The theory of dynamical systems is a very complex subject that has produced several surprises in the recent past in connection with the theory of chaos and fractals. The application of the tools of dynamical systems in cosmological settings is less known, in spite of the number of published scientific papers on this subject. In this paper, a mostly pedagogical introduction to the cosmological application of the basic tools of dynamical systems theory is presented. It is shown that, in spite of their amazing simplicity, these tools allow us to extract essential information on the asymptotic dynamics of a wide variety of cosmological models. The power of these tools is illustrated within the context of the so-called Λ -cold dark matter ( Λ CDM) and scalar field models of dark energy. This paper is suitable for teachers, undergraduate students, and postgraduate students in the disciplines of physics and mathematics.

In this paper we approach to Horndeski and beyond Horndeski theories from the effective fluid perspective. We make explicit the formal equivalence between these theories and general relativity with an effective imperfect fluid. It is shown that, for the viable Horndeski theories, in the general case (arbitrary geometry) the nonvanishing contribution from the higher-order derivative terms to the imperfect fluidlike behavior affects only the heat flux vector but not the anisotropic stresses. The only contribution to the anisotropic stress tensor is due to the nonminimal coupling of the scalar field to the curvature as it is in standard scalar-tensor theories. For the viable beyond Horndeski theories the higher-order derivatives contribute both to the heat flux and to the anisotropic stresses. The effective fluid description is applied to several particular cases of interest. It is corroborated that, in Friedmann-Robertson-Walker background space, due to the underlying symmetries, the effective stress-energy tensor of viable Horndeski and beyond Horndeski theories is formally equivalent to that of a perfect fluid. This result might not be true for other less symmetric backgrounds such as the anisotropic Bianchi I space.

We study the static and dynamic properties of networks of crumpled creases formed in hand crushed sheets of paper. The fractal dimensionalities of crumpling networks in the unfolded (flat) and folded configurations are determined. Some other noteworthy features of crumpling networks are established. The physical implications of these findings are discussed. Specifically, we state that self-avoiding interactions introduce a characteristic length scale of sheet crumpling. A framework to model the crumpling phenomena is suggested. Mechanics of sheet crushing under external confinement is developed. The effect of compaction geometry on the crushing mechanics is revealed.

In this paper we investigate the so-called ‘phantom barrier crossing’ issue in a cosmological model based on the scalar–tensor theory with non-minimal derivative coupling to the Einstein tensor. Special attention will be paid to the physical bounds on the squared sound speed. The numeric results are geometrically illustrated by means of a qualitative procedure of analysis that is based on the mapping of the orbits in the phase plane onto the surfaces that represent physical quantities in the extended phase space, that is: the phase plane complemented with an additional dimension relative to the given physical parameter. We find that the cosmological model based on the non-minimal derivative coupling theory—this includes both the quintessence and the pure derivative coupling cases—has serious causality problems related to superluminal propagation of the scalar and tensor perturbations. Even more disturbing is the finding that, despite the fact that the underlying theory is free of the Ostrogradsky instability, the corresponding cosmological model is plagued by the Laplacian (classical) instability related with negative squared sound speed. This instability leads to an uncontrollable growth of the energy density of the perturbations that is inversely proportional to their wavelength. We show that, independent of the self-interaction potential, for positive coupling the tensor perturbations propagate superluminally, while for negative coupling a Laplacian instability arises. This latter instability invalidates the possibility for the model to describe the primordial inflation.

Analysis of a new spatial interpolation weighting method to estimate missing data applied to rainfall records