Zero‐Hopf bifurcation in the FitzHugh–Nagumo system

We characterize the values of the parameters for which a zero‐Hopf equilibrium point takes place at the singular points, namely, O (the origin), P₊, and P_? in the FitzHugh–Nagumo system. We find two two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at the origin is a zero‐Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero‐Hopf equilibrium point O. We prove that there exist three two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at P₊ and at P_? is a zero‐Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P₊ and P_?. Copyright © 2014 John Wiley & Sons, Ltd.     

On Liouville Transport Equation with Force Field in BV loc

Abstract

We prove the existence and uniqueness of renormalized solutions of the Liouville equation for n particles with an interaction potential in BV _loc except at the origin. This implies the existence and uniqueness of a a.e. flow solution of the associated ODE.     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Nontrivial periodic solutions for strong resonance Hamiltonian systems via a local linking theorem

In this paper, we first establish an existence result of critical points for a class of functionals defined on Hilbert spaces by using a local linking idea. Then as an application of the existence result, we obtain the existence of periodic solutions of strong resonance Hamiltonian systems which are asymptotically linear both at infinity and at origin.     

Dynamics of Quasi-parabolic One-Resonant Biholomorphisms

In this paper we study the dynamics of germs of quasi-parabolic one-resonant biholomorphisms of ? ⁿ⁺¹ fixing the origin, namely, those germs whose differential at the origin has one eigenvalue 1 and the others having a one-dimensional family of resonant relations. We define some invariants and give conditions which ensure the existence of attracting domains for such maps.     

Natural solutions of rational Stieltjes moment problems

In the strong or two-point Stieltjes moment problem, one has to find a positive measure on [0,∞) for which infinitely many moments are prescribed at the origin and at infinity. Here we consider a multipoint version in which the origin and the point at infinity are replaced by sequences of points that may or may not coincide. In the indeterminate case, two natural solutions μ₀ and μ_∞ exist that can be constructed by a limiting process of approximating quadrature formulas. The supports of these natural solutions are disjoint (with possible exception of the origin). The support points are accumulation points of sequences of zeros of even and odd indexed orthogonal rational functions. These functions are recursively computed and appear as denominators in approximants of continued fractions. They replace the orthogonal Laurent polynomials that appear in the two-point case. In this paper we consider the properties of these natural solutions and analyze the precise behavior of which zero sequences converge to which support points.     

Exponential asymptotics of bi-oscillatory integrals. Application to the problem of persistence of homoclinic connections for the (iω0)2iω1 resonance

We explain in this Note how to obtain an exponentially small equivalent of a bioscillatory integral when it involves solutions of a nonlinear differential equation. The method proposed in this Note enables us to study the problem of existence of homoclinic connections for vector fields admitting a (iω₀)²iω₁ resonance at the origin. This problem could not be solved by a direct application of the classical Melnikov method since the Melnikov function is given in this case by an exponentially small bi-oscillatory integral.     

Multiplicity of characteristics with Lagrangian boundary values on symmetric star-shaped hypersurfaces

In this paper, the multiplicity of Lagrangian orbits on C² smooth compact symmetric star-shaped hypersurfaces with respect to the origin in R2n is studied. These Lagrangian orbits begin from one Lagrangian subspace and end on another. An infinitely many existence result is proved via Z₂-index theory. This is a multiplicity result about the Arnold Chord Conjecture in some sense, and is a generalization of the problem about the multiplicity of Lagrangian orbits beginning from and ending on the same Lagrangian subspace which was considered in the authors' previous paper [F. Guo, C. Liu, Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal. 69 (4) (2008) 1425-1436].     

Analyticity of the resolvent for elastic waves in a perturbed isotropic half space

In this article the analytic and asymptotic properties of the resolvent for elastic waves in a three dimensional domain perturbed from the isotropic half space R ³₊ are studied. In this case, the asymptotic expansion of the resolvent at the origin has logarithmic terms. This property is totally different from the case of the whole 3‐dimensional space. The existence of the surface waves like the Rayleigh waves makes this difference. As an application of the asymptotic properties of the resolvent, the rate of the local energy decay estimates for the dynamical equations is obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nontrivial solutions for Neumann problems with an indefinite linear part

In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the Ambrosetti-Rabinowitz condition. Using an abstract existence theorem for C¹-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.     

