The theory of the dynamical systems is a very complex subject which has brought several surprises in the recent past in connection with the theory of chaos and fractals. The application of the tools of the dynamical systems in cosmological settings is less known in spite of the amount of published scientific papers on this subject. In this paper a -mostly pedagogical -introduction to the application in cosmology of the basic tools of the dynamical systems theory is presented. It is shown that, in spite of their amazing simplicity, these allow 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 ΛCDM and scalar field models of dark energy. This paper is suitable for teachers, undergraduate and postgraduate students from physics and mathematics disciplines.

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.

We study the lateral deformations of randomly folded elasto-plastic and predominantly plastic thin sheets under the uniaxial and radial compressions. We found that the lateral deformations of cylinders folded from elasto-plastic sheets of paper obey a power law behavior with the universal Poisson's index 01 . 0 17 . 0 ± = ν , which does not depend neither the paper kind and sheet sizes (thickness, edge length), nor the folding confinement ratio. In contrast to this, the lateral deformations of randomly folded predominantly plastic aluminum foils display the linear dependence on the axial compression with the universal Poisson's ratio 01 . 0 33 . 0 ± = e ν. This difference is consistent with the difference in fractal topology of randomly folded elasto-plastic and predominantly plastic sheets, which is found to belong to different universality classes.The general form of constitutive stress-deformation relations for randomly folded elastoplastic sheets is suggested.

In this paper we generalize the dynamical systems analysis of the cubic galileon model previously investigated in [1] by including self-interaction potentials beyond the exponential one. It will be shown that, consistently with the results of [1], the cubic self-interaction of the galileon vacuum appreciably modifies the late-time cosmic dynamics by the existence of a phantom-like attractor (among other super-accelerated solutions that do not modify in any appreciable way the late-time dynamics and hence are not of interest in the present investigation). In contrast, in the presence of background matter the late-time cosmic dynamics remains practically the same as in the standard quintessence scenario. This means that we can not recover the cubic galileon vacuum continuously from the more general cubic quintessence with background matter, by setting to zero the matter energy density (and the pressure). This happens to be a kind of cosmological vDVZ discontinuity that can be evaded by means of the cosmological version of the Vainshtein screening mechanism. PACS numbers: 02.30.Hq, 04.20.Ha, 04.50.Kd, 05.45.-a, 98.80.-k I. INTRODUCTIONAccording to the increasing set of independent cosmological observations [2-8] the universe today is experiencing an accelerated expansion era. An unknown component dubbed as dark energy has been proposed to explain this recent acceleration in the context of the general relativity. The cosmological constant with equation of state ω = −1, is the simplest and the most accurate candidate according the observations [9]. However, it is plagued by serious theoretical issues such as the vacuum energy problem, the cosmic coincidence problem, the particle nature of dark matter, the validity of general relativity on large scales, and the age problem [10,11]. Since the observations allow the variation in time of the dark energy component, another possibility is to consider the existence of light scalar fields known as "quintessence" [12].Modified gravity represents an alternative approach for addressing the unusual cosmological dynamics at large scales. It is based on the modification of general relativity. We can observe two main streams in this context: introducing a Lagrangian built up of a Ricci, Riemann or another metric tensors as in the case of f (R, G) theories [13] and Brans-Dicke (BD) theories [14], or assuming the existence of additional dimensions that realize cosmic acceleration through the leakage of gravity into the extra-space at cosmological scales as in the Dvali-Gabadadze-Porrati (DGP) braneworld [15,16]. This latter model, however, is plagued by ghost instabilities that cast doubts on its validity. 1 Inspired by the DGP model, in [17] the authors proposed an infrared modification of gravity which is a generalization of the 4D effective theory in the DGP braneworld. The theory is invariant under the Galilean shift symmetry ∂ µ φ → ∂ µ φ + b µ in the Minkowski space-time, which keeps the equations of motion at second order. The scalar field that respects the Galilean symmetry is...

We study the scaling properties of forced folding of thin materials of different geometry. The scaling relations implying the topological crossovers from the folding of three-dimensional plates to the folding of two-dimensional sheets and further to the packing of one-dimensional strings are derived for elastic and plastic manifolds. These topological crossovers in the folding of plastic manifolds were observed in experiments with predominantly plastic aluminum strips of different geometry. Elasto-plastic materials, such as paper sheets during the (fast) folding under increasing confinement force, are expected to obey the scaling force-diameter relation derived for elastic manifolds. However, in experiments with paper strips of different geometry we observed the crossover from packing of one-dimensional strings to folding two dimensional sheets only, because the fractal dimension of the set of folded elasto-plastic sheets is the thickness dependent due to the strain relaxation after a confinement force is withdrawn.