Narrow Results

An expansion of the periodic solution of the van der Pol oscillator for small values of the damping parameter $𝜖$ is shown via a frequency domain method and harmonic balancing technique. An approximation of the amplitude of the first harmonic and the frequency $\omega$ in terms of $𝜖$ is obtained by an analytical form instead of the conventional algorithmic approach used before with this methodology. Its decomposition under different harmonics $sink𝜗$ and $cosk𝜗$ where $k=3,5,7$ is written in terms of their leading coefficients in power series of $𝜖$ to make easy comparisons with other methods in the literature.

In this paper, the buckling of a simply supported stepped periodic column is studied using an analytical method. The column is composed of biperiodic cells of stepped Euler–Bernoulli continuous segments. The deflection solution in each cell can be expressed from the resolution of a fourth-order differential equation. After expressing the continuity conditions between each cell, it is possible to relate the solution of each cell with respect to its neighbors. The differential eigenvalue problem of the bi-periodic structure is converted into a linear difference eigenvalue problem associated to the coefficients of the expressed solution in each cell. A transcendental equation for the buckling load of the continuous biperiodic column is obtained from the resolution of the discrete linear difference eigenvalue problem. This transcendental equation is valid whatever the number $N$ of bi-periodic cells, with $N$ larger than 2. This general expression is corroborated with the buckling values obtained using a direct method for few cells ($N=2$ and $N=3$ for instance). The behavior of the stability limit for large $N$ values is also specifically studied. It is shown that the bi-periodic column asymptotically converges toward a homogenized Euler–Bernoulli column with equivalent stiffness calibrated from Reuss’s averaging method. More refined beam models are also derived using asymptotic arguments. The buckling load converges toward the one of a gradient beam model for sufficiently large number $N$ of cells, which can be equivalently derived from a second-order homogenized beam theory. The convergence of this second-order homogenized beam model toward the equivalent homogenized Euler–Bernoulli column (obtained from Reuss’s averaging method) is from below, as also reported for the exact solution of the biperiodic continuous column. A comparison is also carried out for large values of $N$ with a nonlocal Euler–Bernoulli model, which has the same order of accuracy as obtained from the gradient beam model (second-order homogenized beam model).

We study the two-dimensional Ginzburg–Landau functional in a domain with corners for exterior magnetic field strengths near the critical field where the transition from the superconducting to the normal state occurs. We discuss and clarify the definition of this field and obtain a complete asymptotic expansion for it in the large κ regime. Furthermore, we discuss nucleation of superconductivity at the boundary.

We analyze a general class of self-adjoint difference operators $H_{𝜀}=T_{𝜀}+V_{𝜀}$ on ${\ell }^2({(𝜀\mathbb{Z})}^d)$, where $V_{𝜀}$ is a multi-well potential and $𝜀$ is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30–35]).Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs.

The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian $H_{𝜀}$ is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance $d$ induced by $H_{𝜀}$, we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first $n$ eigenvalues of $H_{𝜀}$ converge to the first $n$ eigenvalues of the direct sum of harmonic oscillators on ${ℝ}^d$ located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of $H_{𝜀}$. These are obtained from eigenfunctions or quasimodes for the operator $H_{𝜀}$, acting on $L^2({ℝ}^d)$, via restriction to the lattice ${(𝜀\mathbb{Z})}^d$.

Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted ${\ell }^2$-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two “wells” (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrödinger operator in [22].

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p₁, p₂, p₃) by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ ℤ. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.

We study the asymptotic behavior of multiple Hurwitz zeta functions and generalized multiple gamma and sine functions as some period parameters tend to 0. We obtain the perfect asymptotic expansions and get some applications in refining a previous paper by the author.

We establish the asymptotic expansion of certain integrals of $\psi$ classes on moduli spaces of curves ${\overline{{ℳ}}}_{g,n}$, when either the $g$ or $n$ goes to infinity. Our main tools are cut-join type recursion formulae from the Witten–Kontsevich theorem, as well as asymptotics of solutions to the first Painlevé equation. We also raise a conjecture on large genus asymptotics for $n$-point functions of $\psi$ classes and partially verify the positivity of coefficients in generalized Mirzakhani’s formula of higher Weil–Petersson volumes.

Let $X$ be a compact connected strongly pseudoconvex Cauchy–Riemann (CR) manifold of real dimension $2n+1,n\ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$th Fourier component of the Szegö kernel function as $m\to \infty$, where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the $S^1$ action. We also obtain explicit formulas for the first three coefficients of the expansion.

The operator expansion of free Green function of Helmholtz equation for arbitrary N-dimensional space leads to asymptotic extension of three dimensions Grimus–Stockinger formula closely related to multipole expansion. Analytical examples inspired by neutrino oscillation and neutrino deficit problems are considered for relevant class of wave packets.

In a wide class of potentials, the exact asymptotic dependence on finite distance R from scattering center is established for outgoing differential flux. It is shown how this dependence is eliminated by integration over solid angle for total flux, unitarity relation, and optical theorem. Thus, their applicability domain extends naturally to the finite R.