Construction of wavelet sets with certain self-similarity properties

A measurable set Q ⊂ R ⁿ is a wavelet set for an expansive matrix A if F ⁻¹ _(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in R _n associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are, up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with the matrix B = A ^t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes.     

The non-linear Schrödinger equation with a periodic δ-interaction

We study the existence and stability of space-periodic standing waves for the space-periodic cubic nonlinear Schrödinger equation with a point defect determined by a space-periodic Dirac distributionat the origin. This equation admits a smooth curve of positive space-periodic solutions with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbationmethod and continuation argument, we prove that in the case of an attractive defect the standing wave solutions are stable in H _per ¹ ([?π, π]) with respect to perturbations which have the same space-periodic as the wave itself. In the case of a repulsive defect, the standing wave solutions are stable in the subspace of even functions of H _per ¹ ([?π, π]) and unstable in H _per ¹ ([?π, π]) with respect to perturbations which have the same space-periodic as the wave itself.     

Local normal forms for constrained systems on 2-manifolds

We give a complete classification in the smooth category of local phase portraits (near the origin) of generic constrained systems of the formA(x)x=f(x), wherex ²,A(x) is a 2×2 matrix-valued function,f is a vector field, the origin is an impasse point (detA(0)=0, and the existence and the uniquiness of solutions breakes down). In the last section we discuss some of the main difficulties to obtain a similar classification in higher dimensions.This work was supported by CNP _q , Brazil.     

On the dynamics of a strongly singular parabolic equation

In this paper we study the semiflow defined by a semilinear parabolic equation, in which both the diffusion and the reaction term present strong order of singularity at the origin. We justify the existence of a global branch of nontrivial equilibrium solutions for subcritical nonlinearities. The approach, based on the critical Caffarelli–Kohn–Nirenberg inequality, follows the arguments of  and .     

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u ₀ satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u ₀. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.     

A partial order in the knot table II

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.     

Derived sequences and reverse mathematics

One of the earliest applications of transfinite numbers is in the construction of derived sequences by Cantor [2]. In [6], the existence of derived sequences for countable closed sets is proved in ATR₀. This existence theorem is an intermediate step in a proof that a statement concerning topological comparability is equivalent to ATR₀. In actuality, the full strength of ATR₀ is used in proving the existence theorem. To show this, we will derive a statement known to be equivalent to ATR₀, using only RCA₀ and the assertion that every countable closed set has a derived sequence. We will use three of the subsystems of second order arithmetic defined by H. Friedman ([3], [4]), which can be roughly characterized by the strength of their set comprehension axioms. RCA₀ includes comprehension for Δ definable sets, ACA₀ includes comprehension for arithmetical sets, and ATR₀ appends to ACA₀ a comprehension scheme for sets defined by transfinite recursion on arithmetical formulas. MSC: 03F35, 54B99.     

Gaussian fluctuations of connectivities in the subcritical regime of percolation

Summary We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp _c : (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL ^–(d–1)/2 . (2) The correlation length, (p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x ₁ =qL,0, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>p _c , the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.Work supported in part by G.N.A.F.A. (C.N.R.)Work supported in part by NSF Grant No. DMS-88-06552     

On the existence of the best discrete approximation in norm by reciprocals of real polynomials

For the given data (w_i,x_i,y_i), i=1,…,M, we consider the problem of existence of the best discrete approximation in l_p norm (1≤p<∞) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l_p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.     

Fixed points and entropy for non solvable dynamics onC, O

This work generalizes for any non-solvable pseudo-groups of Diff_0,C the existence of fixed points arbitrarly close to the origin already proved in the generic case [G-M,Wil]. An elementary proof of the Sherbakov-Nakai's density theorem [Na] is added with moreover some more precision about the derivative of a germ sending a point close to an other one. At last the topological entropy and the entropy of fixed points are strictly positive for such pseudo-groups as in [Wi] and [G-M, Wi2].Dedicated to the memory of R. MañéSupported by CNRS-Dijon and CNPq-IMPA, PICS. n^o. 177.