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# Volume 12, Number 1, 2007

 Danca M. On a Class of Non-Smooth Dynamical Systems: a Sufficient Condition for Smooth Versus Non-Smooth Solutions Abstract In this paper we present a possible classification of the elements of a class of dynamical systems, whose underlying mathematical models contain non-smooth components. For this purpose a sufficient condition is introduced. To illustrate and motivate this classification, three nontrivial and realistic examples are considered. Keywords: non-smooth dynamical systems, Filipov systems, differential inclusions Citation: Danca M., On a Class of Non-Smooth Dynamical Systems: a Sufficient Condition for Smooth Versus Non-Smooth Solutions, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 1-11 DOI:10.1134/S1560354707010017
 Cooley B.,  Newton P. Eigenvalue Distributions from Impacts on a Ring Abstract We consider the collision dynamics produced by three beads with masses ($m_1$, $m_2$, $m_3$) sliding without friction on a ring, where the masses are scaled so that $m_1 = 1/ \epsilon$, $m_2 = 1$, $m_3 = 1- \epsilon$, for $0 \leqslant \epsilon \leqslant 1$. The singular limits $\epsilon = 0$ and $\epsilon = 1$ correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. $0 < \epsilon < 1$, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits $\epsilon \to 0$ and $\epsilon \to 1$ for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit $\epsilon \to 0$, the distribution forms Gaussian peaks around the discrete limiting values $\pm1$, $\pm i$, with variances that scale in power law form as $\sigma^2 \sim \alpha \epsilon^\beta$. By contrast, the convergence in the limit $\epsilon \to 1$ to the discrete values $\pm 1$ is shown to follow a logarithmic power-law $\sigma^2 \sim \log (\epsilon^\beta)$. Keywords: impacts, eigenvalue spectrum, convergence rates Citation: Cooley B.,  Newton P., Eigenvalue Distributions from Impacts on a Ring, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 12-26 DOI:10.1134/S1560354707010029
 Santos A.,   Symmetry of the Restricted 4+1 Body Problem with Equal Masses Abstract We consider the problem of symmetry of the central configurations in the restricted 4+1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1-3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4+1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function $\varphi (s)=-s^k$, with $k<0$) which are crucial in the proof of the symmetry. Keywords: $n$-body problem, central configurations, symmetry Citation: Santos A.,  , Symmetry of the Restricted 4+1 Body Problem with Equal Masses , Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 27-38 DOI:10.1134/S1560354707010030
 Albouy A.,  Fu Y. Euler Configurations and Quasi-Polynomial Systems Abstract Consider the problem of three point vortices (also called Helmholtz' vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to $1/ r$, where $r$ is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to $r^b$, for any $b<0$. For $0 < b < 1$, the optimal upper bound becomes 5. For positive vorticities and any $b<1$, there are exactly 3 collinear normalized relative equilibria. The case $b=-2$ of this last statement is the well-known theorem due to Euler: in the Newtonian 3-body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework. Keywords: relative equilibria, point vortex, real solutions Citation: Albouy A.,  Fu Y., Euler Configurations and Quasi-Polynomial Systems, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 39-55 DOI:10.1134/S1560354707010042
 Cendra H.,  Diaz V. A. The Lagrange–D'Alembert–Poincaré Equations and Integrability for the Euler's Disk Abstract Nonholonomic systems are described by the Lagrange–D'Alembert's principle. The presence of symmetry leads, upon the choice of an arbitrary principal connection, to a reduced D'Alembert's principle and to the Lagrange–D'Alembert–Poincaré reduced equations. The case of rolling constraints has a long history and it has been the purpose of many works in recent times. In this paper we find reduced equations for the case of a thick disk rolling on a rough surface, sometimes called Euler's disk, using a 3-dimensional abelian group of symmetry. We also show how the reduced system can be transformed into a single second order equation, which is an hypergeometric equation. Keywords: nonholonomic systems, symmetry, integrability, Euler's disk Citation: Cendra H.,  Diaz V. A., The Lagrange–D'Alembert–Poincaré Equations and Integrability for the Euler's Disk, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 56-67 DOI:10.1134/S1560354707010054
 Borisov A. V.,  Mamaev I. S. Interaction between Kirchhoff vortices and point vortices in an ideal fluid Abstract We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found by the variable separation method . A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior. Keywords: vortex patch, point vortex, integrability Citation: Borisov A. V.,  Mamaev I. S., Interaction between Kirchhoff vortices and point vortices in an ideal fluid, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 68-80 DOI:10.1134/S1560354707010066
 Zotev D. B. On a Partial Integral which can be Derived from Poisson Matrix Abstract Consider a surface which is a common level of some functions. Suppose that this surface is invariant under a Hamiltonian system. The question is if a partial integral can be derived explicitly from the Poisson matrix of these functions. In some cases such an integral is equal to the determinant of the matrix. This paper establishes a necessary and sufficient condition for this to hold true. The partial integral that results is not trivial if the induced Poisson structure is non-degenerate at one point at least. Therefore, the invariant surface must be even-dimensional. Keywords: Hamiltonian system, invariant submainfold, partial integral, Poisson matrix determinant, trace matrix Citation: Zotev D. B., On a Partial Integral which can be Derived from Poisson Matrix, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 81-85 DOI:10.1134/S1560354707010078
 Bardin B. S. On Nonlinear Motions of Hamiltonian System in Case of Fourth Order Resonance Abstract We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3:1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession. Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angle variables Citation: Bardin B. S., On Nonlinear Motions of Hamiltonian System in Case of Fourth Order Resonance, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 86-100 DOI:10.1134/S156035470701008X
 Chaplygin S. A. On a Pulsating Cylindrical Vortex Abstract This text presents an English translation of the significant paper [5] on vortex dynamics published by outstanding Russian scientist S.A. Chaplygin (1869-1942), which seem to have escaped the attention of later investigators in this field. Chaplygin's solution includes that of an elliptical patch of uniform vorticity in an exterior field of pure shear. Although it was published more than a century ago, in our opinion it is still interesting and valuable. Keywords: two-dimentional vortex structures, Chaplygin–Kida elliptical vortex, Kirchhoff elliptical vortex Citation: Chaplygin S. A., On a Pulsating Cylindrical Vortex, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 101-116 DOI:10.1134/S1560354707010091

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