Transition to chaos and escape phenomenon in two degrees of freedom oscillator with a kinematic excitation

We study the dynamics of a two-degrees-of-freedom (two DOF) nonlinear oscillator representing a quartercar model excited by a road roughness profile. Modelling the road profile by means of a harmonic function we derive the Melnikov criterion for a system transition to chaos or escape. The analytically obtained estimations are confirmed by numerical simulations. To analyze the transient vibrations we used recurrences.Comment: 13 pages, 16 figures, in pres

Do sewn singularities falsify the Palatini cosmology?

We investigate further (cf. arXiv:1512.01199, JCAP01 (2016) 040) Starobinsky cosmological model $R+\gamma R^2$ in the Palatini formalism with Chaplygin gas and baryonic matter as a source. For this aim we use dynamical system theory. The dynamics is reduced to the 2D sewn dynamical system of a Newtonian type (a piecewise-smooth dynamical system). We classify all evolutional paths in the model as well as trajectories in the phase space. We demonstrate the presence of a degenerate freeze singularity (glued freeze type singularities) for the positive $\gamma$. In this case it is a generic feature of early evolution of the universe. We point out that a degenerate type III of singularity can be considered as an endogenous model of inflation between the matter dominating epoch and the dark energy phase. We also investigate cosmological models with negative $\gamma$. It is demonstrated that $\gamma$ equal zero is a bifurcation parameter and dynamics qualitatively changes in comparison to positive $\gamma$. Instead of the big bang the sudden bounce singularity of a finite scale factor appears and there is a generic class of bouncing solutions sewn along the line $a=a_{\text{sing}}$. And we argue that the presence of sudden singularities in an evolutional scenario of the Universe falsifies the negative $\gamma$ in the Palatini cosmology. Only very small values of $\Omega_{\gamma}$ parameter are admissible if we requires that agreements physics with the $\Lambda$CDM model. From the statistical analysis of astronomical observations, we deduce that the case of negative values of $\Omega_\gamma$ can be rejected even if it may fit better to the data.Comment: 26 pages, 15 figures, v3: change of title, more discussion on singularitie

Inflationary cosmology with Chaplygin gas in Palatini formalism

We present a simple generalisation of the $\Lambda$CDM model which on the one hand reaches very good agreement with the present day experimental data and provides an internal inflationary mechanism on the other hand. It is based on Palatini modified gravity with quadratic Starobinsky term and generalized Chaplygin gas as a matter source providing, besides a current accelerated expansion, the epoch of endogenous inflation driven by type III freeze singularity. It follows from our statistical analysis that astronomical data favors negative value of the parameter coupling quadratic term into Einstein-Hilbert Lagrangian and as a consequence the bounce instead of initial Big-Bang singularity is preferred.Comment: 19 pages, 6 figures, typos corrected; version accepted by JCA