Dynamical system approach to cosmological models with a varying speed of light

The methods of dynamical systems have been used to study homogeneous and isotropic cosmological models with a varying speed of light (VSL). We propose two methods for the reduction of the dynamics to the form of planar Hamiltonian dynamical systems for models with a time dependent equation of state. The solutions are analyzed on two-dimensional phase space in the variables $(x,\mathrm{ẋ})$ where x is a function of a scale factor a. Then we show how the horizon problem may be solved on some evolutional paths. It is shown that the models with a negative curvature overcome the horizon and flatness problems. The presented method of reduction can be adapted to the analysis of the dynamics of the Universe with the general form of the equation of state $p=\gamma \mathrm{}\left(a\right)\mathrm{}\epsilon .$ This is demonstrated using as an example the dynamics of VSL models filled with a noninteracting fluid. We demonstrate a new type of evolution near the initial singularity caused by a varying speed of light. Singularity-free oscillating universes are also admitted for a positive cosmological constant. We consider a quantum VSL Friedmann-Robertson-Walker closed model with radiation and show that the highest tunneling rate occurs for a constant velocity of light if $c\left(a\right)\mathrm{}\propto a^n$ and $-1<n<~0.$ It is also proved that the class of models considered is structurally unstable for the case of $n<\mathrm{}0.\mathrm{}$