Volume 31, Number 1
In Memory of Alexey V. Borisov (on his 60th Birthday): Part II (Issue Editors: Ivan Mamaev and Iskander Taimanov)

Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a $SE(2)$ Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors. I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that surfaces of revolution are special: the additional integral of motion is elementary, while for marble rolling it requires special functions. I use the term ``Nose function'' to refer to their expression $N(\theta) = \big(I_1\, \cos^2 \theta + I_3\, \sin^2 \theta + m \, z_C^2(\theta)\big)^{1/2} $ where $\theta$ is the nutation and $z_C(\theta)$ is the center of mass height. $N(\theta)$ appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles $ \phi$ (yaw), $\psi$ roll and $\theta$. The reduction to 1 DoF is done in two stages: first, reduction by the group $SE(2) = \{ (x, y, \, \phi) \} $ to $T^* S^2 $ with almost symplectic 2-form $\Omega_{NH} = dp_\theta \, \wedge d\theta + dp_\psi \, \wedge d\psi + J \cdot K$. The semibasic term is $J \cdot K = - p_\psi \, (d \log\big(N(\theta)\big) \wedge d\psi$. It follows that $\Omega_{NH} $ is conformally symplectic in the sense that $d\left(\frac{1}{N}\, \Omega_{NH}\right) = 0. $ The conserved quantity due to the $S^1$ symmetry about the body axis is $\ell = N(\theta)\,\sin^2 \theta\, \dot{\psi}$, yielding the desired reduction to $(\theta, p_\theta)$. Further simplification results by taking the new time $dt = \sqrt{B(\theta)} \, d\tau$, with $B = I_1 + m \, |CP|^2 $ where $P = (x,y)$ is the point of contact. One gets finally $H = \frac{1}{2} \tilde{p}^2_\theta + V(\theta), \, V(\theta) = \ell^2/2 \sin^2 \theta + m g\, z_C(\theta)$ with $ \tilde{p}_\theta = p_\theta/\sqrt{B} $ and usual symplectic form $ d\tilde{p}_\theta\wedge d\theta$. The moments of inertia $I_1, I_3$ reappear in the reconstruction. As an example, very basic observations are presented for the torus. A detailed study was just finished by A. Kilin and E. Pivovarova in [3].

We analyze the stability of the straight-line motion of the bicycle depending on the mass-geometric parameters of the bicycle and its translational velocity. We construct a region in phase space which corresponds to initial conditions under which the bicycle tends asymptotically to straight-line motion. To investigate the bifurcations of the periodic solutions of the system, we construct a chart of dynamical regimes on the plane of two parameters and a three-dimensional Poincaré map. We analyze the possibility of acceleration or deceleration of the bicycle when the angular velocity of the rotor periodically changes in time.

We study chaotic dynamics and bifurcations in reversible and time-periodic perturbations of the classical Duffing equation in the case when this equation has a homoclinic figure-8 to the saddle zero equilibrium. We consider the perturbation to be nonconservative and show that the phenomenon of mixed dynamics occurs in the corresponding Poincaré map $T$ over the period. However, for small perturbations, the dynamics is predominantly dissipative: here, almost all orbits of $T$ from the interior of the homoclinic-8 (perhaps, except for orbits from small resonant zones) tend to a stable fixed point (sink) at forward iterations and to an unstable fixed point (source) at backward iterations. We show that, when the amplitude of perturbation increases, mixed dynamics, as the phenomenon of the intersection of the attractor with the repeller, becomes quite noticeable and even prevalent. In the paper, we propose two different bifurcation scenarios that lead to such mixed dynamics. In the first scenario, it emerges as a result of the phenomenon known as the attractor-repeller collision. In the second scenario, mixed dynamics arises immediately after a symmetric pair of simple saddle-node bifurcations that, due to reversibility, occur simultaneously with the sink and the source.

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity. We call them Baker functions. The construction is based on the Abel – Jacobi map, which allows us to identify the field of meromorphic functions on the Jacobian variety of the curve with the field of meromorphic functions on the symmetric product of the curve. In our previous paper, a solution to the KP equation was constructed in terms of the Baker function. This paper is devoted to the properties of the Baker functions. In this paper, we construct an entire function whose second logarithmic derivatives are the Baker functions. We prove that the power series expansion of the entire function around the origin is determined only by the coefficients of the defining equation of the curve and a branch point of the curve algebraically.We also describe the quasi-periodicity of the entire function and express the entire function in terms of the Riemann theta function.

