Moscow Mathematical Journal
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ISSN 1609-4514
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2005, Independent University of Moscow
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Volume 5 (2005), Number 1. Abstracts V. Arnold. Ergodic and Arithmetical Properties of Geometrical Progression's Dynamics and of its Orbits [PDF] The multiplication by a constant (say, by 2) acts on the set Z/nZ of residues (mod n) as a dynamical system, whose cycles relatively prime to n all have a common period T(n) and whose orbits consist each of T(n) elements, forming a geometrical progression or residues. The paper provides many new facts on the arithmetical properties of these periods and orbits (generalizing the Fermat's small theorem, extended by Euler to the case where n is not a prime number). The chaoticity of the orbit is measured by some randomness parameter, comparing the distances distribution of neighbouring points of the orbit with a similar distribution for T randomly chosen residues (which is binominal). The calculations show some kind of repulsion of neighbours, avoiding to be close to other members of the same orbit. A similar repulsion is also observed for the prime numbers, providing their distributions nonrandomness, and for the arithmetical progressions of the residues, whose nonrandomness degree is similar to that of the primes. The paper contains also many conjectures, including that of the infinity of the pairs of prime numbers of the form (q,2q+1), like (3,7), (11,23), (23,47) on one side and that on the structure of some ideals in the multiplicative semigroup of odd integers—on the other. Keywords. Geometrical progression, arithmetical dynamics, ergodic properties 2000 Mathematics Subject Classification. 11A07 (11N69 37A45 37B99) M. Briskin and Y. Yomdin. Tangential Version of Hilbert 16th Problem for the Abel Equation [PDF] Two classical problems on plane polynomial vector fields, Hilbert's 16th problem about the maximal number of limit cycles in such a system and Poincaré's center-focus problem about conditions for all trajectories around a critical point to be closed, can be naturally reformulated for the Abel differential equation y'=p(x)y2 + q(x) y3. Recently, the center conditions for the Abel equation have been related to the composition factorization of P=∫ p and Q=∫ q and to the vanishing conditions for the moments mi,j = ∫ Pi Qj q. On the basis of these results we start in the present paper the investigation of the ``Hilbert's tangential problem'' for the Abel equation, which is to find a bound for the number of zeroes of I(t) = ∫ab (q(x)dx)/(1−tP(x)). Keywords. Limit cycles, Abel differential equation, moments, compositions, Bautin ideals 2000 Mathematics Subject Classification. Primary 34C07, 34C08; Secondary 30C05, 30D05 D. Dolgopyat. Livsiĉ Theory for Compact Group Extensions of Hyperbolic Systems [PDF] We prove Livsiĉ-type theorems for rapidly mixing compact group extensions of Anosov diffeomorphisms. Keywords. Cocycle equation, transfer operator, partial hyperbolicity, small divisors 2000 Mathematics Subject Classification. 3730C, 37D30, 37J40 A. Glutsyuk. On the Monodromy Group of Confluent Linear Equations [PDF] We consider a linear analytic ordinary differential equation with complex time having a nonresonant irregular singular point. We study it as a limit of a generic family of equations with confluent Fuchsian singularities. In 1984, V. I. Arnold asked the following question: Is it true that some operators from the monodromy group of the perturbed (Fuchsian) equation tend to Stokes operators of the nonperturbed irregular equation? Another version of this question was also proposed independently by J.-P. Ramis in 1988. We consider only the case of Poincaré rank 1. We show (in dimension two) that, generically, no monodromy operator tends to a Stokes operator; on the other hand, in any dimension, the commutators of appropriate noninteger powers of the monodromy operators around singular points tend to Stokes operators. Keywords. Linear equation, irregular singularity, Stokes operators, Fuchsian singularity, monodromy, confluence 2000 Mathematics Subject Classification. 