Interfacial pattern formation in phase transition and crystal growth and material science is one of the most important subjects in the broad field of nonlinear science. This subject involves the concepts and issues, include the basic states, global stability, limiting-state selection, quantization conditions of eigenvalues, scaling law and free boundary problems of dynamic system far away from the equilibrium state.

This talk attempts to explore these issues through the two prototype problems: (1). dendritic growth from melt; (2). disc-like crystal growth from melt. These problems are highly challenging fundamental problems in condensed matter physics and material science, which have preoccupied many investigators from various areas of science, including applied mathematics for a long period of time.

We shall summarize the major results achieved in terms of a unified systematic asymptotic approach during the last decade. For the case of dendritic growth, these results described the wave-characteristics of interface evolution and led to the so-called interfacial wave (IFW) theory.

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of $100$ photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system.

A new model based on an asymptotic procedure for solving the spinor kinetic equations of carriers and phonons is proposed, which gives naturally the displaced Maxwellian at the leading order. The balance equations for the carrier numbers, total energy density and total momentum for the whole system constitute now a system of four equations for the carrier chemical potentials, the temperature of the system and the drift velocity. In the drift–diffusion approximation, the constitutive laws are derived and the Onsager relations recovered. Moreover, equations for the evolution of the spin densities are added, which account for a general dispersion relation. The treatment of spin-flip processes, derived from first principles, is new and leads to an explicit expression of the relaxation time ${\tau }_{sf}$ as a function of the temperature.

In this paper, we study the number of bifurcated limit cycles from near-Hamiltonian systems where the corresponding Hamiltonian system has a double homoclinic loop passing through a hyperbolic saddle surrounded by a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles of the following system

In this paper, we study the number of bifurcated limit cycles from some polynomial systems with a double homoclinic loop passing through a nilpotent saddle surrounded by a heteroclinic loop, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles in the following system:

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology of several engineering problems. This derivative provides the sensitivity of a problem when a small hole is created at each point of the domain under consideration. In the present work the Topological Derivative for Poisson's problem is calculated using two different approaches: the Domain Truncation Method and a new method based on Shape Sensitivity Analysis concepts. By comparing both approaches it will be shown that the novel approach, which we call Topological-Shape Sensitivity Method, leads to a simpler and more general methodology. To point out the general applicability of this new methodology, the most general set of boundary conditions for Poisson's problem, Dirichlet, Neumann (both homogeneous and nonhomogeneous) and Robin boundary conditions, is considered. Finally, a comparative analysis of these two methodologies will also show that the Topological-Shape Sensitivity Method has an additional advantage of being easily extended to other types of problems.

In this paper, we shall study systems governed by the Neumann problem of second-order elliptic equation with rapidly oscillating coefficients and with control and observations on the boundary. The multiscale asymptotic expansions of the solution for considering problem in the case without any constraints, and homogenized equation in the case with constraints will be given, their rigorous proofs will also be proposed.

The steady motion of a micropolar fluid through a wavy tube with the dimensions depending on a small parameter is studied. An asymptotic expansion is proposed and error estimates are proved by using a boundary layer method. We apply the method of partial asymptotic decomposition of domain and we prove that the solution of the partially decomposed problem represents a good approximation for the solution of the considered problem.

In this paper the flow in a thin tubular structure is considered. The velocity of the flow stands for a coefficient in the convection-diffusion equation set in the thin structure. An asymptotic expansion of solution is constructed. This expansion is used further for justification of an asymptotic domain decomposition strategy essentially reducing the memory and the time of the code. A numerical solution obtained by this strategy is compared to the numerical solution obtained by a direct FEM computation.

We propose kinetic models to describe dust particles in a rarefied atmosphere in order to model the beginning of a Loss of Vacuum Accident (LOVA) in the framework of safety studies in the International Thermonuclear Experimental Reactor (ITER). After having studied characteristic time and length scales at the beginning of a LOVA in ITER and underlined that these characteristic scales justify a kinetic approach, we firstly propose a kinetic model by supposing that the collisions between dust particles and gas molecules are inelastic and are given by a diffuse reflexion mechanism on the surface of dust particles. This collision mechanism allows us to take into account the macroscopic character of dust particles compared to gas molecules. This leads to establish new Boltzmann type kinetic operators that are non-classical. Then, by noting that the mass of a dust particle is huge compared to the mass of a gas molecule, we perform an asymptotic expansion to one of the dust–molecule kinetic operators with respect to the ratio of mass between a gas molecule and a dust particle. This allows us to obtain a dust–molecule kinetic operator of Vlasov type whose any numerical discretization is less expensive than any numerical discretization of the original Boltzmann type operator. At last, we perform numerical simulations with Monte–Carlo and Particle-In-Cell (PIC) methods which validate and justify the derivation of the Vlasov operator. Moreover, examples of 3D numerical simulations of a LOVA in ITER using these kinetic models are presented.