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Computational techniques are extensively applied in nonlinear science. However, while the use of computers for research has been expressive, the evaluation of numerical results does not grow in the same pace. Hammel et al. (Journal of Complexity, 1987, 3(2), 136--145) were pioneers in the numerical reliability field and have proved a theorem that a pseudo-orbit of a logistic map is shadowed by a true orbit within a distance of $10^{-8}$ for $10^{7}$ iterates. But the simulation of the logistic map with less than 100 iterates presents an error greater than $10^{-8}$ in a modern computer, performing a test based on the concept of multiple pseudo-orbits and symbolic computing., Comment: 6th NSC - International Conference on Nonlinear Science and Complexity, NSC, S\~ao Jos\'e dos Campos, Brazil, p.1-4

In this paper, we try to consider the stability conditions of a modified gravity coupled by Weyl tensor. In this way, we indicate the suitable conditions for a successful bounce while the equation of state (EoS) parameter crosses the phantom divider for our new corrected modified gravity. In the spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) universe, according to the ordinary version of the holographic dark energy (HDE) model, describing accelerated expansion of the universe, will be considered. Our model's stability conditions and its general properties of attractors for scalar field scripts which control cosmological acceleration will be checked. The statefinder diagnostic parameters of our model will be compared by today's observational data and defined methods. However its EoS parameter will be obtained, too., Comment: 28 pages and 22 figures. arXiv admin note: text overlap with arXiv:1212.5225 by other authors

In this article in a generalization of our previous work, we investigate the dynamics of the non-local $F(R)$ gravity after casting it into local form. The non-singular bouncing behavior and quintom model of dark energy are achieved without involving negative kinetic energy fields. Two cosmological tests are performed to constrain the model parameters. In case of phantom crossing the distance modulus predicted by the model best-fits the observational data. In comparison with the CPL parametrization for drift velocity, the model in some redshift intervals is in good agreement with the data., Comment: 12 pages, 7 figures, will be published in Astrophysics and space science

In this paper we consider FRW cosmology in modified non-local gravity. The stability analysis shows that there is only one stable critical point for the model and the universe undergoes a quintessence dominated era., Comment: 12 pages, 4 figures, will be published in Int. J. Theor. Phys

In this paper we consider FRW cosmology in $F(R,\phi)$ gravity. It is shown that in particular cases the bouncing behavior may appears in the model whereas the equation of state (EoS) parameter may crosses the phantom divider. For the dynamical universe, quantitatively we also find parameters in the model which satisfies two independent tests:the model independent Cosmological Redshift Drift (CRD) test and the type Ia supernova luminosity distances., Comment: 15 pages, 12 figures

In this paper we consider FRW cosmology in modified gravity which contain arbitrary functions $f(\phi)$. It is shown that the bouncing solution appears in the model whereas the equation of state (EoS) parameter crosses the phantom divider. The reconstruction of the model is also investigated with the aim to reconstruct the arbitrary functions and variables of the model., Comment: 18 pages, 10 figures

In this paper we consider a non-minimally coupled scaler field, and show its equation of state parameter can crossing over -1, $\omega\to -1$, and bouncing condition. Also we obtain the stability conditions and consider reconstructing for our model., Comment: 16 pages, 9 figures. Accepted in MPLA

In this paper we consider a quintom model of dark energy with a single scalar field $T$ given by a Lagrangian which inspired by tachyonic Lagrangian in string theory. We consider non-minimal coupling of tachyon field to the scalar curvature, then we obtain the equation of state (EoS), and the condition required for the model parameters when $\omega$ crosses over -1., Comment: 8 pages, 2 figures