Nonlinear complexification of periodic orbits in the generalized Landau scenario.

We have found a way for penetrating the space of the dynamical systems toward systems of arbitrary dimension exhibiting the nonlinear mixing of a large number of oscillation modes through which extraordinarily complex time evolutions may arise. The system design is based on assuring the occurrence of a number of Hopf bifurcations in a set of fixed points of a relatively generic system of ordinary differential equations, in which the main peculiarity is that the nonlinearities appear through functions of a linear combination of the system variables. The paper outlines the design procedure and presents a selection of numerical simulations with a variety of designed systems whose dynamical behaviors are really rich and full of unknown features. For concreteness, the presentation is focused on illustrating the oscillatory mixing effects on the periodic orbits, through which the harmonic oscillation born in a Hopf bifurcation becomes successively enriched with the intermittent incorporation of other oscillation modes of higher frequencies while the orbit remains periodic and without the necessity of bifurcating instabilities. Even in the absence of a proper mathematical theory covering the nonlinear mixing mechanisms, we find enough evidence to expect that the oscillatory scenario be truly scalable concerning the phase-space dimension, the multiplicity of involved fixed points, and the range of time scales so that extremely complex but ordered dynamical behaviors could be sustained through it.

Ultrafast transient liquid assisted growth of high current density superconducting films.

The achievement of high growth rates in YBa2Cu3O7 epitaxial high-temperature superconducting films has become strategic to enable high-throughput manufacturing of long length coated conductors for energy and large magnet applications. We report on a transient liquid assisted growth process capable of achieving ultrafast growth rates (100 nm s-1) and high critical current densities (5 MA cm-2 at 77 K). This is based on the kinetic preference of Ba-Cu-O to form transient liquids prior to crystalline thermodynamic equilibrium phases, and as such is a non-equilibrium approach. The transient liquid-assisted growth process is combined with chemical solution deposition, proposing a scalable method for superconducting tapes manufacturing. Additionally, using colloidal solutions, the growth process is extended towards fabrication of nanocomposite films for enhanced superconducting properties at high magnetic fields. Fast acquisition in situ synchrotron X-ray diffraction and high resolution scanning transmission electron microscopy (STEM) become crucial measurements in disentangling key aspects of the growth process.

Nonlinear oscillatory mixing in the generalized Landau scenario.

We present a set of phase-space portraits illustrating the extraordinary oscillatory possibilities of the dynamical systems through the so-called generalized Landau scenario. In its simplest form the scenario develops in N dimensions around a saddle-node pair of fixed points experiencing successive Hopf bifurcations up to exhausting their stable manifolds and generating N-1 different limit cycles. The oscillation modes associated with these cycles extend over a wide phase-space region by mixing ones within the others and by affecting both the transient trajectories and the periodic orbits themselves. A mathematical theory covering the mode-mixing mechanisms is lacking, and our aim is to provide an overview of their main qualitative features in order to stimulate research on it.

Can the crystallization rate be independent from the crystallization enthalpy? The case of amorphous silicon.

The crystallization enthalpy measured in a large series of amorphous silicon (a-Si) materials varies within a factor of 2 from sample to sample (Kail et al 2011 Phys. Status Solidi RRL 5 361). According to the classical theory of nucleation, this variation should produce large differences in the crystallization kinetics leading to crystallization temperatures and activation energies exceeding 550 °C and 1.7 eV, respectively, the 'standard' values measured for a-Si obtained by self-implantation. In contrast, the observed crystallization kinetics is very similar for all the samples studied and has no correlation with the crystallization enthalpy. This discrepancy has led us to propose that crystallization in a-Si begins in microscopic domains that are almost identical in all samples, independently of their crystallization enthalpy. Probably the existence of microscopic inhomogeneities also plays a crucial role in the crystallization kinetics of other amorphous materials and glasses.

Restricted epitaxial growth of crystallites in hydrogenated nanocrystalline silicon during thermal crystallization experiments.

Thermal crystallization experiments have been carried out on nanocrystalline silicon films. From the thermal activation of the process, it is concluded that the amorphous phase crystallizes by solid phase epitaxy around the preexisting crystallites. However, and in contrast with the usual epitaxial crystallization of wafers partially amorphized by ion implantation or ball-milled powders, the epitaxial growth is inhibited for most of the amorphous-crystalline interface. Our analysis indicates that a small fraction of the interface contributes to the epitaxial growth. Although the influence of oxidation and of Si-H groups located at the interface cannot be completely excluded, this behavior is explained in terms of a high density of microvoids located at the a-c interface. This result has implications for the models of electrical conduction of nc-Si:H.

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which self-similarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N-1 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features.

N-dimensional dynamical systems exploiting instabilities in full.

We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions. (c) 2000 American Institute of Physics.

Speed of wave-front solutions to hyperbolic reaction-diffusion equations.

The asymptotic speed problem of front solutions to hyperbolic reaction-diffusion (HRD) equations is studied in detail. We perform linear and variational analyses to obtain bounds for the speed. In contrast to what has been done in previous work, here we derive upper bounds in addition to lower ones in such a way that we can obtain improved bounds. For some functions it is possible to determine the speed without any uncertainty. This is also achieved for some systems of HRD (i.e., time-delayed Lotka-Volterra) equations that take into account the interaction among different species. An analytical analysis is performed for several systems of biological interest, and we find good agreement with the results of numerical simulations as well as with available observations for a system discussed recently.