Volume 30, Number 6
Special Issue: In Memory of Alexey V. Borisov (on his 60th Birthday): Part I (Issue Editors: Ivan Mamaev and Iskander Taimanov)

In this paper we investigate rhomboidal central configurations for the planar fivebody problem. These are configurations whose convex hulls are rhombuses, and one interior mass lies on a diagonal of the rhombus. Consider partition of the configuration space by orderings of mutual distances, our main theorem provides criteria for central configurations in every region of this partition. For some regions we obtain existence and uniqueness of central configurations, for some we obtain nonexistence, for others our criteria are necessary conditions on masses. Our proofs are elementary and completely analytic.

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.

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.

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 $c>0$.

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.

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 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.

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].

The purpose of this study is to investigate a simple model for the half-center oscillator (HCO), which consists of two nonidentical phase oscillators ($\phi$-neurons) with mutual chemical excitatory couplings. In the absence of couplings, each element is in an excitable state. Using one- and two-parameter bifurcation analysis, we determine regions of stability of various synchronous temporal patterns typical for HCO and describe in detail bifurcation transitions between them. The proposed simple HCO model allows one to reproduce the main effects observed in biologically plausible HCO models, and can be easily implemented in locomotor units in neuromorphic robotics.

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.

In this paper, we prove the theorem given in the title and a similar theorem for generalized pseudo-Anosov homeomorphisms of the Klein bottle with only one “needle” singularity. We reformulate both statements in terms of the invariant foliation singularity type and orientability of foliations. The method we use to prove the theorems implies using the construction of band surface. Every band surface is represented with its combinatorial description, called the configuration. Applying a series of Rauzy transformations to all possible configurations in the cases considered, we show that the necessary conditions imposed on the invariant foliations are violated whence the results follow.

An original effective method for constructing explicit solutions of integrable Davey – Stewartson type equations is proposed, based on the use of dressing chains. The main difficulty arising when using the symmetry approach in 3D is associated with nonlocal variables entering the equation. To solve the nonlocality problem, it is proposed to replace the infinite dressing chain with its finite-field reductions preserving the integrability property. The application of the method is illustrated by the DS I equation, for which a new class of explicit solutions is constructed that depend on two arbitrary functions. In this example, the dressing chain is replaced by a finite-field reduction of the Toda lattice corresponding to a simple Lie algebra $A_2$.

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.

In this study, we analyze a planar mathematical pendulum whose suspension point oscillates vertically according to a harmonic law. The pendulum bob is electrically charged and positioned slightly above two electric charges of equal sign and intensity, which are equidistant from the suspension point and separated by a distance of $2d$. Here, $d$ denotes the distance from each charge to the orthogonal projection of the suspension point onto the horizontal line where the charges lie. We formulate the Hamiltonian structure of this mechanical system, identify two equilibrium points, and examine the system’s linear stability. The dynamics are governed by three dimensionless parameters: $\mu$ which relates to the electric charges; $\varepsilon$, associated with the amplitude of oscillation of the suspension point; and $\alpha$, determined by the frequency of the system. We then investigate the parametric stability of the equilibrium points. Finally, we present the boundary surfaces that separate regions of stability and instability in the parameter space. For specific values of $\mu$, we derive cross-sectional curves that delineate these regions, using results from the Krein – Gelfand – Lidskii theorem and the Deprit – Hori method.

We study bifurcation scenarios leading to the so-called Shilnikov singular attractors as a result of breakdown of closed invariant curves in the Chialvo map, which is a twodimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariant curve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relates to the immediately occurring big invariant curve after disappearance of a stable fixed point at the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenarios of breakdown of the invariant curve and chaos development leading to the emergence of Shilnikov singular attractors. Additionally, we study geometrical peculiarities of these attractors, such as structures of rotating patterns of orbits inside the Shilnikov singular funnel and present a two-parametric analysis with Lyapunov exponents and minimal a distance between the chaotic attractor and the unstable focus.

This paper is a continuation of our previous work where we investigated the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable connected surfaces such that their nonwandering sets consist of one-dimensional basic sets (attractors and repellers). In that work, we showed that the dynamical properties of each diffeomorphism from a given class define a collection consisting of nonempty multisets of natural numbers (each such collection contains at least two multisets). These multisets are topological invariants of the diffeomorphism and uniquely determine the topology of the ambient surface. In this paper, we solve the problem of realization of diffeomorphisms from the class $\mathbb G(M^2)$ with respect to a given collection of multisets of natural numbers. We describe all possible collections of multisets from which one can construct a diffeomorphism from the class $\mathbb G(M^2)$, presenting a step-by-step algorithm of construction.

This paper establishes the existence of periodic response solutions for multidimensional nonlinear Schrödinger equations (NLS) with unbounded perturbations. We adapt the Craig – Wayne – Bourgain (CWB) method to this context. The central point is that the core separation lemma remains valid for sufficiently weak unbounded perturbations, which allows the CWB method to be successfully applied. This work extends the applicability of the CWB method to problems with unbounded perturbations.

The dynamics that necessarily coexists with a homoclinic orbit is captured by its dynamical core. In this work we characterize the dynamical core of a broad class of homoclinic orbits in the Smale horseshoe, specifically those with decorations of three types: maximal, P-lists and star decorations. For each of these families, we construct an explicit pruning region whose survival set — consisting of all symbolic sequences whose orbits avoid the region under the shift — coincides with the dynamical core. This provides a unified symbolic description of the forced dynamics and establishes a framework for computing dynamical invariants such as topological entropy.