Physics

This paper is devoted to a discussion of the Discrete Fourier Transform (DFT) representation of a chaotic finite-duration sequence. This representation has the advantage that is itself a finite-duration sequence corresponding to samples equally spaced in the frequency domain. The Fast Fourier Transform (FFT) algoritm allows us an effective computation, and it can be applied to a relatively short time series. DFT representation requirements were analized and applied for determining the order-chaos transition in a nonlinear system described by the equation $x[n+1]=rx[n](1-x[n])$. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent for the time series set, obtained from the logistic equation.

The DFT representation is applied, to a ordered-chaotic finite-duration sequences set generated by the H\'enon mapping, for describing the order-chaos transition. This representation has the advantage that it may be applied to a relatively shorter discret-time series. The bifurcation diagram is produced and the largest Lyapounov exponent is calculated for each time series of the set, showing a good agreement with the results obtained by the DFT representation. The threshold value of the control parameter for the order-chaos transition was determined to be A=1.0532.

We investigate the phase-space of a flat FRW universe including both a scalar field, $\phi,$ coupled to matter, and radiation. The model is inspired in scalar-tensor theories of gravity, and thus, related with $F(R)$ theories through conformal transformation. The aim of the chapter is to extent several results to the more realistic situation when radiation is included in the cosmic budget particularly for studying the early time dynamics. Under mild conditions on the potential we prove that the equilibrium points corresponding to the non-negative local minima for $V(\phi)$ are asymptotically stable. Normal forms are employed to obtain approximated solutions associated to the inflection points and the strict degenerate local minimum of the potential. We prove for arbitrary potentials and arbitrary coupling functions $\chi(\phi),$ of appropriate differentiable class, that the scalar field almost always diverges into the past. It is designed a dynamical system adequate to studying the stability of the critical points in the limit $|\phi|\to\infty.$ We obtain there: radiation-dominated cosmological solutions; power-law scalar-field dominated inflationary cosmological solutions; matter-kinetic-potential scaling solutions and radiation-kinetic-potential scaling solutions. Using the mathematical apparatus developed here, we investigate the important examples of higher order gravity theories $F(R) = R + \alpha R^2$ (quadratic gravity) and $F(R) =R^n.$ We illustrated both analytically and numerically our principal results. In the case of quadratic gravity we prove, by an explicit computation of the center manifold, that the equilibrium point corresponding to de Sitter solution is locally asymptotically unstable (saddle point).

This paper is devoted to show the results obtained by using the magnitude-squared coherence for determining order-chaos transition in a system described by the logistic equation dynamics. For determining the power spectral density of a chaotic finite-duration discrete-time sequence the Welch average periodogram method was used. This method has the advantage that can be applied to any stationary signal by using the discrete Fourier transform (DFT) representation of a discrete-time series which allows an effective computation via fast Fourier transform (FFT) algorithm, and that can be applied to a discrete-time series shorter than that required by nonlinear dynamical analysis methods. The estimate of the Inverse Average Magnitude-Squared Coherence Index (IAMSCI) for each discrete-time series in the set obtained from the logistic mapping was calculated. The control parameter, {\bf r}, ranges in the interval [2.8,4] for producing the discrete-time series set. When the condition $r \geq 3.57$ is satisfied, each discrete-time series exhibited a positive value for IAMSCI estimate, indicating a high level of coherence loss of the signal and corresponding to a chaotic behavior of the dynamical system. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent of the time series set obtained from the logistic equation.

This paper is devoted to show the results obtained by using the magnitude-squared coherence for determining order-chaos transition in a system described by the logistic equation dynamics. For determining the power spectral density of a chaotic finite-duration discrete-time sequence the Welch average periodogram method was used. This method has the advantage that can be applied to any stationary signal by using the discrete Fourier transform (DFT) representation of a discrete-time series which allows an effective computation via fast Fourier transform (FFT) algorithm, and that can be applied to a discrete-time series shorter than that required by nonlinear dynamical analysis methods. The estimate of the Inverse Average Magnitude-Squared Coherence Index (IAMSCI) for each discrete-time series in the set obtained from the logistic mapping was calculated. The control parameter, {\bf r}, ranges in the interval [2.8,4] for producing the discrete-time series set. When the condition $r \geq 3.57$ is satisfied, each discrete-time series exhibited a positive value for IAMSCI estimate, indicating a high level of coherence loss of the signal and corresponding to a chaotic behavior of the dynamical system. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent of the time series set obtained from the logistic equation.

