arXiv subjects
Publications and source records attributed to A. V. Toporensky.
We have investigated the decay statistics of triple systems with different masses in Newtonial dynamics. We demonstrate that in a broad interval of mass ratios this statistics has good approximation by power-law tails. The power indices do not show any significant dependence on mass ratios.
We investigated a flat multidimensional cosmological model in Gauss-Bonnet gravity in presence of a matter in form of perfect fluid. We found analytically new stationary regimes (these results are valid for arbitrary number of spatial dimensions) and studied their stability by means of numerical recipes in 4+1-dimensional case. In the vicinity of the stationary regime we discovered numerically another non-singular regime which appears to be periodical. Finally, we demonstrated that the presence of matter in form of a perfect fluid lifts some constraints on the dynamics of the 4+1-dimensional model which have been found earlier.
We investigate dynamics of (4+1) and (5+1) dimensional flat anisotropic Universe filled by a perfect fluid in the Gauss-Bonnet gravity. An analytical solutions valid for particular values of the equation of state parameter $w=1/3$ have been found. For other values of $w$ structure of cosmological singularity have been studied numerically. We found that for $w > 1/3$ the singularity is isotropic. Several important differences between (4+1) and (5+1) dimensional cases are discussed.
We investigate a flat anisotropic (5+1)-dimensional cosmological model in the presence of the Gauss-Bonnet (GB) contribution in addition to usual Einstein term in the action. We compared it with (4+1)-dimensional case and found a substantial difference in corresponding cosmological dynamics. This difference is manifested in the probability of the model to have smooth transition from GB-term-dominated to Einstein-term-dominated phases--this probability in a reasonable measure on the initial condition space is almost zero for (4+1) case and about 60% for (5+1) case. We discuss this difference as well as some features of the dynamics of the considered model.
We consider perturbations of the complex quadratic map $ z \to z^2 +c$ and corresponding changes in their quasi-Mandelbrot sets. Depending on particular perturbation, visual forms of quasi-Mandelbrot set changes either sharply (when the perturbation reaches some critical value) or continuously. In the latter case we have a smooth transition from the classical form of the set to some forms, constructed from mostly linear structures, as it is typical for two-dimensional real number dynamics. Two examples of continuous evolution of the quasi-Mandelbrot set are described.
In this paper, we analyze the viability of a vacuum Gauss-Bonnet cosmology by examining the dynamics of the homogeneous and anisotropic background in 4+1 dimensions. The trajectories of the system either originate from the standard singularity or from non-standard type, the later is characterized by the divergence of time derivative of the Hubble parameters for its finite value. At the onset, the system should relax to Einstein phase at late times as the effect of Gauss-Bonnet term becomes negligible in the low energy regime. However, we find that most of the trajectories emerging from the standard big-bang singularity lead to future re-collapse whereas the system beginning its evolution from the non-standard singularity enters the Kasner regime at late times. This leads to the conclusion that the measure of trajectories giving rise to a smooth evolution from a standard singularity to the Einstein phase is negligibly small for generic initial conditions.
Quadratic curvature corrections to Einstein-Hilbert action lead in general to higher-order equations of motion, which can induced instability of some unperturbed solutions of General Relativity. We study conditions for stability of de Sitter cosmological solution. We argue that simple form of this condition known for FRW background in 3+1 dimensions changes seriously if at least one of these two assumptions is violated. In the present paper the stability conditions for de Sitter solution have been found for multidimensional FRW background and for Bianchi I metrics in 3+1 dimensions.
We investigate the cosmological dynamics of a brane Universe when quantum corrections from vacuum polarization are taken into account. New vacuum de Sitter points existing on Randall-Sundrum brane are described. We show also that quantum correction can destroy the DGP de Sitter solution on induced gravity brane.
We investigate the possibility for a flat Bianchi I brane Universe to recollaps due to the presence of a negative "dark radiation" and an anisotropic stress in the form of a homogeneous magnetic field, localized on the brane.
