### Article

## Спиральный хаос в моделях типа Лотки-Вольтерры

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a ﬁxed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-ﬁxed axis is equal to zero. Depending on the system parameters, we ﬁnd cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the eﬀect of reversal, which was observed previously in the motion of rattlebacks.

Based on the results of numerical simulations we discuss and illustrate dynamical phenomena characteristic for the rattleback, a solid body of convex surface moving on a rough horizontal plane, which are associated with the lack of conservation for the phase volume in the nonholonomic mechanical system. Due to local compression of the phase volume, behaviors can be realized, analogous to those occurring in dissipative systems, like stable equilibrium points, corresponding to stationary rotations, limit cycles (rotations with oscillations), strange attractors. A chart of dynamical regimes in the parameter plane of a total mechanical energy and a relative angle between the geometric and dynamic principal axes of the body is plotted and discussed. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. Several examples of strange attractors are considered; their phase portraits, Lyapunov exponents, and Fourier spectra are discussed.

Investigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus born from stable equilibrium. Then the unstable invariant manifold of saddle-focus winds on the stable limit cycle and forms a whirlpool. For some parameter's value the unstable invariant manifold touches one-dimensional stable invariant manifold and forms homoclinic trajectory to saddle-focus. If in this case the limit cycle loses stability (for example, as result of sequence of period doubling bifurcations) and saddle value of saddle-focus is negative then strange attractor appears on base of homoclinic trajectory.

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.