This chapter describes a dynamical scenario in which an autonomous system can incorporate many oscillation modes of different frequencies, and where the oscillations can intermittently combine with each other to produce extremely complex self-sustained behaviors with multiple time scales. The complex oscillations are illustrated by means of the experimental devices with which they were discovered, which are simple enough to offer a comprehensive view of what is happening at the physical level. At the same time, we consider what remains to be understood in order to explain the observed behavior and how the qualitative methods of nonlinear dynamics can be used to provide such an understanding. Although a proper mathematical theory for the oscillatory mixing mechanism has yet to be developed, its features suggest significant possibilities for scalability, making it potentially relevant for the study of systems exhibiting rich oscillatory dynamics.

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.

R. Herrero, J. Farjas, F. Pi, G. Orriols ⁎ Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, 08222 Terrassa, Spain ramon.herrero@upc.edu † Departament de Física, Campus Montilivi, Universitat de Girona, 17003 Girona, Spain jordi.farjas@udg.edu ‡ Departament de Física, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain francesc.pi@uab.cat gaspar.orriols@uab.cat

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.

The well-defined but intricate course of time evolution exhibited by many naturally occurring phenomena suggests some source of dynamic order sustaining it. In spite of its obviousness as a problem, it has remained absent from the scientific debate due to the lack of candidates for a proper answer. Here we propose a general explanation based on the oscillatory possibilities of the dynamical systems, as demonstrated with a family of differential equations exhibiting self-sustained oscillations of unbounded complexity: complex evolutions by nonlinear mixing of an arbitrarily large number of oscillation modes, in which the different modes describe specific dynamical activities and their combination articulates the interactive conjunction of such activities into the whole functioning. The dynamical scenario is rather generic since it is exclusively based on the reiterative occurrence of the two most standard mechanisms of nonlinear dynamics: the saddle-node and Hopf bifurcations, and its extraordinary richness makes feasible the well-defined occurrence of ordered features over enormously complex dynamical activities as simply arising from the proper structure of dynamical relations among the system components, i.e., without requiring any other physical cause than those involved in such relations.

This paper may be ultimately described as an attempt to make feasible the evolutionary emergence of novelty in a supposedly deterministic world whose behavior is associated with that of the mathematical dynamical systems. It means philosophical implications that the paper needs to address, subsidiarily at least. The work was motivated by the observation of complex oscillatory behaviors in a family of physical devices and related mathematical models, for which there is no known explanation in the mainstream of nonlinear dynamics. The paper begins by describing a nonlinear mechanism of oscillatory mode mixing explaining such behaviors and, through its generalization to richer nonlinear vector fields, establishes a generic dynamical scenario with extraordinary oscillatory possibilities, including expansive growing scalability toward high dimensionalities and through nonlinear multiplicities. The scenario is then used to tentatively explain complex oscillatory behaviors observed in nature like those of turbulent fluids and living brains. Finally, by considering the scenario as a dynamic substrate underlying generic aspects of both the functioning and the genesis of complex behaviors in a supposedly deterministic world, a theoretical framework covering the evolutionary development of structural transformations in the time evolution of that world is built up. The analysis includes attempts to clarify the roles of items often invoked apropos of pathways to complexity like chaos, pattern formation, externally-driven bifurcations, hysteresis, irreversibility, and order through random fluctuations. Thermodynamics, as the exclusive field of physics in providing generic evolutionary criteria, is briefly and synthetically considered from the dynamical systems point of view by trying to elucidate its explanatory possibilities concerning the emergence of complexity. Quantum mechanics gets involved in two different ways: the lack of a dynamical systems perspective in the currently accepted interpretations of that fundamental theory and the indeterminacy issues, and both questions are discussed to point out their consequences. The reported evolutionary framework is far from a complete theory but includes both the elements and the skeleton for its tentative building within feasible philosophical grounds. In the lack of alternatives, one should imagine how it could be one of such theories and how it could be built, in order to evaluate our approach. In particular, notice that our approach is to a theory of nothing of the physical world but of the underlying reasons for its ordered and creative functioning, which we interpret to be independent of that world, i.e., a theory of what the Catalan expression ”l’entrellat del món” describes so well.

A significant reduction in the Neel temperature of CoO ultrathin films is revealed by highly sensitive specific heat measurement. It is found that the films consist of weakly coupled antiferromagnetic (AF) grains. The T(N) reduction from large to small grain samples scales with the grain size reduction, according to the Binder theory of critical phenomena in systems of reduced dimensions. In these finite AF nanosystems, the intergrain exchange interaction is reduced by the presence of surface Co spins, weakly coupled to Co moments inside the grains.

We report on in situ heat capacity measurements (370–800K) using quasiadiabatic ultrafast differential scanning nanocalorimetry in thin films (1–200nm) of Ni grown by electron beam evaporation. The heat capacity shows a broad peak with a rounded maximum that is attributed to the decrease of long-range interactions in the ferromagnetic to paramagnetic phase transition of Ni. The calorimetric data exhibit a reduction of the Curie temperature as the thickness of the films (or the average grain size) decreases. The magnitude of the jump in specific heat at TC scales with the number of surface or interface atoms.