We investigate the dynamics of a flat isotropic brane Universe with two-component matter source: perfect fluid with the equation of state $p=(γ-1) ρ$ and a scalar field with a power-law potential $V \sim ϕ^α$. The index $α$ can be either positive or negative. We describe solutions for which the scalar field energy density scales as a power-law of the scale factor (so called scaling solutions). In the nonstandard brane regime when the brane is driven by energy density square term these solutions are rather different from their analogs in the standard cosmology. A particular attention is paid to the inverse square potential. Its dynamical properties in the nonstandard brane regime are in some sense analogous to those of the exponential potential in the standard cosmology. Stability analysis of the scaling solutions are provided. We also describe solutions existing in regions of the parameter space where the scaling solutions are unstable or do not exist.
Several aspects of scalar field dynamics on a brane which differs from corresponding regimes in the standard cosmology are investigated. We consider asymptotic solution near a singularity, condition for inflation and bounces and some detail of chaotic behavior in the brane model. Each results are compared with those known in the standard cosmology.
The generality of inflation in closed FRW Universe is studied for the models with a scalar field on a brane and with a complex scalar field. The results obtained are compared with the previously known results for the model with a scalar field and a perfect fluid. The influence of the measure chosen in the initial condition space on the ratio of inflationary solution is described.
We investigate the dynamics of a Bianchi I brane Universe in the presence of a nonlocal anisotropic stress ${\cal P}_{μν}$ proportional to a "dark energy" ${\cal U}$. Using this ansatz for the case ${\cal U} > 0$ we prove that if a matter on a brane satisfies the equation of state $p=(γ-1)ρ$ with $γ\le 4/3$ then all such models isotropize. For $γ> 4/3$ anisotropic future asymptotic states are found. We also describe the past asymptotic regimes for this model.
The shear dynamics in Bianchi I cosmological model on the brane with a perfect fluid (the equation of state is $p=(γ-1) μ$) is studied. It is shown that for $1 < γ< 2$ the shear parameter has maximum at some moment during a transition period from nonstandard to standard cosmology. An exact formula for the matter density $μ$ in the epoch of maximum shear parameter as a function of the equation of state is obtained.
The results on chaos in FRW cosmology with a massive scalar field are extended to another scalar field potential. It is shown that for sufficiently steep potentials the chaos disappears. A simple and rather accurate analytical criterion for the chaos to disappear is given. On the contrary, for gently sloping potentials the transition to a strong chaotic regime can occur. Two examples, concerning asymptotically flat and Damour-Mukhanov potentials are given.
The dynamics of closed scalar field FRW cosmological models is studied for several types of exponentially and more than exponentially steep potentials. The parameters of scalar field potentials which allow a chaotic behaviour are found from numerical investigations. It is argued that analytical studies of equation of motion at the Euclidean boundary can provide an important information about the properties of chaotic dynamics. Several types of transition from chaotic to regular dynamics are described.
We investigate FRW cosmological solutions in the theory of modulus field coupled to gravity through a Gauss-Bonnet term. The explicit analytical forms of nonsingular asymptotics are presented for power-law and exponentially steep modulus coupling functions. We study the influence of modulus field potential on these asymptotical regimes and find some forms of the potential which do not destroy the nonsingular behavior. In particular, we obtain that exponentially steep coupling functions arising from the string theory do not allow nonsingular past asymptotic unless modulus field potential tends to zero for modulus field $ψ\to \pm \infty$. Finally, the modification of the chaotic dynamics in the closed FRW universe due to presence of the Gauss-Bonnet term is discussed.
The chaotic behavior in FRW cosmology with a scalar field is studied for scalar field potentials less steep than quadratic. We describe a transition to much stronger chaos for appropriate parameters of such potentials. The range of parameters which allows this transition is specified. The influence of ordinary matter on the chaotic properties of this model is also discussed.