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Alexei Tsygvintsev

Publications and source records attributed to Alexei Tsygvintsev.

12 recordsLinked to original sources
arXiv2024

Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem

We consider the Newtonian planar three-body problem, defining a syzygy (velocity syzygy) as a configuration where the positions (velocities) of the three bodies become collinear. We demonstrate that if the total energy is negative, every collision-free solution has an infinite number of velocity syzygies. Specifically, the velocities of the three bodies become parallel within every interval of time containing three consecutive syzygies. Using comparison theory for matrix Riccati equations, we derive new upper and lower bounds on the moments when syzygies occur.

math.DS↗
arXiv2023

On some collinear configurations in the planar three-body problem

In this paper, we further investigate the planar Newtonian three-body problem with a focus on collinear configurations, where either the three bodies or their velocities are aligned. We provide an independent proof of Montgomery's result, stating that apart from the Lagrange solution, all negative energy solutions to the zero angular momentum case result in syzygies, i.e., collinear configurations of positions. The concept of generalised syzygies, inclusive of velocity alignments, was previously explored by the author for bounded solutions. In this study, we broaden our scope to encompass negative energy cases and provide new bounds. Our methodology builds upon the elementary Sturm-Liouville theory and the Wintner-Conley "linear" form of the three-body problem, as previously explored in the works of Albouy and Chenciner.

math.DS↗
arXiv2018

On the overfly algorithm in deep learning of neural networks

In this paper we investigate the supervised backpropagation training of multilayer neural networks from a dynamical systems point of view. We discuss some links with the qualitative theory of differential equations and introduce the overfly algorithm to tackle the local minima problem. Our approach is based on the existence of first integrals of the generalised gradient system with build-in dissipation.

cs.LG↗
arXiv2013

Bounded analytic maps, Wall fractions and ABC flow

In this work we study the qualitative properties of real analytic bounded maps defined in the infinite complex strip. The main tool is approximation by continued g-fractions of Wall. As an application, the ABC flow system is considered which is essential to the origin of the solar magnetic field.

nlin.CD↗
arXiv2013

Continued g-fractions and geometry of bounded analytic maps

In this work we study qualitative properties of real analytic bounded maps. The main tool is approximation of real valued functions analytic in rectangular domains of the complex plane by continued g-fractions of Wall. As an application, the Sundman-Poincaré method in the Newtonian three-body problem is revisited and applications to collision detection problem are considered.

math.DS↗
arXiv2006

On some exceptional cases in the integrability of the three-body problem

We consider the Newtonian planar three--body problem with positive masses $m_1$, $m_2$, $m_3$. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases $ \sum m_i m_j/(\sum m_k)^2= 1/3$, $2^3/3^3$, $2/3^2$ where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in our previous papers and based of the Morales-Ramis-Ziglin approach.

math.DS↗
arXiv2006

On the analytic non-integrability of the Rattleback problem

We establish the analytic non-integrability of the nonholonomic ellipsoidal rattleback model for a large class of parameter values. Our approach is based on the study of the monodromy group of the normal variational equations around a particular orbit. The embedding of the equations of the heavy rigid body into the rattleback model is discussed.

math.DS↗
arXiv2004

On the convergence of continued fractions at Runckel's points and the Ramanujan conjecture

We consider the limit periodic continued fractions of Stieltjes $$ \frac{1}{1-} \frac{g_1 z}{1-} \frac{g_2(1-g_1) z}{1-} \frac{g_3(1-g_2)z}{1-...,}, z\in \mathbb C, g_i\in(0,1), \lim\limits_{i\to \infty} g_i=1/2, \quad (1) $$ appearing as Shur--Wall $g$-fraction representations of certain analytic self maps of the unit disc $|w|< 1$, $w \in \mathbb C$. We precise the convergence behavior and prove the general convergence [2, p. 564 ] of (1) at the Runckel's points of the singular line $(1,+\infty)$ It is shown that in some cases the convergence holds in the classical sense. As a result a counterexample to the Ramanujan conjecture [1, p. 38-39] stating the divergence of a certain class of limit periodic continued fractions is constructed.

math.CV↗
arXiv2000

On the algebraic invariant curves of plane polynomial differential systems

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $ω=dx/P=dy/Q$. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate was already found by D. Cerveau and A. Lins Neto [Ann. Inst. Fourier Grenoble 41, 883-903] in a different way.

math.DS↗
arXiv2000

On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations

We consider systems of ordinary differential equations with quadratic homogeneous right hand side. We give a new simple proof of a result already obtained in [8,10] which gives the necessary conditions for the existence of polynomial first integrals. The necessary conditions for the existence of a polynomial symmetry field are given. It is proved that an arbitrary homogeneous first integral of a given degree is a linear combination of a fixed set of polynomials.

math.DS↗
arXiv2000

The meromorphic non-integrability of the three-body problem

We study the planar three-body problem and prove the absence of a complete set of complex meromorphic first integrals in a neighborhood of the Lagrangian solution. We use the Ziglin's method and study the monodromy group of the corresponding normal variational equations.

math.DS↗