In this work, we show the suitability of the thin film membrane-based calorimetric technique to measure kinetically limited phase transitions such as the dehydrogenation of metallic hydrides. Different compounds such as Mg, Mg/Al and Mg80Ti20 have been deposited over the active area of the microchip by electron beam evaporation. After several hydrogenation treatments at different temperatures to induce the hydride formation, calorimetric measurements on the dehydrogenation process of those thin films, either in vacuum or in air, are performed at a heating rate of 10°C/min. We observe a significant reduction in the onset of dehydrogenation for Mg80Ti20 compared with pure Mg or Mg/Al layers, which confirms the beneficial effect of Ti on dehydrogenation. We also show the suitability of the membrane-based nanocalorimeters to be used in parallel with optical methods. Quantification of the energy released during hydrogen desorption remains elusive due to the semi-insulating to metallic transition of the film which affects the calorimetric trace.

We have investigated by differential scanning calorimetry the thermal evolution of Cu∕Mg multilayers with different modulation lengths, ranging from 7∕28to30∕120nm. The Cu and Mg layers were grown by sequential evaporation in an electron beam deposition system. The phase identification and layer microstructure were determined by cross-section transmission electron microscopy, Rutherford backscattering, and scanning electron microscopy with focused ion beam for sample preparation. Upon heating, the intermetallic CuMg2 forms at the interfaces until coalescence is reached and thickens through a diffusion-limited process. Cross-section transmission electron microscopy observations show a distinct microstructure at the top and bottom of the as-prepared Mg layers, while no significant differences were seen in the Cu layers. We show that this effect is responsible for the observed asymmetry in the nucleation process between the Cu on Mg and the Mg on Cu interfaces. By modeling the calorimetric data we determine the role of both interfaces in the nucleation and lateral growth stages. We also show that vertical growth proceeds by grain development of the product phase, increasing significantly the roughness of the interfaces.

The crystallization temperature of ultrathin films of a-Ge sandwiched between SiO2 layers increases with decreasing thickness and also with the heating rate. Based on this premise we used ultrafast heating to probe the amorphous-to-liquid transition in ultrathin films of amorphous Ge. Upon cooling, the melt solidifies into nanocrystalline Ge. We show that sensitive nanocalorimetric measurements can provide kinetic and thermodynamic information involved during the amorphous-to-liquid- and nanocrystalline-to-liquid-phase transitions in the Ge system. A 3-nm a-Ge ultrathin layer is condensed from an e-beam evaporator under high vacuum conditions onto the 180nm thick free-standing SiNx membranes of the calorimeters that form the calorimetric cells. The a-Ge film is sandwiched between a-SiO2 layers to minimize heterogeneous nucleation at the interfaces. Nanocalorimetric measurements are carried out in-situ inside the e-beam setup. The apparent heat capacity (Cp) of the a-Ge layers is extracted measuring the power released to the sample at fast heating rates (104–105K/s), in the temperature range from 350 to 1200K.

We report the observation of temperature-dependent nucleation and lateral growth behavior of the intermediate product phase ${\mathrm{CuMg}}_{2}$ in Cu/Mg multilayers depending if the interface is Mg on Cu or Cu on Mg. The addition of an intermediate Fe layer inhibiting ${\mathrm{CuMg}}_{2}$ nucleation on one of the Cu/Mg interfaces demonstrates the important role of the nature of the interface on the nucleation events of the intermediate phase. This unexpected behavior is related to the mode of growth of the thin films. A kinetic model with independent nucleation and lateral growth stages at each interface is used to fit the calorimetric curves.

Bulk second-order nonlinearity was generated in BK7 glass at a higher temperature and with a longer poling time than near-surface second-order nonlinearity. The temporal decay of the bulk second-order nonlinearity was slower than that of the near-surface second-order non-linearity. The thickness of the near-surface nonlinear layer increased with poling time. Poled BK7 glass was also measured by x-ray photoelectron spectroscopy. Depletion of Na at the anodic surface and its accumulation at the cathodic surface was observed. At the cathodic surface, a higher-energy peak near O (1s) appeared, which shows peroxy-radical defects. At the anodic surface, a lower-energy peak near Si (2p) appeared, which may be attributed to E' centers or to two coordinated Si defects. The mechanisms of generation of these defects and of the second-order nonlinearities are discussed. (C) 2002 Optical Society of America.

Second harmonic generation from a poled glass is studied for its application prospects and its mechanisms. By poling the second order susceptibility (SOS) in the surface layer of glass, the anode side can be comparable with that in a KDP crystal. We report the SOS in poled ZF/sub 7/ lead silica is as large as 7 pm/V by calibrating with a quartz plate. As we know, it is the first time that a poled glass is used to generate the sum-frequency at 355 nm.

We present the experimental observation of phase synchronization transitions in the bidirectional coupling of chaotic and nonchaotic oscillators. A variety of transitions are characterized and compared to numerical simulations of a time delayed model. The characteristic 2pi phase jumps usually appear during the transitions, specially in those clearly associated with a saddle-node bifurcation. The study is done with pairs of optothermal oscillators linearly coupled by heat transfer.

The amplitude death phenomenon has been experimentally observed with a pair of nonlinear oscillators based on the thermo-optic effect and on the linearly coupled by heat transfer. A parametric analysis has been done both experimentally and numerically, pointing out that the death effect occurs for strong couplings and similar input powers.

An experimental analysis of the distribution and thickness of the bulk nonlinearity induced in poled silica is reported. The second-order susceptibility decreases exponentially from the anodic interface. Maker fringe patterns showing a double structure are interpreted in relation to the presence of two nonlinear profiles, one concentrated near the anodic surface and another extending into the bulk of the sample. The Maker fringe theory is properly generalized and a double fitting technique reproducing well the experimental results is used to characterize the induced nonlinearities. The dependence of the second-harmonic signal on the poling temperature is given, which is different from that of sol-gel silica.

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.