Ross. Akad. Nauk Ser. Mat., 71 (2007), no. 2, 173–222.

On the diversity of nondegeneracy conditions in KAM theory Sevryuk M. B. (Russia) Institute of Energy Problems of Chemical Physics sevryuk@mccme.ru Consider a completely integrable Hamiltonian system with n degrees of freedom and its Hamiltonian perturbation. The phase space of the unperturbed system is foliated into Lagrangian invariant n-tori I = const carrying conditionally periodic motions with frequency vectors (I) = H0(I)/I, where (I1,..., In) G Rn are the action variables and His the unperturbed Hamilton function. If the unperturbed system is Kolmogorov nondegenerate (the frequency map : G Rn is a local diffeomorphism), then according to KAM theory, each unperturbed torus with Diophantine frequencies gives rise to a perturbed torus with the same frequencies (provided that the perturbation magnitude is sufficiently small).

If the unperturbed system is isoenergetically nondegenerate ( = 0 in G and the projectification [1 : · · · : n]: G RPn-1 of the map is a local diffeomorphism on every energy level hypersurface H0 = const), then each unperturbed torus with Diophantine frequencies gives rise to a perturbed torus with the same frequency ratios and with the same energy value. On the other hand, if the unperturbed system is Rssmann nondegenerate (there exists a positive integer K such that at every point I G, the n linear hull of all the partial derivatives ||(I)/I1 · · · In of all the orders 0 || = 1 + · · · + n K is Rn), then a perturbed system admits many invariant n-tori carrying quasi-periodic motions but there is, generally speaking, no connection between the unperturbed frequencies and the perturbed ones.

The same three types of nondegeneracy can be defined for the socalled upper dimensional coisotropic invariant M-tori (whose dimension M is greater than the number n of degrees of freedom). However, in the upper dimensional Hamiltonian KAM theory, it is Rssmann nondegeneracy that was historically considered first [1] (in contrast to the “classical” Lagrangian case) and is usually regarded as the “main” one.

Recently, in the Lagrangian framework, there were introduced nondegeneracy conditions intermediate between the Kolmogorov and Rssmann conditions [2,3]. For example, one can formulate nondegeneracy conditions that guarantee the following partial preservation of frequencies: each Dio phantine collection of frequencies 1,..., k (k < n being fixed) gives rise to an (n - k)-parameter Cantor family of perturbed invariant tori whose first k frequencies coincide with 1,..., k. Similarly, there are nondegeneracy conditions that are intermediate between the isoenergetic and Rssmann conditions and lead to partial preservation of frequency ratios [2,3].

The case of the lower dimensional isotropic invariant m-tori (whose dimension m is less than the number n of degrees of freedom) is more complicated. In the lower dimensional Hamiltonian KAM theory, one should take into account not only the frequencies 1,..., m of the invariant tori but also their Floquet exponents 1,..., n-m, i.e., the eigenvalues of the coefficient matrix for the variational equation along the torus. However, the Kolmogorov, isoenergetic, and Rssmann nondegeneracy conditions can be carried over to lower dimensional invariant tori (not in a unique way), all the lower dimensional versions of these conditions involving the unperturbed Floquet exponents, and sometimes the dependence of the system on several external parameters is required. Partial preservation of the frequencies or frequency ratios in the lower dimensional context has been explored in the works [3–5]. Very recently, partial preservation of frequencies and Floquet exponents was also treated [6].

References [1] Parasyuk I.O. Conservation of multidimensional invariant tori of Hamiltonian systems // Ukrain. Math. J. – 1984. – V. 36, N 4.

– P. 380–385.

[2] Chow S.-N., Li Y., Yi Y. Persistence of invariant tori on submanifolds in Hamiltonian systems // J. Nonlinear Sci. – 2002. – V. 12, N 6. – P. 585–617.

[3] Sevryuk M.B. Partial preservation of frequencies in KAM theory // Nonlinearity. – 2006. – V. 19, N 5. – P. 1099–1140.

[4] Li Y., Yi Y. Persistence of hyperbolic tori in Hamiltonian systems // J. Differ. Equations. – 2005. – V. 208, N 2. – P. 344–387.

[5] Liu Zh. Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems // Nonlinear Anal. – 2005. – V. 61, N 8. – P. 1319–1342.

[6] Sevryuk M.B. Partial preservation of frequencies and Floquet exponents in KAM theory // Proc. Steklov Inst. Math. (submitted).

Three classical problems of parametric resonance Seyranian A. P. (Russia) Institute of Mechanics, Moscow State Lomonosov University seyran@imec.msu.ru A problem of stabilization of a vertical (inverted) position of a pendulum under high frequency vibration of the suspension point is considered. Small viscous damping is taken into account, and periodic excitation function describing vibration of the suspension point is assumed to be arbitrary. A formula for stability region of Hill’s equation with damping near zero frequency is obtained. For several examples it is shown that analytical and numerical results are in a good agreement with each other.

