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Let J0, and let p = (). Consider all vector fields X in the base passing through p. The isotropy algebra of is defined by the formula (0) g = [X]2 X = 0, where [X]2 is the 2-jet of X at p. Its subalgebra p p g = [X]2 g [X]1 = 0 is independent of J0.

p p A first nontrivial differential invariant of point transformations of equations (1) is a horizontal 2-form 2 on J2 with values in g. This invariant is a unique obstruction to a linearizability of these equations by point transformations.

A next differential invariant appears on J3. It is a horizontal 3-form 3 with values in g.

Applying operations of tensor algebra to 2 and 3, we construct new differential invariants on J3.

Finally, we construct differential invariants on Jk, k 4. In particular, we construct scalar differential invariants.

Obtained invariants make possible to solve the equivalence problem for equations E under consideration so that all their 3-jets [SE]3 belong to a p generic orbit of the action of point transformations on J3. In particular, we solve the equivalence problem for generic equations (1).

References [1] Liouville R. Sur les invariantes de certaines equationes differentielles // Jour. de lEcole Politechnique, 59 (1889) 788.

[2] Yumaguzhin V.A. On the obstruction to linearizability of 2-order ordinary differential equations // Acta Applicandae Mathematicae, Vol. 83, No. 1-2, 2004. pp.133-148.

Lyapunov problem and spectrum of dynamical system Zadorozhny V. F. (Ukraine) Glushkov Institute of Cybernetics, Kiev zvf@compuserv.com.ua Approaches of determination ofthe Lyapunov function have been proposed in many issues [1] and further developed by many authors. But these very special approaches can not be extended to the case of general dynamical system, or cannot be used to obtain an algorithm of approximation of the region of attraction in the case of asymptotic stability. Thus the problem arises of finding the Lyapunov function (Lyapunov problem).

Consider the autonomous differential system equations as follows = X(x) (1) where X : Rn Rn is an R-analytic function on Rn with X(0) = 0 (i.e.

x = 0 is a steady state of (1)). Let the one is asymptotically stabile and Rn is region of attraction and on there is a couple function (V, W ) that theorem of Lyapunov for asymptotic stability is satisfied. It can be shown that in this case reasoning yields a simple inequality Rg g(xt(x))w(xt(x))dt N g 2 V (x) a. e. (2) where xt(x) is solution of Eq. (1), xo = x is initial value, N is any number and 2 is the norm of the element g L2(), i.e. g 2 = |g|2 dx. In this case, the operator R is the Hilbert-Schmidt operator in L2() (see [2] p.136) i.e. Rg (x, y)g(y)dy, x, y. In order to construct the solution the following equation Lf Xxf = gw (3) Let us apply this result to Eq.(1).Write down this formula as follows f = (x, y)g(y)dy, or f = Rg (4) Differentiating formula (4) in view of (3) we obtain the formula gw = (x, y)g(y)dy, where Xx. We will consider the general case of L2-decomposition. Let us transform gw to the sum g + vo into L2().

Here is absolute number and vo is given function, for example the one is a local Lyapunovs function and gvodx = 0. This reasoning yields an integral equation g + vo = (x, y)g(y)dy.

THEOREM [3,4].

1. A domain will be the region of attraction of the asymptotically stable steady-state x = 0 iff the real part of eigenvalues of the operator L is negative.

2. The (L) defined by the eigenvalues of the uniform Eq. (6).

3. Suppose the dynamical system (1) satisfies the condition ( (L)) 0. In this case, there exists the function V 0 whose time derivative with respect to the given system (1) is positive almost everywhere, i.e. V satisfies a necessary and sufficient condition of asymptotically stable the steady state x = 0.

In the following the kernel reduces to the degenerate kernel N and the formal linearization of Poincare-Siegel [5] to a large extent. The spectrum approach enables us to suggest a new method for solving Lyapunov problem by means of Hille Yosida theory [3]. We claim that the Lyapunov problem is the problem of the existence of the dissipation operator L : [Lf, f]+[f, Lf] 0. A new approach to studying a nonlinear bunched beam dynamics based on the self-consistent Vlasov-Maxwell model. In the framework of this scheme, a new approach based on such property as universality of Maxwell equations and methods of control theory is applied [4-6].

References [1] L. Yu. Anapolskyi, V. D. Irtegov, V.M. Matrosov Different ways of the construction of Lyapunov function. VINITNI GENERAL MEKHANICA, V. 2, 1975, pp. 53-112.

