Complex statistics and diffusion in nonlinear disordered particle chains.

We investigate dynamically and statistically diffusive motion in a Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy (subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is always observed. We then carry out a statistical analysis of the motion, in both cases, in the sense of the Central Limit Theorem and present evidence of different chaos behaviors, for various groups of particles. Integrating the equations of motion for times as long as 10(9), our probability distribution functions always tend to Gaussians and show that the dynamics does not relax onto a quasi-periodic Kolmogorov-Arnold-Moser torus and that diffusion continues to spread chaotically for arbitrarily long times.

Mathematical modeling suggests that periodontitis behaves as a non-linear chaotic dynamical process.

BACKGROUND: This study aims to expand on a previously presented cellular automata model and further explore the non-linear dynamics of periodontitis. Additionally the authors investigated whether their mathematical model could predict the two known types of periodontitis, aggressive (AgP) and chronic periodontitis (CP). METHODS: The time evolution of periodontitis was modeled by an iterative function, based on the hypothesis that the host immune response level determines the rate of periodontitis progression. The chaotic properties of this function were investigated by direct iteration, and the model was validated by immunologic and clinical parameters derived from two clinical study populations. RESULTS: Periodontitis can be described as chaos with the level of the host immune response determining its progression rate; the dynamics of the proposed model suggest that by increasing the host immune response level, periodontitis progression rate decreases. Renormalization transformations show the presence of two overlapping zones of disease activity corresponding to AgP and CP. By k-means cluster analysis, immunologic parameters corroborated the findings of the renormalization transformations. Periodontitis progression rates are modeled to scale with a power law of 1.3, and the mean exponential speed of the system is found to be 1.85 (metric entropy); clinical datasets confirmed the mathematical estimates. CONCLUSIONS: This study introduces a mathematical model that identifies periodontitis as a non-linear chaotic process. It offers a quantitative assessment of the disease progression rate and identifies two zones of disease activity that correspond to the existing classification of periodontitis in the AgP and CP types.

Aggressive periodontitis defined by recursive partitioning analysis of immunologic factors.

BACKGROUND: The present study aims to extend recent findings of a non-linear model of the progression of periodontitis supporting the notion that aggressive periodontitis (AgP) and chronic periodontitis (CP) are distinct clinical entities. This approach is based on the implementation of recursive partitioning analysis (RPA) to evaluate a series of immunologic parameters acting as predictors of AgP and CP. METHODS: RPA was applied to three population samples, that were retrieved from previous studies, using 17 immunologic parameters. The mean values of the parameters in control subjects were used as the cut-off points. Leave-one-out cross-validation (LOOCV) prediction errors were estimated in the proposed models, as well as the Kullback-Leibler divergence (DKL) of the distribution of positive results in AgP compared to CP and negative results in CP compared to AgP. RESULTS: Seven classification trees were derived showing that the relationship of interleukin (IL)-4, IL-1, IL-2 has the highest potential to rule out or rule in AgP. On the other hand, immunoglobulin (Ig)A, IgM used to rule out AgP and cluster of differentiation 4 (CD4)/CD8, CD20 used to rule in AgP showed the least LOOCV cost. Penalizing DKL with LOOCV cost promotes the IL-4, IL-1, IL-2 model for ruling out AgP, whereas the single CD4/CD8 ratio with a lowered discrimination cut-off point was used to rule in AgP. CONCLUSIONS: Although a test is unlikely to have both high sensitivity and high specificity, the use of immunologic parameters in the right model can efficiently complement a clinical examination for ruling out or ruling in AgP.

Breathers in a nonautonomous Toda lattice with pulsating coupling.

We study a nonautonomous Toda lattice, with a periodically switched on-off coupling coefficient, describing a pulsating strength of neighbor particle interaction. It is shown that when the uncoupled oscillations are linear and under appropriate conditions for the duration of the time intervals where the coupling is switched off, breather solutions can be obtained analytically. Their dynamics and collisions are related to the soliton dynamics of the corresponding autonomous Toda lattice, while a "ratchet" effect is shown to result in breather deceleration, providing a mechanism for breather velocity and collision control.

Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences.

We focus on two approaches that have been proposed in recent years for the explanation of the so-called Fermi-Pasta-Ulam (FPU) paradox, i.e., the persistence of energy localization in the "low-q " Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of "natural packets" exhibiting exponential stability, while in the second, emphasis is placed on the existence of "q breathers," i.e., periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of " q-tori" representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.

