A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

Differential growth responses to water balance of coexisting deciduous tree species are linked to wood density in a Bolivian tropical dry forest

A seasonal period of water deficit characterizes tropical dry forests (TDFs). There, sympatric tree species exhibit a diversity of growth rates, functional traits, and responses to drought, suggesting that each species may possess different strategies to grow under different conditions of water availability. The evaluation of the long-term growth responses to changes in the soil water balance should provide an understanding of how and when coexisting tree species respond to water deficit in TDFs. Furthermore, such differential growth responses may be linked to functional traits related to water storage and conductance. We used dendrochronology and climate data to retrospectively assess how the radial growth of seven coexisting deciduous tree species responded to the seasonal soil water balance in a Bolivian TDF. Linear mixed-effects models were used to quantify the relationships between basal area increment and seasonal water balance. We related these relationships with wood density and sapwood production to assess if they affect the growth responses to climate. The growth of all species responded positively to water balance during the wet season, but such responses differed among species as a function of their wood density. For instance, species with a strong growth response to water availability averaged a low wood density which may facilitate the storage of water in the stem. By contrast, species with very dense wood were those whose growth was less sensitive to water availability. Coexisting tree species thus show differential growth responses to changes in soil water balance during the wet season. Our findings also provide a link between wood density, a trait related to the ability of trees to store water in the stem, and wood formation in response to water availability.

Disentangling the formation of contrasting tree line physiognomies combining model selection and Bayesian parameterization for simulation models.

Alpine tree-line ecotones are characterized by marked changes at small spatial scales that may result in a variety of physiognomies. A set of alternative individual-based models was tested with data from four contrasting Pinus uncinata ecotones in the central Spanish Pyrenees to reveal the minimal subset of processes required for tree-line formation. A Bayesian approach combined with Markov chain Monte Carlo methods was employed to obtain the posterior distribution of model parameters, allowing the use of model selection procedures. The main features of real tree lines emerged only in models considering nonlinear responses in individual rates of growth or mortality with respect to the altitudinal gradient. Variation in tree-line physiognomy reflected mainly changes in the relative importance of these nonlinear responses, while other processes, such as dispersal limitation and facilitation, played a secondary role. Different nonlinear responses also determined the presence or absence of krummholz, in agreement with recent findings highlighting a different response of diffuse and abrupt or krummholz tree lines to climate change. The method presented here can be widely applied in individual-based simulation models and will turn model selection and evaluation in this type of models into a more transparent, effective, and efficient exercise.

Reliable computation of robust response tori on the verge of breakdown

We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy-Hammel route for the creation of strange non- chaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computer- assisted and are based on a tailored version of the Newton-Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques.

Estimating 750 years of temperature variations and uncertainties in the Pyrenees by tree-ring reconstructions and climate simulations.

Past temperature variations are usually inferred from proxy data or estimated using general circulation models. Comparisons between climate estimations derived from proxy records and from model simulations help to better understand mechanisms driving climate variations, and also offer the possibility to identify deficiencies in both approaches. This paper presents regional temperature reconstructions based on tree-ring maximum density series in the Pyrenees, and compares them with the output of global simulations for this region and with regional climate model simulations conducted for the target region. An ensemble of 24 reconstructions of May-to-September regional mean temperature was derived from 22 maximum density tree-ring site chronologies distributed over the larger Pyrenees area. Four different tree-ring series standardization procedures were applied, combining two detrending methods: 300-yr spline and the regional curve standardization (RCS). Additionally, different methodological variants for the regional chronology were generated by using three different aggregation methods. Calibration verification trials were performed in split periods and using two methods: regression and a simple variance matching. The resulting set of temperature reconstructions was compared with climate simulations performed with global (ECHO-G) and regional (MM5) climate models. The 24 variants of May-to-September temperature reconstructions reveal a generally coherent pattern of inter-annual to multi-centennial temperature variations in the Pyrenees region for the last 750 yr. However, some reconstructions display a marked positive trend for the entire length of the reconstruction, pointing out that the application of the RCS method to a suboptimal set of samples may lead to unreliable results. Climate model simulations agree with the tree-ring based reconstructions at multi-decadal time scales, suggesting solar variability and volcanism as the main factors controlling preindustrial mean temperature variations in the Pyrenees. Nevertheless, the comparison also highlights differences with the reconstructions, mainly in the amplitude of past temperature variations and in the 20th century trends. Neither proxy-based reconstructions nor model simulations are able to perfectly track the temperature variations of the instrumental record, suggesting that both approximations still need further improvements.

Computer-aided system for morphometric mandibular index computation (Using dental panoramic radiographs)

Objective: We propose and validate a computer aided system to measure three different mandibular indexes: cortical width, panoramic mandibular index and, mandibular alveolar bone resorption index. Study Design: Repeatability and reproducibility of the measurements are analyzed and compared to the manual estimation of the same indexes. Results: The proposed computerized system exhibits superior repeatability and reproducibility rates compared to standard manual methods. Moreover, the time required to perform the measurements using the proposed method is negligible compared to perform the measurements manually. Conclusions: We have proposed a very user friendly computerized method to measure three different morphometric mandibular indexes. From the results we can conclude that the system provides a practical manner to perform these measurements. It does not require an expert examiner and does not take more than 16 seconds per analysis. Thus, it may be suitable to diagnose osteoporosis using dental panoramic radiographs

On the computation of reducible invariant tori on a parallel computer

We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.

Modeling and manufacturability assessment of bistable quantum-dot cells

We have investigated the behavior of bistable cells made up of four quantum dots and occupied by two electrons, in the presence of realistic confinement potentials produced by depletion gates on top of a GaAs/AlGaAs heterostructure. Such a cell represents the basic building block for logic architectures based on the concept of quantum cellular automata (QCA) and of ground state computation, which have been proposed as an alternative to traditional transistor-based logic circuits. We have focused on the robustness of the operation of such cells with respect to asymmetries derived from fabrication tolerances. We have developed a two-dimensional model for the calculation of the electron density in a driven cell in response to the polarization state of a driver cell. Our method is based on the one-shot configuration-interaction technique, adapted from molecular chemistry. From the results of our simulations, we conclude that an implementation of QCA logic based on simple ¿hole arrays¿ is not feasible, because of the extreme sensitivity to fabrication tolerances. As an alternative, we propose cells defined by multiple gates, where geometrical asymmetries can be compensated for by adjusting the bias voltages. Even though not immediately applicable to the implementation of logic gates and not suitable for large scale integration, the proposed cell layout should allow an experimental demonstration of a chain of QCA cells.

Concurrent : Easy to use Python computing environment

Concurrent aims to be a different type of task distribution system compared to what MPI like system do. It adds a simple but powerful application abstraction layer to distribute the logic of an entire application onto a swarm of clusters holding similarities with volunteer computing systems. Traditional task distributed systems will just perform simple tasks onto the distributed system and wait for results. Concurrent goes one step further by letting the tasks and the application decide what to do. The programming paradigm is then totally async without any waits for results and based on notifications once a computation has been performed.