The Burgers – Gardner equation with a nonlinear source is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for the traveling wave solutions. The Painlevé test is used to find constraints on the parameters of the equation for its integrability. The first integral of the nonlinear ordinary differential equation is obtained taking into account the results of the Painlevé test. The first integral thus found is used to obtain the general solution of the equation. Some exact solutions are found by means of the simplest equation method.

We provide a new expansion of the Fourier coefficient of the perturbing function of the PCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e., small value of eccentricity and semimajor axis. See, e.g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \mathbb{Z}^2$ and the presence of common zeros as functions of actions between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is the first step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called ``non-torus'' set from $O(1-\sqrt{\varepsilon})$ (implied by standard KAM theory) to $O(1-\varepsilon |\log\varepsilon|^c )$ for some $c>0$.

Within the framework of the two-layer quasi-geostrophic model on a rotating plane, the motions for a special case of three point vortices with intensities $\big(\kappa_1^1, \kappa_2^1, \kappa_2^2\big)=(4,1,-2)$ are considered (here, the subscript denotes the layer number: 1 is the upper layer, 2 is the lower layer, and the superscript denotes the vortex number in the layer). Thus, it is assumed that one cyclonic vortex is located in the upper layer, and two vortices, cyclonic and anticyclonic, are located in the lower layer. It is shown that in the general case each vortex performs periodic motions in such a way that every half-period the vortex structure takes a collinear state. In this case, over time, the trajectory of each vortex completely fills a certain ring region around the vorticity center. However, among the continuum of these quasi-ordered trajectories, one can always find a family of closed periodic solutions, both purely circular (preserving the collinear structure) and more complex, so-called $N$-modal ($N$-symmetric) stationary solutions. In this paper, these solutions are constructed and their main properties are described.

A caustic of a billiard is a curve whose tangent lines are reflected to its own tangent lines. A billiard is called Birkhoff caustic-integrable if there exists a topological annulus adjacent to its boundary from inside that is foliated by closed caustics. The famous Birkhoff Conjecture, studied by many mathematicians, states that the only Birkhoff caustic-integrable billiards are ellipses. The conjecture is open even for billiards whose boundaries are ovals of algebraic curves. In this case the billiard is known to have a dense family of so-called rational caustics that are also ovals of algebraic curves. We introduce the notion of a complex caustic: a complex algebraic curve whose complex tangent lines are sent by complexified reflection to its own complex tangent lines.We show that the usual billiard on a real planar curve $\gamma$ has a complex caustic if and only if $\gamma$ is a conic. We prove an analogous result for billiards on all the surfaces of constant curvature. These results are corollaries of the solution of S. Bolotin’s polynomial integrability conjecture: a joint result by M. Bialy, A. Mironov and the author. We extend them to the projective billiards introduced by S. Tabachnikov, which are a common generalization of billiards on surfaces of constant curvature. We also deal with a well-known class of projective billiards on conics that are defined to have caustics forming a dual conical pencil. We show that, up to restriction to a finite union of arcs, each of them is equivalent to a billiard on an appropriate surface of constant curvature.

In this paper we study Riemannian metrics on 2-surfaces with integrable geodesic flows by means of an additional rational-in-momenta first integral. This problem is reduced to a quasi-linear system of PDEs. We construct solutions to this system via the classical hodograph method. These solutions give rise to local examples of metrics and rational integrals. Some of the constructed metrics have a very simple form. A family of implicit integrable examples parameterized by two arbitrary functions of one variable is also provided.

The problem of motion of a heavy gyrostat with a fixed point under the action of gyroscopic forces, corresponding to the classical Hess case in the problem of motion of a heavy rigid body with a fixed point, is considered. We derive that the problem of motion of a gyrostat is reduced to solving the second-order linear differential equation with rational coefficients. Using the Kovacic algorithm, we obtain the conditions under which the general solution of the corresponding second-order linear differential equation is expressed in terms of Liouvillian functions and, therefore, it can be presented in explicit form. We prove that under the obtained conditions the equations of motion can be integrated by quadratures.