Volume 17, Numbers 3-4

Volume 17, Numbers 3-4, 2012
On the 70th birthday of professor A.P. Markeev

Citation: Anatoly Pavlovich Markeev. On his 70th Birthday, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 219-233
Markeev A. P.
In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases.
In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler’s case.
It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.
Keywords: rigid body dynamics, collision, periodic motion, stability
Citation: Markeev A. P., On the Dynamics of a Rigid Body Carrying a Material Point, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 234-242
Bardin B. S.,  Savin A. A.
We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position.
Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.
Keywords: Hamiltonian system, periodic motions, normal form, resonance, action–angle variables, orbital stability
Citation: Bardin B. S.,  Savin A. A., On the Orbital Stability of Pendulum-like Oscillations and Rotations of a Symmetric Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 243-257
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
Keywords: non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., How to Control Chaplygin’s Sphere Using Rotors, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 258-272
Celletti A.,  Lhotka C.
We consider a dissipative vector field which is represented by a nearly-integrable Hamiltonian flow to which a dissipative contribution is added. The vector field depends upon two parameters, namely the perturbing and dissipative parameters, and by a drift term. We study an $\mathcal{l}$-dimensional, time-dependent vector field, which is motivated by mathematical models in Celestial Mechanics. Assuming to start with non-resonant initial conditions, we provide the construction of the normal form up to an arbitrary order. To construct the normal form, a suitable choice of the drift parameter must be performed. The normal form allows also to provide an explicit expression of the frequency associated to the normalized coordinates. We also give an example in which we construct explicitly the normal form, we make a comparison with a numerical integration, and we determine the parameter values and the time interval of validity of the normal form.
Keywords: dissipative system, normal form, non-resonant motion
Citation: Celletti A.,  Lhotka C., Normal Form Construction for Nearly-integrable Systems with Dissipation, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 273-292
Ivanov A. P.
Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddlenode, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known dry friction oscillator, which serves as a popular model for the description of self-excited frictional oscillations of a braking system. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.
Keywords: nonsmooth dynamical systems, discontinuous bifurcations, oscillators with dry friction
Citation: Ivanov A. P., Analysis of Discontinuous Bifurcations in Nonsmooth Dynamical Systems, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 293-306
Lanchares V.,  Pascual A. I.,  Elipe A.
We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under low order resonances. For resonances of order bigger than two there are several results giving stability conditions, in particular one based on the geometry of the phase flow and a set of invariants. In this paper we show that this geometric criterion is still valid for low order resonances, that is, resonances of order two and resonances of order one. This approach provides necessary stability conditions for both the semisimple and non-semisimple cases, with an appropriate choice of invariants.
Keywords: nonlinear stability, resonances, normal forms
Citation: Lanchares V.,  Pascual A. I.,  Elipe A., Determination of Nonlinear Stability for Low Order Resonances by a Geometric Criterion, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 307-317
Lerman L. M.,  Turaev D. V.
We review results on local bifurcations of codimension 1 in reversible systems (flows and diffeomorphisms) which lead to the birth of attractor-repeller pairs from symmetric equilibria (for flows) or periodic points (for diffeomorphisms).
Keywords: reversible system, reversible diffeomorphism, bifurcation, symmetry, equilibrium state, periodic point
Citation: Lerman L. M.,  Turaev D. V., Breakdown of Symmetry in Reversible Systems, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 318-336
Maciejewski A. J.,  Przybylska M.
Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler–Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler–Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler–Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.
Keywords: rigid body, Euler–Poisson equations, solvability in special functions, differential Galois group
Citation: Maciejewski A. J.,  Przybylska M., Integrable Variational Equations of Non-integrable Systems, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 337-358
Neishtadt A. I.,  Su T.
Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. Discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances that is absent in the original system.
Keywords: systems with rotating phases, passage through a resonance, numerical integration, discretisation
Citation: Neishtadt A. I.,  Su T., On Phenomenon of Scattering on Resonances Associated with Discretisation of Systems with Fast Rotating Phase, Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 359-366
Zobova A. A.
The present comments on the recent paper by M. C. Ciocci, B.Malengier, B. Langerock, and B. Grimonprez "Towards a Prototype of a Spherical Tippe Top" (Journal of Applied Mathematics, 2012) [1] discuss theoretical results of the paper in the context of previous publications, which are apparently unknown to the above-mentioned authors.
Citation: Zobova A. A., Comments on the Paper by M.C. Ciocci, B. Malengier, B. Langerock, and B. Grimonprez "Towards a Prototype of a Spherical Tippe Top", Regular and Chaotic Dynamics, 2012, vol. 17, nos. 3-4, pp. 367-369

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