A generalization of Sturm's comparison theorem

The Riccati equation method is used to establish a new comparison theorem for systems of two linear first order ordinary differential equation. This result is based on a, so called, concept of "null-classes", and is a generalization of Sturm's comparison theorem.Comment: 8 page

Oscillatory and non oscillatory criteria for linear four dimensional hamiltonian systems

The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of linear four dimensional hamiltonian systems. An oscillatory and two non oscillatory criteria are proved. On an example the obtained oscillatory criterion is compared with some well known results.Comment: 12 page

Stability criteria for second order linear ordinary differential equations

We use some properties of solutions of Riccati equation for establishing boundedness and stability criteria for solutions of second order linear ordinary differential equations. We show that the conditions on coefficients of the equations, appearing in the proven criteria, do not follow from the conditions, which ensure the application of the WKB approximation to the second order linear equations. On these examples we compare the obtained results wit the results obtained by the Liapunov and Bogdanov methods, by a method involving estimates of solutions in the Lozinski's logarithmic norms, and by the freezing method. We compare these results with the Wazevski's theorem as well.Comment: 15 page

Oscillatory criteria for the second order linear ordinary differential equations in the marginal sub extremal and extremal cases

The Riccati equation method is used to establish three new oscillatory criteria for the second order linear ordinary differential equations in the marginal, sub extremal and extremal cases.We show that the first of these criteria implies the J. Deng's oscillatory criterion. An extremal oscillatory condition for the Mathieu's equation is obtained. The obtained results are compared with some known oscillatory criteria.Comment: 18 page

New oscillation criteria for linear matrix Hamiltonian systems

By the use of Riccati equation technique new approaches (in particular a unitary transformation approach) are used to obtain new oscillation criteria for linear matrix Hamiltonian systems in a new direction. That direction is to break the positive definiteness restriction, imposed on one of coeffcients of the Hamiltonian system.Comment: 9 page

Phase Diagram for Spinning and Accreting Neutron Stars

Neutron star configurations are considered as thermodynamical systems for which a phase diagram in the angular velocity (Omega) - baryon number (N) plane is obtained with a dividing line N_{crit}(Omega) for quark core configurations. Trajectories of neutron star evolution in this diagram are studied for different scenarios defined by the external torque acting on the star due to radiation and/or mass accretion. They show a characteristic change in the rotational kinematics when the star enters the quark core regime. For isolated pulsars the braking index signal for deconfinement has been studied in its dependence on the mass of the star. Model calculations of the spin evolution of accreting low-mass X-ray binaries in the phase diagram have been performed for different values of the initial magnetic field, its decay time as well as initial mass and mass accretion rate. Population clustering of these objects at the line N_{crit}(Omega) in the phase diagram is suggested as an observable signal for the deconfinement phase transition if it exists for spinnning and accreting neutron stars.Comment: 20 pages, 10 figure

Deconfinement transition in rotating compact stars

Using the formalism of general relativity for axially symmetric gravitational fields and their sources - rotating compact stars - a perturbation theory with respect to angular velocity is developed and physical quantities such as mass, shape, momentum of inertia and total energy of the star are defined. The change of the internal structure of the star due to rotation has been investigated and the different contributions to the moment of inertia have been evaluated separately. Numerical solutions have been performed using a two-flavor model equation of state describing the deconfinement phase transition as constrained by the conservation of total baryon number and electric charge. During the spin down evolution of the rotating neutron star, below critical values of angular velocity a quark matter core can appear which might be detected as a characteristic signal in the pulsar timing. Within the spin-down scenario due to magnetic dipole radiation it is shown that the deviation of the braking index from $n=3$ could signal not only the occurrence but also the size of a quark core in the pulsar. A new scenario is proposed where, due to mass accretion onto the rapidly rotating compact star, a spin-down to spin-up transition might signal a deconfinement transition in its interior.Comment: 10 pages, 9 figures; A&A accepte

The behavior of solutions of the systems of two first order linear ordinary differential equations

The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of Leighton's theorem is obtained. Three new principles for the second order linear differential equations are derived. Stability and non conjugation criteria are proved for the mentioned systems, as well as estimates are obtained for the solutions of the last ones.Comment: 61 pages, 18 figures, 3 paragraph

On the stability of systems of two linear first-order ordinary differential equations

The Riccati equation method is used to establish some new stability criteria for systems of two linear first-order ordinary differential equations. It is shown that two of these criteria in the two dimensional case imply the Routh - Hurwitz's criterion.Comment: 14 page

Global solvability criteria for some classes of nonlinear second order ordinary differential equations

The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.Comment: 26 page