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Volume 19, Number 5

Volume 19, Number 5, 2014

O'Neil K. A.,  Cox-Steib N.
Generalized Adler–Moser and Loutsenko Polynomials for Point Vortex Equilibria
Abstract
Equilibrium configurations of point vortices with circulations of two discrete values are associated with the zeros of a sequence of polynomials having many continuous parameters: the Adler–Moser polynomials in the case of circulation ratio −1, and the Loutsenko polynomials in the case of ratio −2. In this paper a new set of polynomial sequences generalizing the vortex system to three values of circulations is constructed. These polynomials are extensions of the previously known polynomials in the sense that they are special cases of the latter when the third circulation is zero. The polynomials are naturally connected with rational functions that satisfy a second-order differential equation.
Keywords: point vortex, equilibrium, polynomial method
Citation: O'Neil K. A.,  Cox-Steib N., Generalized Adler–Moser and Loutsenko Polynomials for Point Vortex Equilibria, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 523-532
DOI:10.1134/S1560354714050013
Schmidt D. S.,  Vidal C.
Stability of the Planar Equilibrium Solutions of a Restricted $1+N$ Body Problem
Abstract
We started our studies with a planar Eulerian restricted four-body problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin and two masses of size $\mu$ rotate around it uniformly. The problem was studied in [3], where it was shown that there exist noncollinear equilibria, which are Lyapunov stable for small values of $\mu$. KAM theory is used to establish the stability of the equilibria. Our computations do not agree with those given in [3], although our conclusions are similar. The ERFBP is a special case of the $1+N$ restricted body problem with $N=2$. We are able to do the computations for any $N$ and find that the stability results are very similar to those for $N=2$. Since the $1+N$ body configuration can be stable when $N>6$, these results could be of more significance than for the case $N=2$.
Keywords: $1+N$ body problem, relative equilibria, normal form, KAM stability
Citation: Schmidt D. S.,  Vidal C., Stability of the Planar Equilibrium Solutions of a Restricted $1+N$ Body Problem , Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 533-547
DOI:10.1134/S1560354714050025
Caşu I.
Symmetries of the Maxwell–Bloch Equations with the Rotating Wave Approximation
Abstract
In this paper a symplectic realization for the Maxwell–Bloch equations with the rotating wave approximation is given, which also leads to a Lagrangian formulation. We show how Lie point symmetries generate a third constant of motion for the dynamical system considered.
Keywords: Maxwell–Bloch equations, symmetries, Hamiltonian dynamics, Lagrangian dynamics
Citation: Caşu I., Symmetries of the Maxwell–Bloch Equations with the Rotating Wave Approximation, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 548-555
DOI:10.1134/S1560354714050037
Chang D. E.
On the Method of Interconnection and Damping Assignment Passivity-Based Control for the Stabilization of Mechanical Systems
Abstract
Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetric interconnection submatrix. Reduction of the number of matching partial differential equations in IDA-PBC and simplification of the structure of the matching partial differential equations are achieved by eliminating the gyroscopic force from the matching partial differential equations. In addition, easily verifiable criteria are provided for Lyapunov/exponential stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with arbitrary degrees of underactuation and for all nonlinear controlled Hamiltonian systems with one degree of underactuation. A general design procedure for IDA-PBC is given and illustrated with examples. The duality of the new IDAPBC method to the method of controlled Lagrangians is discussed. This paper renders the IDA-PBC method as powerful as the controlled Lagrangian method.
Keywords: feedback control, stabilization, energy shaping, mechanical system
Citation: Chang D. E., On the Method of Interconnection and Damping Assignment Passivity-Based Control for the Stabilization of Mechanical Systems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 556-575
DOI:10.1134/S1560354714050049
Kudryashov N. A.,  Sinelshchikov D. I.
Special Solutions of a High-order Equation for Waves in a Liquid with Gas Bubbles
Abstract
A fifth-order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling wave solutions are constructed. Connection of selfsimilar solutions with Painlevé transcendents and their high-order analogs is discussed.
Keywords: waves in a liquid with gas bubbles, evolution equations, exact solutions, meromorphic solutions, fifth-order KdV equation, Kaup–Kupershmidt equation
Citation: Kudryashov N. A.,  Sinelshchikov D. I., Special Solutions of a High-order Equation for Waves in a Liquid with Gas Bubbles, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 576-585
DOI:10.1134/S1560354714050050
Fortunati A.,  Wiggins S.
Persistence of Diophantine Flows for Quadratic Nearly Integrable Hamiltonians under Slowly Decaying Aperiodic Time Dependence
Abstract
The aim of this paper is to prove a Kolmogorov type result for a nearly integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists in the possibility to choose an arbitrarily small decaying coefficient consistently with the perturbation size.
The proof, based on the Lie series formalism, is a generalization of a work by A. Giorgilli.
Keywords: Hamiltonian systems, Kolmogorov theorem, aperiodic time dependence
Citation: Fortunati A.,  Wiggins S., Persistence of Diophantine Flows for Quadratic Nearly Integrable Hamiltonians under Slowly Decaying Aperiodic Time Dependence, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 586-600
DOI:10.1134/S1560354714050062

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