Lie transforms and the Hamiltonization of non-Hamiltonian systems

To develop the perturbation solution of the non-Hamiltonian system of differential equationsy=g(y, t; ), it is sufficient to obtain the perturbation solution of a Hamiltonian system represented by the HamiltonianK=Y·g(y, t; ) which is linear in the adjoint vectorY. This Hamiltonization allows the direct use of the perturbation methods already established for Hamiltonian systems. To demonstrate this fact, a Hamiltonian algorithm developed by this author and based on the Lie-Deprit transform is applied to the Hamiltonized system and is shown to be equivalent to the application of the non-Hamiltonian form of this same algorithm to the original non-Hamiltonian system.     

The equivalence of von Zeipel mappings and Lie transforms

An order independent method is used to identify the transformation obtained from Von Zeipel's method with that obtained from Lie transforms. The correspondence between the generators is given in explicit form.     

Hyperboloidal precession of a dynamically symmetric satellite. Construction of normal forms of the Hamiltonian

The Norma specialized program package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is used in studies of small-amplitude periodic motions in the neighbourhood of regular precessions of a dynamically symmetric satellite on a circular orbit. The case of hyperboloidal precession is considered. Analytical expressions for normal forms and generating functions depending on frequencies of the system as on parameters are derived. Possible resonances are considered in particular. The 6th order of normalization is achieved. Though the intermediate analytical expressions occupy megabytes of computer's main memory, final ones are quite compact. Obtained analytical expressions are applied to the analysis of stability of small-amplitude periodic motions in the neighbourhood of hyperboloidal precession.     

Singularly weighted symplectic forms and applications to asteroid motion

New techniques to study Hamiltonian systems with Hamiltonian forcing are proposed. They are based on singularly weighted symplectic forms and transformations which preserve these forms. Applications pertaining to asteroid motion are outlined. These involve the presence of both Jupiter and Saturn.     

The concept of integrability on cantor sets for Hamiltonian systems

Differentiable Hamiltonian systems close to nondegenerate, integrable Hamiltonian systems are shown to be integrable on a Cantor set in the sense that on some Cantor set, (i) the invariant KAM-tori form a smooth foliation, (ii) there exist smooth, independent integrals in involution, and (iii) there exists a complete solution of the Hamilton Jacobi equation. The complement of the Cantor set is shown to be small in measure.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Wavelet analysis and applications to some dynamical systems

For about five years the wavelet transform technique has given interesting results in a great number of different fields such as mathematics, quantum mechanics, signal analysis and image processing, cluster analysis... The wavelet transform appears as a new time-frequency method which is particularly well-suited to detect and to localize discontinuities and scaling behaviours in signals. The main properties of the wavelet transform and its improvements over classical analyzing methods are summarized. Some results among the first applications to the dynamical systems are presented: solution of partial differential equations, fractal and turbulence characterization, and asteroid family determination from cluster analysis.     

A note on the application of a generalized canonical approach to non-linear Hamiltonian systems

In the author's treatment of the ideal resonance problem (1988), a non-canonical transformation was employed to bring the original Hamiltonian to a form amenable to the use of standard action-angle variables. Though the strictly Hamiltonian form of equations of motion was thus compromised, their general form was maintained, allowing transformation of the system to arbitrary order and forestalling the introduction of elliptic functions until a final explicit integration required in this approach. The general theory of such transformations is presented, and some points regarding their application are discussed, leading to the conclusion that the approach is practically limited to systems with a single degree of freedom only.     

The stability and uniqueness of coronal loops

A hydrodynamic model of high resolution is used to examine the stability of coronal loops to finite amplitude perturbations. The loop is heated by means of a low-amplitude energy input and its subsequent dynamic relaxation is followed.Firstly, the initial atmosphere is generated by solving the time independent form of the hydrodynamic equations. It is shown that the loop structure depends critically on the balance between the radiative losses and the quiescent heating at the base of the transition zone, i.e. on the concavity of the temperature profile in this region. This result already anticipates the need for high spatial resolution across the model transition zone.The dynamic evolution of the loop is then investigated for two classes of lower boundary conditions. In one case the chromospheric temperature is fixed throughout the simulation; in the other the low chromosphere is represented by a rigid insulating barrier. In both cases the loop is found to be stable: The loop is also unique to the extent that it relaxes to a state which is physically indistinguishable from its initial configuration. It is pointed out however, that a loop whose chromosphere is only marginally stable can evolve dynamically away from the initial static configuration.Finally, the observational consequences of the analysis are discussed. The differential emission measure profile is found to change its form as the loop cools, firstly, through an evaporative phase in which the coronal density increases; secondly, through a quasi-steady relaxation in which the enhanced coronal density gradually drains away to the chromosphere. This behaviour represents a possible observational test of the model.     

