We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the t
urbulence, also representing a non-trivial testing object for studying non-linear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory we reformulate the Lorenz system as a set of two second order non-linear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise, which accoun
ts for the general memory and retarded effects of the frictional force, and on the fluctuation-dissipation theorem. The presence of the memory effects influences the response of the disk to external random interactions, and modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The Power Spectral Distribution (PSD) of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the Intra Day Variability (IDV) of the Active Galactic Nuclei (AGN) may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability.
We present a general solution of the Einstein gravitational field equations for the static spherically symmetric gravitational interior spacetime of an isotropic fluid sphere. The solution is obtained by transforming the pressure isotropy condition,
a second order ordinary differential equation, into a Riccati type first order differential equation, and using a general integrability condition for the Riccati equation. This allows us to obtain an exact non-singular solution of the interior field equations for a fluid sphere, expressed in the form of infinite power series. The physical features of the solution are studied in detail numerically by cutting the infinite series expansions, and restricting our numerical analysis by taking into account only $n=21$ terms in the power series representations of the relevant astrophysical parameters. In the present model all physical quantities (density, pressure, speed of sound etc.) are finite at the center of the sphere. The physical behavior of the solution essentially depends on the equation of state of the dense matter at the center of the star. The stability properties of the model are also analyzed in detail for a number of central equations of state, and it is shown that it is stable with respect to the radial adiabatic perturbations. The astrophysical analysis indicates that this solution can be used as a realistic model for static general relativistic high density objects, like neutron stars.
New further integrability conditions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The first case corresponds to fixed functional forms of the coefficients $a(x)$ and $c(x)$ of the Riccati equation, and of the function $F(x)=a(x
)+[f(x)-b^{2}(x)]/4c(x)$, where $f(x)$ is an arbitrary function. The second integrability case is obtained for the reduced Riccati equation with $b(x)equiv 0$. If the coefficients $a(x)$ and $c(x)$ satisfy the condition $pm dsqrt{f(x)/c(x)}/dx=a(x)+f(x)$, where $f(x)$ is an arbitrary function, then the general solution of the reduced Riccati equation can be obtained by quadratures. The applications of the integrability condition of the reduced Riccati equation for the integration of the Schrodinger and Navier-Stokes equations are briefly discussed.
We study the dynamical evolution of a phase-transition-induced collapse neutron star to a hybrid star, which consists of a mixture of hadronic matter and strange quark matter. The collapse is triggered by a sudden change of equation of state, which r
esult in a large amplitude stellar oscillation. The evolution of the system is simulated by using a 3D Newtonian hydrodynamic code with a high resolution shock capture scheme. We find that both the temperature and the density at the neutrinosphere are oscillating with acoustic frequency. However, they are nearly 180$^{circ}$ out of phase. Consequently, extremely intense, pulsating neutrino/antineutrino fluxes will be emitted periodically. Since the energy and density of neutrinos at the peaks of the pulsating fluxes are much higher than the non-oscillating case, the electron/positron pair creation rate can be enhanced dramatically. Some mass layers on the stellar surface can be ejected by absorbing energy of neutrinos and pairs. These mass ejecta can be further accelerated to relativistic speeds by absorbing electron/positron pairs, created by the neutrino and antineutrino annihilation outside the stellar surface. The possible connection between this process and the cosmological Gamma-ray Bursts is discussed.
The generalized Chaplygin gas, which interpolates between a high density relativistic era and a non-relativistic matter phase, is a popular dark energy candidate. We consider a generalization of the Chaplygin gas model, by assuming the presence of a
bulk viscous type dissipative term in the effective thermodynamic pressure of the gas. The dissipative effects are described by using the truncated Israel-Stewart model, with the bulk viscosity coefficient and the relaxation time functions of the energy density only. The corresponding cosmological dynamics of the bulk viscous Chaplygin gas dominated universe is considered in detail for a flat homogeneous isotropic Friedmann-Robertson-Walker geometry. For different values of the model parameters we consider the evolution of the cosmological parameters (scale factor, energy density, Hubble function, deceleration parameter and luminosity distance, respectively), by using both analytical and numerical methods. In the large time limit the model describes an accelerating universe, with the effective negative pressure induced by the Chaplygin gas and the bulk viscous pressure driving the acceleration. The theoretical predictions of the luminosity distance of our model are compared with the observations of the type Ia supernovae. The model fits well the recent supernova data. From the fitting we determine both the equation of state of the Chaplygin gas, and the parameters characterizing the bulk viscosity. The evolution of the scalar field associated to the viscous Chaplygin fluid is also considered, and the corresponding potential is obtained. Hence the viscous Chaplygin gas model offers an effective dynamical possibility for replacing the cosmological constant, and to explain the recent acceleration of the universe.
