Regular and Chaotic Dynamics最新文献

This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski’s seminal work on the motions of a rigid body around a fixed point under the influence of gravity.The point of departure for understanding Kowalewski’s workbegins with Kirchhoff’s model for the equilibrium configurations of an elastic rod in ({mathbb{R}}^{3}) subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I. V. Komarov and V. B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere (S^{3}) and the hyperboloid (H^{3}) [17] and, secondly, it showsthat these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of (so(4,mathbb{C})) generated byan affine quadratic Hamiltonian (H) (Kirchhoff – Kowalewski type).

The paper shows that the passage to complex variablesis synonymous with the representation of (so(4,mathbb{C})) as (sl(2,mathbb{C})times sl(2,mathbb{C})) and the embedding of (H) into (sp(4,mathbb{C})), an important intermediate step towards uncovering the origins of Kowalewski’s integral. There is a quintessential Kowalewski type integral of motion on (sp(4,mathbb{C})) that appears as a spectral invariant for the Poisson system associated with a Hamiltonian (mathcal{H}) (a natural extension of (H)) that satisfies Kowalewski’s conditions.

The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski’s ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.

The Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid are considered. The Cauchy problems for all these partial differential equations are not solved by theinverse scattering transform. Reductions of these equations to nonlinear ordinary differentialequations do not pass the Painlevé test. However, there are local expansions of the generalsolutions in the Laurent series near movable singular points.We demonstrate that the obtained information from the Painlevé test for reductions ofnonlinear evolution dissipative differential equations can be used to construct thenonautonomous first integrals of nonlinear ordinary differential equations. Taking intoaccount the found first integrals, we also obtain analytical solutions of nonlinear evolutiondissipative differential equations. Our approach is illustrated to obtain thenonautonomous first integrals for reduction of the Korteweg – de Vries – Burgers equation,the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation andthe nonlinear differential equation for description surface waves in a convecting fluid.The obtained first integrals are used to construct exact solutions of the above-mentionednonlinear evolution equations with as many arbitrary constants as possible. It is shown thatsome exact solutions of the equation for description of nonlinear waves in a convectingliquid are expressed via the Painlevé transcendents.

This paper establishes the existence of periodic response solutions for multi-dimensional 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 circular restricted three-body problem is used as an approximate model in space mission planning. Its periodic solutions around equilibrium points, which are referred to as the libration points, are utilized for exploration of possible spacecraft trajectories in the preliminary stages of mission design. In this paper, a numerical methodology for periodic libration point orbits (LPOs) computation is introduced and applied to the construction and study of N-periodic (up to (N=6)) quasi-planar orbit families in the Earth-Moon system. The stability and the bifurcation points of these families are determined. The proposed method is based on an iterative algorithm searching for the initial state vector of periodic LPOs, which allows computing unstable long-periodic and large-amplitude orbits. The method is suited to perform a straightforward switch to bifurcating branches of periodic orbits.

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 dimensionlessparameters: (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 – Horimethod.

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 consider motion of a material point placed in a constant homogeneous magnetic field in (mathbb{R}^{n}) and also motion restricted to the sphere (S^{n-1}).While there is an obvious integrability of the magnetic system in (mathbb{R}^{n}), the integrability of the system restricted to the sphere (S^{n-1}) is highly nontrivial. We provecomplete integrability of the obtained restricted magnetic systems for (nleqslant 6). The first integrals of motion of the magnetic flows on the spheres (S^{n-1}), for (n=5) and (n=6), are polynomials of degree(1), (2), and (3) in momenta.We prove noncommutative integrability of the obtained magnetic flows for any (ngeqslant 7) when the systems allow a reduction to the cases with (nleqslant 6). We conjecture that the restricted magnetic systems on (S^{n-1}) are integrable for all (n).

This paper addresses the spatial restricted elliptic problemof three bodies (material points) gravitating toward each other under Newton’s law ofgravitation. The eccentricity of the orbit of the main attracting bodies is assumed to besmall, and nonlinear oscillations ofa passively gravitating body near a Lagrangian triangular libration point are studied.It is assumed that in the limiting case of the circular problem the ratioof the frequency of rotation of the main bodies about their common center of massto the value of one of the frequencies of small linear oscillations of the passive bodyis exactly equal to three. A detailed analysis is made of two different particular cases ofinfluence of the three-dimensionality of theproblem on the characteristics of nonlinear oscillations of the passive body.

We show that the equations of motion of a rigid body about a fixed point in the Newtonianfield with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schrödinger operators are discussed.

We explicitly compute the maximal Lyapunov exponent for a switched system on (mathrm{SL}_{2}(mathbb{R})) and the corresponding switching function which realizes the maximal exponent. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.