34M35 (34M40) J. Guckenheimer and R. Haiduc. Canards at Folded Nodes [PDF] Folded singularities occur generically in singularly perturbed systems of differential equations with two slow variables and one fast variable. The folded singularities can be saddles, nodes or foci. Canards are trajectories that flow from the stable sheet of the slow manifold of these systems to the unstable sheet of their slow manifold. Benoît has given a comprehensive description of the flow near a folded saddle, but the phase portraits near folded nodes have been only partially described. This paper examines these phase portraits, presenting a picture of the flows in the case of a model system with a folded node. We prove that the number of canard solutions in these systems is unbounded. Keywords. Folded node, singularly perturbed system, slow-fast vector field 2000 Mathematics Subject Classification. 34E15 J. Hubbard. Parametrizing Unstable and Very Unstable Manifolds [PDF] Existence and uniqueness theorems for unstable manifolds are well-known. Here we prove certain refinements. Let f: (Cn,0) → Cn be a germ of an analytic diffeomorphism, whose derivative Df(0) has eigenvalues λ1, …, λn such that |λ1| ≥ … ≥ |λk| > |λk+1| ≥ … ≥ |λn|, with |λk| > 1. Then there is a unique k-dimensional invariant submanifold whose tangent space is spanned by the generalized eigenvectors associated to the eigenvalues λ1, …, λk, and it depends analytically on f. Further, there is a natural parametrization of this ``very unstable manifold,'' which can be extended to an analytic map Ck → Cn when f is defined on all of Cn, and is an injective immersion if f is a global diffeomorphism. We also give the corresponding statements for stable manifolds, which are analogous locally but quite different globally. Keywords. Invariant manifold, resonance 2000 Mathematics Subject Classification. Primary 37D10; Secondary 37F15, 37G05 V. Kaloshin. A Geometric Proof of the Existence of Whitney Stratifications [PDF] In this paper we give a simple geometric proof of existence of so-called Whitney stratification for (semi)analytic and (semi)algebraic sets. Roughly, stratification is a partition of a singular set into manifolds so that these manifolds fit together ``regularly''. The proof presented here does not use analytic formulas only qualitative considerations. It is based on a remark that if there are two manifolds of the partition V and W of different dimension and V ⊂ \bar W, then irregularity of the partition at a point x in V corresponds to the existence of nonunique limits of tangent planes TyW as y approaches x. Keywords. Stratifications, (semi)algebraic sets, (semi)analytic sets, Wing lemma 2000 Mathematics Subject Classification. Primary: 14F45, 32C42, 57N80, 58A35, 58C27 S. Katok and I. Ugarcovici. Geometrically Markov Geodesics on the Modular Surface [PDF] The Morse method of coding geodesics on a surface of constant negative curvature consists of recording the sides of a given fundamental region cut by the geodesic. For the modular surface with the standard fundamental region each geodesic (which does not go to the cusp in either direction) is represented by a bi-infinite sequence of non-zero integers called its geometric code. In this paper we show that the set of all geometric codes is not a finite-step Markov chain, and identify a maximal 1-step topological Markov chain of admissible geometric codes which we call, as well as the corresponding geodesics, geometrically Markov. We also show that the set of geometrically Markov codes is the maximal symmetric 1-step topological Markov chain of admissible geometric codes, and obtain an estimate from below for the topological entropy of the geodesic flow restricted to this set. Keywords. Modular surface, geodesic flow, topological entropy, topological Markov chain 2000 Mathematics Subject Classification. Primary 37D40, 37B40; Secondary 11A55, 20H05 K. Khanin, D. Khmelev, and A. Sobolevskii. On the Velocities of Lagrangian Minimizers [PDF] We consider minimizers for the natural time-dependent Lagrangian system in Rd with Lagrangian L(x,v,t) = |v|β/β − U(x,t), where β>1. For minimizers on a time interval of length T with one end-point fixed, we prove that the absolute values of velocities are bounded by K log2/β T, provided that the potential U(x,t) and its gradient are uniformly bounded. We also show that the above estimate is asymptotically sharp. Keywords. Action-minimizing trajectories, time-dependent Lagrangian systems, variational problems in unbounded domains 2000 Mathematics Subject Classification. 