Unique Stable Point Research Articles

Dynamical analysis in scalar field cosmology

We give a general method to find exact cosmological solutions for scalar-field dark energy in the presence of perfect fluids. We use the existence of invariant transformations for the Wheeler De Witt (WdW) equation. We show that the existence of a point transformation under which the WdW equation is invariant is equivalent to the existence of conservation laws for the field equations, which indicates the existence of analytical solutions. We extend previous work by providing exact solutions for the Hubble parameter and the effective dark-energy equation of state parameter for cosmologies containing a combination of perfect fluid and a scalar field whose self-interaction potential is a power of hyperbolic functions. We find solutions explicitly when the perfect fluid is radiation or cold dark matter and determine the effects of non-zero spatial curvature. Using the Planck 2015 data, we determine the evolution of the effective equation of state of the dark energy. Finally, we study the global dynamics using dimensionless variables. We find that if the current cosmological model is Liouville integrable (admits conservation laws) then there is a unique stable point which describes the de-Sitter phase of the universe.

First exit times for Lévy-driven diffusions with exponentially light jumps

We consider a dynamical system described by the differential equation Ẏt=−U'(Yt) with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity ɛ to obtain the stochastic differential equation dXtɛ=−U'(Xt−ɛ) dt+ɛ dLt. The process L is a symmetric Lévy process whose jump measure ν has exponentially light tails, ν([u, ∞))∼exp(−uα), α>0, u→∞. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (−1, 1). In the small noise limit ɛ→0, the law of the first exit time σx, x∈(−1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index α=1, namely, ln Eσ∼ɛ−α for 0<α<1, whereas ln Eσ∼ɛ−1|ln ɛ|1−1/α for α>1.