On chaos control of nonlinear fractional chaotic systems via a neural collocation optimization scheme and some applications

In this paper, we study chaos control of a class of fractional-order chaotic systems where the dynamic control system depends on the Caputo fractional derivatives. We first propose an infinite horizon optimal control problem related to the given fractional chaotic system. With the help of an approximation, we replace the Caputo derivative to integer order derivative. We then convert the obtained infinite horizon optimal control problem into an equivalent finite horizon one. Based on the Pontryagin minimum principle (PMP) for optimal control problems and by constructing an error function, we define an unconstrained minimization problem. In the optimization problem, we use trial solutions for state, costate and control functions where these trial solutions are constructed by using a two-layered perceptron neural network. A learning procedure of the proposed neural network with convergence properties are also given. Some numerical results are introduced to explain our main results. Three applicable examples on chaos control of Malkus waterwheel, finance fractional chaotic models and fractional-order Geomagnetic Field models are finally considered.     

A numerical study of the size of the homoclinic tangle of hyperbolic tori and its correlation with Arnold diffusion in Hamiltonian systems

Using a three degrees of freedom quasi-integrable Hamiltonian as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to single resonances. We measure an exponential dependence of the splitting of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. We also detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. The variation of the size of the homoclinic tangle as well as the topological transitions turn out to be correlated to the speed of Arnold diffusion.     

The INPOP10a planetary ephemeris and its applications in fundamental physics

Compared to the previous INPOP versions, the INPOP10a planetary and lunar ephemeris has several improvements. For the planets of our Solar System, no big change was brought in the dynamics but improvements were implemented in the fitting process, the data sets used in the fit and in the selection of fitted parameters. We report here the main characteristics of the planetary part of INPOP10a like the fit of the product of the solar mass with the gravitational constant ( GM_\odot{{\rm GM}_{\odot}}) instead of the astronomical unit. Determinations of PPN parameters as well as adjustments of the Sun J2 and of asteroid masses are also presented. New advances of nodes and perihelia of planets were also estimated and are given here. As for INPOP08, INPOP10a provides to the user, positions and velocities of the planets, the Moon, the rotation angles of the Earth and the Moon as well as TT-TDB Chebyshev polynomials at .     

The applications of radio-wave scattering theory to radio-meteor observations

The observation of radio meteors is an important technique providing information on the neutral atmosphere in the height range 80–110 km. Reliable interpretations of such observations require a knowledge of the process of radio-wave reflection from meteoric ionization. It is shown that recent theoretical results modify the interpretations based on approximate scattering models whose use may result in serious errors in the values of such parameters as ambipolar diffusion coefficients, diffusion scale heights, neutral atmosphere wind velocities and associated wind shears.     

Multi-cloud model (MCM) method and its applications to the study on asymmetric spectral profiles

A method of multi-cloud model (MCM) is proposed in this paper in order to research asymmetric profiles of spectral line formed by solar discrete active objects aligned along the line-of-sight difection. Based on the MCM method, under the conditions of certain assumptions and approximations, the line-of-sight velocityV and three other physical parameter approximation values, (i.e. Doppler width _D , source functionS and optical depth at line center ₀) within different clouds may be derived simultaneously by fitting both profiles theoretical and observational. An application example of the method withm=3 shows that MCM method is suitable to measure the velocity fields of multi-object at the same time from their non-Gaussian profiles of complex. TheV and _D derived from the method are reliable,S and ₀ are approximation. An influence of the variations of initial values in the parameter on the solution is given as well.     

e-VLBI技术及其在卫星定轨中的应用概况

e-VLBI技术继承了VLBI本身具有的极高角分辨率,且其利用高速通讯网络传送观测数据,能够快速得到观测结果,这些优点对于卫星的快速测定轨和提高卫星现有的定轨精度是十分有利的。通过对e-VLBI技术特点的分析及其发展历程的回顾,并结合卫星的差分VLBI观测原理,重点阐述和讨论了e-VLBI技术在我国的应用现状、发展及要求,展望了我国未来e-VLBI的发展前景。