### Article

## On diagrammatic technique for nonlinear dynamical systems

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev{V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has two purely imaginary eigenvalues, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to pseudonormal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations L(p)x=f(x)+b(t), p=d/dt, with 2pi-periodic forcing b and periodic f we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of 2pi-periodic solutions.

A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.