Regular and Chaotic Dynamics最新文献

We provide a new expansion of the Fourier coefficient of the perturbing function of thePCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formulafor the coefficient in the region of application of KAM theory (i. e., small value of eccentricity and semimajoraxis. See, e. g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes ((m,k)inmathbb{Z}^{2}) and the presence of common zeros as functions of actions between coefficients relative to modes ((m,k)),((2m,2k)) and ((m,k)),((2m,2k)),((3m,3k)).Thanks to the previous expansion, this numerical analysis is done up to order (60) in the powerof eccentricity and semimajor axis. This is thefirst step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called “non-torus” set from (O(1-sqrt{varepsilon})) (implied by standard KAM theory) to (O(1-varepsilon|logvarepsilon|^{c})) for some (c>0).

Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a (SE(2)) Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors.I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that surfaces of revolution are special: the additional integral of motion is elementary, while for marble rolling it requires special functions. I use the term “Nose function” to refer to their expression (N(theta)=big{(}I_{1}cos^{2}theta+I_{3}sin^{2}theta+mz_{C}^{2}(theta)big{)}^{1/2}) where (theta) is the nutation and (z_{C}(theta)) is the center of mass height. (N(theta)) appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles (phi text{(yaw)}, psi text{ roll and } theta). The reduction to 1 DoF is done in two stages: first, reduction by the group (SE(2)={(x,y,phi)}) to (T^{*}S^{2}) with almost symplectic 2-form (Omega_{NH}=dp_{theta}wedge dtheta+dp_{psi}wedge dpsi+Jcdot K). The semibasic term is (Jcdot K=-p_{psi}(dlogbig{(}N(theta)big{)}wedge dpsi). It follows that (Omega_{NH}) is conformally symplectic in the sense that (dleft(frac{1}{N}Omega_{NH}right)=0.)The conserved quantity due to the (S^{1}) symmetry about the body axis is (ell=N(theta)sin^{2}thetadot{psi}), yielding the desired reduction to ((theta,p_{theta})). Further simplification results by taking the new time (dt=sqrt{B(theta)}dtau, text{ with } B=I_{1}+m|CP|^{2}) where (P=(x,y)) is the point of contact.One gets finally (H=frac{1}{2}tilde{p}^{2}_{theta}+V(theta),V(theta)=ell^{2}/2sin^{2}theta+mgz_{C}(theta) text{ with } tilde{p}_{theta}=p_{theta}/sqrt{B}) and usual symplectic form (dtilde{p}_{theta}wedge dtheta). The moments of inertia (I_{1},I_{3}) reappear in the reconstruction.As an example, very basic observations are presented for the torus.A detailed study wasjust finished by A. Kilin and E. Pivovarova in [3].

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity.We call them Baker functions.The construction is based on the Abel – Jacobi map, which allows us to identify the field of meromorphic functions on the Jacobian variety of the curve with the field of meromorphic functions on the symmetric product of the curve.In our previous paper, a solution to the KP equation was constructed in terms of the Baker function.This paper is devoted to the properties of the Baker functions.In this paper, we construct an entire function whose second logarithmic derivatives are the Baker functions.We prove that the power series expansion of the entire function around the origin is determined only by the coefficients of the defining equation of the curve and a branch point of the curve algebraically.We also describe the quasi-periodicity of the entire function and express the entire function in terms of the Riemann theta function.

For a (mathcal{PT})-symmetric periodic Schrödinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.

A Jacobi field is a potential force field whose potential is a homogeneous function of degree(-2). The problem of the motion of a particle in such a field admits an additional integralquadratic in velocities. It can be used to reduce the number of degrees of freedom and topass to the study of a reduced system with spherical configuration space. These resultsare extended to the more general case of the motion of a particlein spaces of constant curvature. An analysis is made of particle motionon a cone whose vertex coincides with the singular point of theJacobi potential. A lower estimate of the distance from the moving particle to the vertex of the cone is given. This approach is also applicable to a more general case where the charged particleis additionally located in the magnetic field of a monopole. A billiard inside the conewith a particle bouncing elastically off its boundary isconsidered.

A system of two diffusively coupled Bautin (generalized Stuart – Landau) oscillators is considered. Using a specially designed reduced system, the existence and stability of homogeneous solutions are investigated. Such solutions represent oscillatory regimes in which the amplitudes of different oscillators are identical to each other and coincide at any given time. A partition of “coupling strength — frequency mismatch” parameter plane into regions with different dynamical behavior of the oscillators is obtained. It is established that the phase space of the system has a foliation into a continuum of two-dimensional invariant manifolds. It is shown that oscillation quenching in the system, in contrast to systems of diffusively coupled Stuart – Landau oscillators, is determined by new mechanisms and is associated with the bifurcation of merger of invariant tori and the saddle-node (tangent) bifurcations of limit cycles. At the same time, the quenching does not occur monotonously with a change in the coupling strength, but abruptly, and the critical value of the coupling strength depends on the frequency mismatch between the oscillators.

We consider the problem of spectral stability of traveling wave solutions (u=gamma(x-Wt))for a system of viscous conservation laws (partial_{t}u+partial_{x}F(u)=partial^{2}_{x}u).Such solutions correspond to heteroclinic trajectories (gamma) of a system of ODEs.In general conditions of stability can be obtained only numerically.We propose a model class of piecewise linear (discontinuous) vector fields (F) for which thestabilityproblem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and constructseveral examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.

A treatment is given of the spatial restricted elliptic problem of threebodies interacting under Newtonian gravity. The problem depends on two parameters:the ratio between the masses of the main attracting bodies and the eccentricity of theirelliptic orbits. The eccentricity is assumed to be small. Nonlinear equations of motion ofthe test mass near a triangular libration point are analyzed. It is assumed that theparameters of the problem lie on the curves of third-order resonances corresponding tothe planar restricted problem.In addition to these resonances (their number is equal to five), the spatial problemhas a resonance that takes place at any parameter values since thethe frequency of small linear oscillations of the test mass along the axis perpendicular tothe plane of the orbit of the main bodies is equal to the frequency of Keplerian motion ofthese bodies.In this paper, the normal form of the Hamiltonian function of perturbed motionthrough fourth-degree terms relative to deviations from the libration point is obtained.Explicit expressions for the coefficients of normal form up to and including the seconddegree of eccentricity are found.

The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced.It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines theresults of the problem of gravitation (without vibration) and that of vibration(ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.

For an autonomous dynamical system of (n) differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of ((n-1)) integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in (2n)-dimensional phase space only (n) integrals are sufficient as equations are generated by 1 function — the Hamiltonian.

There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations (ddot{boldsymbol{q}}=boldsymbol{M}(boldsymbol{q})), (boldsymbol{q}inmathbb{R}^{n}), which allows for integration by quadratures.

In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.