The stability of a class of reversible systems

The problem of the stability of the point of rest of an autonomous system of ordinary differential equations from a class of reversible systems [1] characterized by the critical case of m zero roots and n pairs of pure imaginary roots is considered. When there are no internal resonances [2, 3], the point of rest always has Birkhoff complete stability [2]. Internal resonances may lead to Lyapunov instability. The conditions of stability and instability of the model system when there are third-order resonances may be obtained from a criterion previously developed [4] for the case of pure imaginary roots. The results are used to analyse the stability of the translational-rotational motion of an active artificial satellite in a non-Keplerian circular orbit, including a geostationary satellite in any latitude [4, 5]. The region of stability of relative equilibria and regular precession of the satellite is constructed assuming a central gravitational field and the resonance modes are analysed.     

Resonant instability in multidimensional Lyapunov systems

The stability of a point of rest of a Lyapunov system, which describes the perturbed motion of a dynamical system with an arbitrary number of degrees of freedom, is investigated. It is assumed that the characteristic equation of the system of the first approximation has a number of imaginary roots, while the quadratic part of the integral is not sign-definite. The case of third-order resonance is considered, when the stability problem is solved by the first non-linear (quadratic) terms.     

Modeling the ionospheric propagation of acoustic gravity waves from the Tohoku tsunami of 2011

The observations of the upper atmosphere after the severe Tohoku earthquake of March 11, 2011 are analyzed. The observations cover the Hawaii region and span the time interval from about 7 h after the main shock to about half an hour before the arrival of the tsunami wave generated this earthquake. We explore the factors that are responsible for the emergence of the ionospheric response, which comprises a series of waves propagating ahead of the tsunami. The contribution of the moving tsunami wave into the formation of such a pre-tsunami response is established by the numerical simulation of the equations of geophysical hydrodynamics with two-dimensional geometry.