The quantitative determination of the order-chaos transition in a nonlinear dynamical system described by H\'enon mapping defined as $x[n+1]=1.0-A*x[n]^2+B*y[n], y[n+1]=B*x[n]$, where $B=0.3$, and $A$ is an adjustable control parameter, was made. This was achieved by applying the Inverse Average Magnitude-Squared Coherence Index (IAMSCI). This method is based on the Welch average periodogram technique and it has the advantage respect to nonlinear dynamical methods that it may be applied to any stationary signal by using discrete Fourier transform (DFT) representation which allows to operate on a short discrete-time series. Its effectiveness was demonstrated by comparing the results obtained by applying IAMSCI method with those obtained by calculating the largest Lyapounov exponent (LLE), both applied to the same discrete-time series set derived from the H\'enon mapping.

The quantitative determination of the order-chaos transition in a nonlinear dynamical system described by H\'enon mapping defined as $x[n+1]=1.0-A*x[n]^2+B*y[n], y[n+1]=B*x[n]$, where $B=0.3$, and $A$ is an adjustable control parameter, was made. This was achieved by applying the Inverse Average Magnitude-Squared Coherence Index (IAMSCI). This method is based on the Welch average periodogram technique and it has the advantage respect to nonlinear dynamical methods that it may be applied to any stationary signal by using discrete Fourier transform (DFT) representation which allows to operate on a short discrete-time series. Its effectiveness was demonstrated by comparing the results obtained by applying IAMSCI method with those obtained by calculating the largest Lyapounov exponent (LLE), both applied to the same discrete-time series set derived from the H\'enon mapping.

From experimental data points we evaluated the parameters of the model and obtained a fitting function for describing the dependence between the effective ultrasonic pulse propagation velocity and the free water quantity in concrete. We also analyzed the error propagation through the algorithm used. It allowed to decide under which conditions the linear approach between the effective ultrasonic pulse propagation velocity and the free water quantity in concrete can be applied.

We perform a detailed dynamical analysis of anisotropic scalar-field cosmologies, and in particular of the most significant Kantowski-Sachs, Bianchi I and Bianchi III cases. We follow the new and powerful method of $f$-devisers, which allows us to perform the whole analysis for arbitrary potentials. Thus, one can just substitute the specific potential form in the final results and obtain the corresponding behavior, without the need of new calculations. We find a very rich behavior, and amongst others the universe can result in isotropized solutions with observables in agreement with observations, such as de Sitter, quintessence-like, or stiff-dark energy solutions. Additionally, in the case of Kantowski-Sachs geometry we find that a cosmological bounce and turnaround are realized in a part of the parameter-space. Finally, applying the general results to the well-studied exponential and power-law potentials, we find that some of the general stable solutions disappear. This feature ma...

In this book are studied, from the perspective of the dynamical systems, several Universe models. In chapter 1 we give a bird's eye view on cosmology and cosmological problems. Chapter 2 is devoted to a brief review on some results and useful tools from the qualitative theory of dynamical systems. They provide the theoretical basis for the qualitative study of concrete cosmological models. Chapters 1 and 2 are a review of well-known results. Chapters 3, 4, 5 and 6 are devoted to our main results. In these chapters are extended and settled in a substantially different, more strict mathematical language, several results obtained by one of us in arXiv:0812.1013 [gr-qc]; arXiv:1009.0689 [gr-qc]; arXiv:0904.1577[gr-qc]; and arXiv:0909.3571 [hep-th]. In chapter 6, we provide a different approach to the subject discussed in astro-ph/0503478. Additionally, we perform a Poincar\'e compactification process allowing to construct a global phase space containing all the cosmological info...

The estimation of mechanical properties of concrete is one of the most important applications of the ultrasonic method. This technique often involves the measuring of the ultrasonic pulse propagation velocity. However, there exist some factors influencing the effective ultrasonic pulse propagation velocity in concrete. This paper is devoted to a discussion of the behavior of the effective ultrasonic pulse propagation velocity as a function of the free water contained in concrete. The travelling of an ultrasonic ray through a heterogeneous medium composed by cement and aggregates(solid phase), water(liquid phase) and air(gaseous phase) was empirically modelled. A relationship between the transition time of the ultrasonic ray in the testing piece and the free water quantity was considered. We assumed an empirical intermediate relationship between the ultrasonic path length in porous space in concrete filled with water and the free water quantity in the testing piece. From experimental...