An asymptotic formula for stabilization region of the inverted pendulum is derived. It is shown that the effect of small viscous damping is of the third order, and taking it into account leads to increasing critical stabilization frequency. The method of stability analysis is based on calculation of derivatives of the Floquet multipliers.

The swing problem is undoubtedly among the classical problems of mechanics. It is known from practice that to set a swing into motion one should erect when the swing is in limit positions and squat when it is in the middle vertical position, i.e., carry out oscillations with double the natural frequency of the swing. However in the literature you can not find formulae for instability regions explaining the phenomenon of swinging. In the present paper the simplest model of the swing is described by a massless rod with a concentrated mass periodically sliding along the rod axis. Based on analysis of multipliers the asymptotic formulae for instability (parametric resonance) domains in the three-dimensional parameter space are derived and analyzed.

The third classical problem is the problem of finding instability regions for a system with periodically varying moment of inertia. An equation describing small torsional oscillations of the system with periodic coefficients dependent on four parameters including damping is derived. Analytical results for instability (parametric resonance) regions in parameter space are obtained. Numerical examples are presented.

Asymptotics of eigenfunctions to linear ordinary differential operators and Stokes lines Shapiro B. (Sweden) Stockholm University shapiro@math.su.se In the talk I describe the asymptotic root distribution for the eigenfunctions of the classical univariate Schroedinger operator with polynomial coefficients as well simliar asymptotics for a general class of the so-called exactly solvable operators. A (partially conjectural) relation to the Stokes lines will be pointed out. All the necessary notions will be defined during the talk.

Inverse problem for a finite semiconductor network on the annulus Shapiro Michael (USA) Michigan State University mshapiro@math.msu.edu This is a joint work with Mikhail Gekhtman (University of Notre Dame) and Alexander Vainshtein (Haifa University).

A directed graph G whose arcs are equipped with positive numbers (conductivities) xe, e Edge(G) is called a semiconductor network. For a given semiconductor network we define the boundary measurement between a source node u and a node v as the total conductivity between sink these two nodes, namely, xe, where the sum is taken over P :uv eP all directed paths P connecting u and v while the product is taken over all arcs e of path P.

The inverse problem for semiconductor networks can be formulated as follows. Is it possible to restore a semiconductor network (up to some natural equivalence relations) qiven a complete collection of boundary measurements A.Postnikov in a recent preprint [1] considered the inverse problem for semiconductor networks on the disk with sources and sinks on the boundary circle. He described all elementary transformation of networks generating a natural equivalence relation and proved that the inverse problem has a unique solution up to this equivalence.

In the current paper we consider a semiconductor network on the annulus where all sources belong to the inner circle and all sinks are on the outer circle. To make this problem meaningful we need to introduce a spectral parameter in the definition of boundary measurements. Unlike networks on the disk the inverse problem on the annulus does not have a unique solution up to a natural equivalence. We describe all inverse problem solutions for networks on the annulus with only one source and one sink.

References [1] Postnikov A. Total positivity, Grassmannians, and networks // arXiv:math/0609764.

Fluid models and phase transitions in the large queuing networks Shlosman S. B. (Russia) IITP RAS shlosman@gmail.com We show that in some models of large networks it is possible to observe the onset of coherent behavior. The corresponding long range memory effect can be a source of slowing down of the network performance. The phase transition is turned on once the load exceeds the critical level. It is similar to the low temperature breaking of continuous symmetry in statistical physics.

This is a joint work with A. Rybko and A. Vladimirov.

Enumeration of real rational curves on Del Pezzo surfaces Shustin Eugenii (Israel) Tel Aviv University shustin@post.tau.ac.il A systematic development of the enumerative geometry of real algebraic curves has been initiated by the discovery of Welschinger invariants [5] and the appearance of tropical enumerative geometry [4]. In a particular case of a real Del Pezzo surface with a non-empty real part (R), with any real ample divisor D and a component of (R), one can associate the Welschinger invariant W(, D), which counts with appropriate weights ±1 the real rational curves in the linear system |D|, passing through r(, D) := c1()·D-1 generic points in. The invariant does not depend on the choice of fixed points, and provides a uniform lower bound the the number of real rational curves in |D| passing through any chosen configuration of r(, D) real points in, whereas the corresponding genus zero Gromov-Witten invariant GW0(, D) gives an upper bound. The tropical geometry converts the computation of the Welschinger invariants into a count of certain plane tropical curves (finite planar graphs), passing through configurations of r(, D) points in the plane. This allows one to establish interesting phenomena in the behavior of Welschinger invariants.

Among them Conjecture 1. Welschinger invariants of all real Del Pezzo surfaces with a non-empty real part are positive.