Quasiperiodic and chaotic discrete breathers in a parametrically driven system without linear dispersion.

We study a one-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DB's) can be explicitly constructed by an exact separation of their time and space dependence. Introducing parametric periodic driving, we first show how a variety of such DB's can be obtained by selecting spatial profiles from the homoclinic orbits of an invertible map and combining them with initial conditions chosen from the Poincaré surface of section of a simple Duffing's equation. Placing then our initial conditions at the center of the islands of a major resonance, we demonstrate how the corresponding DB can be stabilized by varying the amplitude of the driving. We thus discover around elliptic points a large region of quasiperiodic breathers, which are stable for very long times. Starting with initial conditions close to the elliptic point at the origin, we find that as we approach the main chaotic layer, a quasiperiodic breather either destabilizes by delocalization or turns into a chaotic breather, with an evidently broadbanded Fourier spectrum before it collapses. For some breather profiles stable quasiperiodic breathers exist all the way to the separatrix of the Duffing equation, indicating the presence of large regions of tori around the DB solution in the multidimensional phase space. We argue that these strong localization phenomena are due to the absence of phonon resonances, as there are no linear dispersion terms in our lattices. We also show, however, that these phenomena persist in more realistic physical models, in which weak linear dispersion is included in the equations of motion, with a sufficiently small coefficient.

Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation.

We study an integrable discretization of the nonlinear Schrödinger equation (NLS) under the effects of damping and periodic driving, from the point of view of spatially localized solutions oscillating in time with the driver's frequency. We locate the equilibrium states of the discretized (DNLS) system in the plane of its dissipation gamma and forcing amplitude H parameters and use a shooting algorithm to construct the desired solutions psi(n)(t)=phi(n) exp(it) as homoclinic orbits of a four-dimensional symplectic map in the complex phi(n),phi(n+1) space, for -infinity

A study of pharmacological vs. pathological autonomic nervous system blockade in humans, using heart rate chaotic dynamics.

Pathological blocking of the Autonomous Nervous System (ANS) is diagnosed for patients having Autonomic Neuropathy passing a well-defined set of criteria. In this work, It is shown that standard analysis can be complemented by a study of the chaotic dynamics of the Heart Rate (HR), over a period of 12 min in the resting position. It was found that patients who suffering from ANS blockade, typically exhibit a smaller degree of fractality and complexity of the chaotic attractor reconstructed from the time series of the HR signal. These dynamical measures are more evident in normal human subjects that have been subjected to pharmacological ANS blockade.

Strategy for the pursuit of spiral waves in excitable media.

Spiral waves are though to be the underlying mechanism of re-entrant ventricular and atrial tachycardias. In such cases, one is generally interested in eliminating spiral wave activity from the medium. In this paper, solve a cubic FitzHugh-Nagumo system of PDEs is solved in two dimensions with initial conditions such that a spiral wave is formed at the center of a rectangular grid. Then the effect of a spatially-localized step-like periodic forcing placed at different positions around the spiral tip is studied. Due to this forcing, the tip begins to drift away from the perturbation in a direction that depends on their relative location. By shifting successively the location of the perturbation relative to that of the tip strategy is developed which it is possible to pursuit the spiral wave away from the center of the grid accelerating its drift with every shift of the perturbation.

Nonlinear time series analysis of electrocardiograms.

In recent years there has been an increasing number of papers in the literature, applying the methods and techniques of Nonlinear Dynamics to the time series of electrical activity in normal electrocardiograms (ECGs) of various human subjects. Most of these studies are based primarily on correlation dimension estimates, and conclude that the dynamics of the ECG signal is deterministic and occurs on a chaotic attractor, whose dimension can distinguish between healthy and severely malfunctioning cases. In this paper, we first demonstrate that correlation dimension calculations must be used with care, as they do not always yield reliable estimates of the attractor's "dimension." We then carry out a number of additional tests (time differencing, smoothing, principal component analysis, surrogate data analysis, etc.) on the ECGs of three "normal" subjects and three "heavy smokers" at rest and after mild exercising, whose cardiac rhythms look very similar. Our main conclusion is that no major dynamical differences are evident in these signals. A preliminary estimate of three to four basic variables governing the dynamics (based on correlation dimension calculations) is updated to five to six, when temporal correlations between points are removed. Finally, in almost all cases, the transition between resting and mild exercising seems to imply a small increase in the complexity of cardiac dynamics. (c) 1995 American Institute of Physics.