37J50 L. Ortiz-Bobadilla, E. Rosales-González, and S. Voronin. Rigidity Theorems for Generic Holomorphic Germs of Dicritic Foliations and Vector Fields in (C2, 0) [PDF] We consider the class Vn+1d of dicritical germs of holomorphic vector fields in (C2, 0) with vanishing n-jet at the origin for n ≥ 1. We prove, under some genericity assumptions, that the formal equivalence of two generic germs implies their analytic equivalence. A similar result is also established for orbital equivalence. Moreover, we give formal, orbitally formal, and orbitally analytic classifications of generic germs in Vn+1d up to a change of coordinates with identity linear part. Keywords. Dicritic foliations, dicritic vector fields, rigidity, formal equivalence, analytic equivalence 2000 Mathematics Subject Classification. 32S70, 32S05, 32S30, 34A25, 34C20, 57R30 D. Panazzolo and R. Roussarie. Bifurcations of Cuspidal Loops Preserving Nilpotent Singularities [PDF] A cuspidal loop L of a smooth planar vector field X0 is a singular cycle formed by the union of a cuspidal singularity with 2-jet equivalent to y(∂/∂x) + (x2+b0xy)(∂/∂y) and a connection between its two local separatrices. We consider smooth unfoldings Xλ along cuspidal loop L of X0 parameterized by λ ∈ (Rp,0). We assume that the cuspidal point exists at all parameter values. Let P0 be the Poincaré map of X0 along L. If this map is not formally equal to the identity, then it has the asymptotic expansion P0: u → u+a± |u|τ+…, where ± is the sign of u, a±≠0, and τ is a coefficient belonging to the sequence S = {1,7/6,11/6,2,&hellip} = {n∈N} ∪ {m+1/6, m∈N, m≥1} ∪ {p−1/6, p∈N, p≥2}. In this case we say that (X0,L) has finite codimension equal to the order of τ in the sequence S. The main result of this paper is that the cyclicity of the unfolding Xλ has an explicit bound of e.o.H0(s), where s is the codimension of (X0,L) (e.o.H0(s) ∼ 5s/3 when s → ∞). This bound is sharp for generic unfoldings. For analytic unfoldings, the cyclicity is always finite and is given by the codimension of the related Abelian integral in the case of a non-conservative perturbation of a Hamiltonian vector field. Keywords. Bifurcations, cuspidal loop, nilpotent singularity 2000 Mathematics Subject Classification. Primary 34C23; Secondary 34C25, 34C37, 37F75 Ch. Rousseau. Modulus of Orbital Analytic Classification for a Family Unfolding a Saddle-Node [PDF] In this paper we consider generic families of 2-dimensional analytic vector fields unfolding a generic (codimension 1) saddle-node at the origin. We show that a complete modulus of orbital analytic classification for the family is given by an unfolding of the Martinet–Ramis modulus of the saddle-node. The Martinet–Ramis modulus is given by a pair of germs of diffeomorphisms, one of which is an affine map. We show that the unfolding of this diffeomorphism in the modulus of the family is again an affine map. The point of view taken is to compare the family with the ``model family" (x2−ε)(∂/∂x) +y(1+a(ε)x)(∂/∂y). The nontriviality of the Martinet–Ramis modulus implies geometric ``pathologies" for the perturbed vector fields, in the sense that the deformed family does not behave as the standard family. Keywords. Saddle-node, orbital analytic classification, modulus 2000 Mathematics Subject Classification. 34M35 A. Shilnikov, L. Shilnikov, and D. Turaev. Blue-Sky Catastrophe in Singularly Perturbed Systems [PDF] We show that the blue-sky catastrophe, which creates a stable periodic orbit of unboundedly increasing length, is a typical phenomenon for singularly perturbed (multi-scale) systems with at least two fast variables. Three distinct mechanisms of this bifurcation are described. We argue that it is behind the transition from periodic spiking to periodic bursting oscillations. Keywords. Saddle-node, global bifurcations, stability boundaries, slow-fast system, bursting oscillations, spikes, excitability 2000 Mathematics Subject Classification. 37G15, 34E15, 37C27, 34C26 D. Turaev. An Example of a Resonant Homoclinic Loop of Infinite Cyclicity [PDF] We describe a codimension-3 bifurcation surface in the space of Cr-smooth (r≥3) dynamical systems (with phase space of dimension 4 or higher) having an attractive two-dimensional invariant manifold with an infinite sequence of periodic orbits of alternating stability which converge to a homoclinic loop. Keywords. Codimension-3 homoclinic bifurcation, invariant manifold, swallow tail, limit cycle 2000 Mathematics Subject Classification. 37G20, 34C07, 34C37 |
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