This, in particular, would imply that: Given a real Del Pezzo surface and a real ample divisor D, through ant r(, D) generic points, belonging to the same connected component of (R), one can trace at least one real rational curve C |D|.

Another observation is Conjecture 2. For any real Del Pezzo surface with (R) =, any component of (R), and any real ample divisor D, log W(, nD) log GW0(, nD) lim = lim = c1() · D.

n n n log n n log n Using the tropical geometry techniques, we prove Theorem 1 ([1,2]). Conjectures 1 and 2 hold for the plane P2, the quadric (P1)2, the plane P2 blown up at k generic real points, 1 k 4, k the quadric S2 with S2(R) S2, and for that quadric S2, S2, S2 blown 1,0 2,0 0,up at one or two real points, or at two imaginary conjugate points.

An important problem is to find recursive formulas for the Welschinger invariants similar to that for the Gromov-Witten ones, like the Kontsevich formula (WDVV equation), or the Caporaso-Harris formula. No analogue of the Kontsevich formula is known. One, however, can prove Theorem 2 ([3]). Welschinger invariants of P2 and P2, 1 k 4, k satisfy a Caporaso-Harris type recursive formula.

It is especially interesting that this formula includes tropical Welschinger invariants, which so far have no algebraic analogue.

References [1] I. Itenberg, V. Kharlamov, and E. Shustin. Logarithmic equivalence of Welschinger and Gromov-Witten invariants. Russian Math.

Surveys 59 (2004), no. 6, 1093–1116.

[2] I. Itenberg, V. Kharlamov, and E. Shustin. New cases of logarithmic equivalence of Welschinger and Gromov-Witten invariants. Preprint at arXiv:math.AG/0612782.

[3] I. Itenberg, V. Kharlamov, and E. Shustin. A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces.

Preprint at arXiv:math.AG/0608549.

[4] G. Mikhalkin. Enumerative tropical algebraic geometry in R2. J.

Amer. Math. Soc. 18 (2005), 313–377.

[5] J.-Y. Welschinger. Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. Invent. Math. (2005), no. 1, 195–234.

Long-term evolution of the asteroid orbits at the 3:1 mean motion resonance with Jupiter planar problem Sidorenko V. V. (Russia) Keldysh Institute of Applied Mathematics, Moscow sidorenk@spp.keldysh.ru We consider the 3:1 mean-motion resonance of the planar elliptic restricted three body problem (Sun-Jupiter-asteroid). Using the numeric averaging both over the orbital motion and resonant angle librations/oscillations, we obtained the evolutionary equations which describe the long-term behavior of the asteroid’s argument of pericentre and eccentricity. The detailed classification of the possible evolution paths was developed. It generalized significantly the similar results on secular effects in the discussed problem recently obtained under the scope of the well known Wisdom model [1].

The validity of the averaged equations is closely connected with conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the resonant angle cause the violations of the adiabatic invariance and the regions of the adiabatic chaos appear in the system’s phase space.

A special attention was given to the very-high-eccentricity asteroidal motion. Being limited to relatively small values of the eccentricity the Wisdom model did not allow to study the the transitions from the moderate values of the eccentricity (e 0.2 - 0.3) to the values 0.9 - 0.although their existence was demonstrated [2]. We found that under the certain conditions the region of the adiabatic chaos can be a place where such transitions are permissible.

References [1] Neishtadt A.I., Sidorenko V.V. Wisdom system: dynamics in the adiabatic approximation. // Celest. Mech. Dyn. Astron. – 2004.

90, p. 307–330.

[2] Ferraz-Mello S., Klafke J.C. A model for the study of very-higheccentricity asteroidal motion. The 3:1 resonance. Predictability, Stability and Chaos in N-body Dynamical Systems. Ed. Roy A.E.

New York: Plenum Press (1991), p. 177–184.

Geometry of singular manifolds in a zero-pressure adhesive flow Sobolevski A. N. (Russia) Moscow State University sobolevski@phys.msu.ru Free inertial flow of a continuous fluid with no pressure or viscosity develops singularities when and where trajectories of particles cross.

To continue the motion after a singularity, one needs to know what happens to colliding particles. In a number of applications varying from cosmology to numerical fluid dynamics, particles are assumed to be sticky, i.e., to collide absolutely inelastically. In spite of the apparent simplicity of this model constructing the flow beyond singularities is a challenge. We suggest a mathematical framework for multidimensional adhesive ballistic flows and characterize their singularities geometrically.

Ordinary double solids Steenbrink Joseph H. M. (Netherlands) Radboud University J.Steenbrink@math.ru.nl This is a report on joint work with Martijn Grooten. It concerns double covers of projective three-space whose ramification surface has degree four and has only generic projection singularities. We analyze their Abel– Jacobi mappings.

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