperbolic evolution groups and dichotomic evolution families

Recently, Ben-Artzi and Gohberg [2] used the concept ofC ₀-semigroups in order to characterize the existence of dichotomies for nonautonomous differential equations on _n. A similar task was performed by Latushkin and Stepin [11] for dichotomies of linear skew-product flows. In this paper we will useC _o-semigroups to characterize existence of dichotomies for strongly continuous evolution families (U(t,s)) _t.s on Hilbert and Banach spaces. Under an exponential growth condition we show that the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.     

Well-Posedness of Surface Wave Equations Above a Viscoelastic Fluid

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson–Segalman type in a domain with a free surface. Managing more general constitutive laws is also briefly depicted. The 2D geometry is defined by a solid fixed bottom and an upper free boundary submitted to surface tension. The proof relies on a Lagrangian formulation. First we solve two intermediate problems through a fixed point using mainly (Allain in Appl Math Optim 16:37–50, 1987) for the Navier–Stokes part. Then we solve the whole Lagrangian problem on [0, T ₀] for T ₀ small enough through a contraction mapping. Since the Lagrangian solution is regular enough and the change of coordinates invertible, we can come back to an Eulerian one.     

Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise

A class of semilinear nonautonomous parabolic equations subjected to additive white noise is considered. The existence of a family of randomN-dimensional approximate inertial manifolds (AIMs) whose neighborhoods of thickness of order exp(- _N+1 ) attract exponentially in the mean all the trajectories is proved forN large enough. Here _N+1 is the (N+1)th eigenvalue of the corresponding linear problem, and and are positive constants. We also construct a sequence of AIMs which converges to the exact inertial manifold, when a spectral gap condition is satisfied. These results remain true for deterministic autonomous and nonautonomous cases.     

On a Crystalline Variational Problem, Part I:¶First Variation and Global L∞ Regularity

Let φ:ℝ ⁿ → [0,+∞[ be a given positively one-homogeneous convex function, and let ?_φ≔{φ≤ 1 }. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ?_φ (ℝ ⁿ ) of “smooth” boundaries in the relative geometry induced by the ambient Banach space (ℝ ⁿ , φ). It can be seen that, even when ?_φ is a polytope, ?_φ(ℝ ⁿ ) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂?_φ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂?_φ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary δE in the class ?_φ(ℝ ⁿ ), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on δE. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on δE. We define such a divergence to be the φ-mean curvature κ_φ of δE; the function κ_φ is expected to be the initial velocity of δE, whenever δE is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that κ_φ is bounded on δE and that its sublevel sets are characterized through a variational inequality.     

A Confocally Multicoated Elliptical Inclusion under Antiplane Shear: Some New Results

The problem of a confocally multicoated elliptical inclusion in an unbounded matrix subjected to an antiplane shear is studied. Making use of the complex potentials and conformal mapping techniques, we show that the multiple coatings can be analyzed through a recurrence procedure in the transformed domain, while remaining explicit in detail and transparent overall. Particularly, the effect of the multiple confocal coatings is mathematically represented by a (2×2) array alone, resulting from a serial multiplication of matrices of the same order. Further we prove the following proposition. If the displacement prescribed at the remote boundary of the matrix is a polynomial of degree j in the position coordinates x _i , the stresses at the innermost core are polynomials of degree j–1,j–3,..., in x _i . This result is universally true provided that all elliptical interfaces are confocal, while no regard is paid to the number of coatings, their constituent properties and area fractions. Explicit expressions for the stresses at the innermost core are obtained in simple, closed forms.     

Decay rates for viscoelastic plates with memory

We consider the viscoelastic plate equation and we prove that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel g satisfiesg(t) -c_0g(t)^1+1/p and g,g^1+1/p L¹() with p > 2, then the energy decays as 1/(1+t)^p.IM, Federal University of Rio de Janeiro, Supported by a grant of CNPq.     

Static Solutions of the Vlasov-Einstein System

Static solutions of the SO(3)-symmetric Vlasov-Einstein system are studied via a variational approach. For the constitutive relation of the Emden-Fowler type φ(E,F)≡E ^{σ+ 1} F ^k we prove the existence of such solutions of sufficiently small mass-energy, provided 0<σ < k+3/2. These solutions are local minimizers of the energy-Casimir functional, subjected to a variational barrier. Accepted July 16, 2000?Published online January 22, 2001     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

On a Crystalline Variational Problem, Part II:¶BV Regularity and Structure of Minimizers on Facets

For a nonsmooth positively one-homogeneous convex function φ:ℝ ⁿ → [0,+∞[, it is possible to introduce the class ?_φ (ℝ ⁿ ) of smooth boundaries with respect to φ, to define their φ-mean curvature κ_φ, and to prove that, for E∈?_φ (ℝ ⁿ ), κ_φ∈L ^∞(δE) [9]. Based on these results, we continue the analysis on the structure of δE and on the regularity properties of κ_φ. We prove that a facet F of δE is Lipschitz (up to negligible sets) and that κ_φ has bounded variation on F. Further properties of the jump set of κ_φ are inspected: in particular, in three space dimensions, we relate the sublevel sets of κ_φ on F to the geometry of the Wulff shape ?_φ≔{φ≤ 1 }. Accepted October 11, 2000?Published online 14 February, 2001     

Levitan Almost Periodic and Almost Automorphic Solutions of <Emphasis Type="Italic">V</Emphasis>-monotone Differential Equations

In the present article we consider a special class of equations

Generalized Homoclinic Solutions of a Coupled Schr?dinger System Under a Small Perturbation

This paper is devoted to study a coupled Schr?dinger system with a small perturbation $$\begin{array}{ll}u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R} \end{array}$$ where β is a constant and ε is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schr?dinger-KdV system.     

Weak Singularities of Planar Vector Fields Under Weak Perturbation

We characterize the elements of the set H _n of degree n homogeneous polynomial vector fields that are structurally stable with respect to perturbation in H _n , both on the plane and on the Poincaré sphere. We use this information to characterize elements of the set W _n of smooth vector fields on ² beginning with terms of order n at (0, 0) that are structurally stable in a neighborhood of (0, 0) under perturbation in W _n . We also determine the set of elements of H _n that are determining for topological equivalence at (0, 0), in the sense that the topological type of the singularity at (0, 0) is invariant under the addition of higher order terms.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

The index of lyapunov stable fixed points in two dimensions

In this paper, we prove that a stable isolated fixed point of an orientation preserving local homeomorphism onR ₂ has fixed point index 1. We also give a number of applications to differential equations. In particular, we deduce that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.     

Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

Ergodicity of the Infinite Dimensional Fractional Brownian Motion

In this short note we prove that an infinite dimensional fractional Brownian motion B _H of any Hurst parameter ${H \in (0, 1)}In this short note we prove that an infinite dimensional fractional Brownian motion B _H of any Hurst parameter H ? (0, 1){H \in (0, 1)} forms an ergodic metric dynamical system. For the proof we mainly use the fundamental theorems of Kolmogorov.