[2] P.R.Halmos, V. S. Sunder Bounded Intergral Operators on LSpaces, Springer Verlag, Berlin Heidelberg New York 1978 p.[3] V.F. Zadorozhny Fridrichs Method in a Lyapunov Problem, Applicable Analysis V. 81, No 3, 2002 pp.529-[4] V. F. Zadorozhny. Lyapunov Problem in Dynamical Control Systems Kibernetick i Sistemny Analiz, No. 6, 2002, pp. 133-142. [In Russian] [5] V. I. Arnold. The additional chapters of the theory of the ordinary differential equations M.: Nauka, 1978, 304 p. [In Russian] Quasi-projections Zakalyukin V. M. (Russia) Moscow State University The University of Liverpool The starting point of the singularity theory was classical Whitney theorem saying that generic singularities of a projection of a two-surface in the three space are folds along lines and pleats at isolated points.

The classification of singularities of projections of a two-surface embedded into RP to a plane obtained by V. I. Arnold, O. Platonova, V. Goryunov and O. Scherback at the beginning of eighty-th was a nice generalization of Whitney theorem. The surface is assumed to be generic, and centre of projection can vary in RP. The list contains 14 simple classes Pi, i = 1,..., 11 (see [1]).

P1 P2 P3 P6 P P4 P7 P P5 P Pwhich are equivalent to a projection of a germ at the origin of surface z = f(x, y) in R3 with coordinates (x, y, z) by a sheaf of rays parallel to x axis with the following functions P1 : f = x, P2 : f = x2, P3 : f = x3 + xy, P4 : f = x3 xy2, P5 : f = x3 + xy3, P6 : f = x4 + xy, P7 : f = x4 x3y + xy, P8 : f = x5 x3y + xy, P9 : f = x3 xy4, P10 : f = x4 + x2y + xy3, P11 : f = x5 + xy.

Some of these classes are non weighted homogeneous however all are simple. The equivalence here is the diffeomorphism of the domain of the ambient space containing the germ of the surface and does not containing the center of projection which preserve the fibration over the plane base of the projection.

These singularities were used later intensively.

We suggest another more rough classification which provides less number of classes. Namely two surfaces are called pseudo-equivalent if there is a diffeomorphism of the domain of the ambient space mapping one surface onto the other and satisfying the following property: if the projection ray is tangent to one of the surface at a point then at the image (or at the inverse image, respectively) of the point the other surface is also tangent to the ray passing through it. After a modification of this equivalence to get geometrical in J. Damon sense relation we get the following list of generic quasi-singularities of projections Qi, i = 1,..., 9.

Q1 Q2 Q3 Q6 Q Q4 Q Q QWhere Q1 : f = x, Q2 : f = x2, Q3 : f = x3 + xy, Q4 : f = x3 xy2, Q5 : f = x3 + xy3, Q6 : f = x4 + xy, Q7 : f = x4 + x2y, Q8 : f = x5 + xy, Q9 : f = x3 xy4.

Comparing these relations, the P8 and P11 merge into single class Q8, as well as P7 and P10 merge into Q7. All remaining classes with equal subscripts coincide. Differentiation with respect to x of Qi provides a normal form of a simple boundary singularity in the plane (x, y) with the boundary y = 0.

We discuss other nice properties of quasi singularities. In particular, the discriminants of quasi projections are isomorphic to some strata of the discriminats of the ordinary ones.

Partially supported by RFBR050100458 grant.

References [1] V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vassiliev, Singularities II. Classification and Applications, Encyclopaedia of Mathematical Sciences, vol. 39, Dynamical Systems VIII, SpringerVerlag, Berlin a.o., 1993.

V. I. Arnolds hypothesis on congruences for the traces of iterations of integer-valued matrixes and some dynamical zeta functions Zarelua A. V. (Russia) MSTU zarelua@higeom.math.msu.su Analyzing experimental data, V. I. Arnold in 20042005 formulated several questions on congruences for the traces of integer-valued matrixes.

An authors theorem from algebraic number theory (2006) generalizes a C. J. Smyths theorem (1986) that gives a generalization of the Gauss version of Fermats Little theorem. These theorems imply the positive answer on some Arnolds questions. We show that some known results on dynamical zeta functions are rather simple consequences of congruences supposed to be true by V. I. Arnold.

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