Equilibration of energy in slow-fast systems.

Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.

Quantum pseudointegrable Hamiltonian impact system.

Quantization of a toy model of a pseudointegrable Hamiltonian impact system is introduced, including Einstein-Brillouin-Keller quantization conditions, a verification of Weyl's law, the study of their wave functions, and a study of their energy levels properties. It is demonstrated that the energy level statistics are similar to those of pseudointegrable billiards. Yet, here, the density of wave functions which concentrate on projections of classical level sets to the configuration space does not disappear at large energies, suggesting that there is no equidistribution in the configuration space in the large energy limit; this is shown analytically for some limit symmetric cases and is demonstrated numerically for some nonsymmetric cases.

Robust exponential acceleration in time-dependent billiards.

A class of nonrelativistic particle accelerators in which the majority of particles gain energy at an exponential rate is constructed. The class includes ergodic billiards with a piston that moves adiabatically and is removed adiabatically in a periodic fashion. The phenomenon is robust: deformations that keep the chaotic character of the billiard retain the exponential energy growth. The growth rate is found analytically and is, thus, controllable. Numerical simulations corroborate the analytic predictions with good precision. The acceleration mechanism has a natural thermodynamical interpretation and is applied to a hot dilute gas of repelling particles.

Bistable Bacterial Growth Dynamics in the Presence of Antimicrobial Agents.

The outcome of an antibiotic treatment on the growth capacity of bacteria is largely dependent on the initial population size (Inoculum Effect). We characterized and built a model of this effect in E. coli cultures using a large variety of antimicrobials, including conventional antibiotics, and for the first time, cationic antimicrobial peptides (CAMPs). Our results show that all classes of antimicrobial drugs induce an inoculum effect, which, as we explain, implies that the dynamic is bistable: For a range of anti-microbial densities, a very small inoculum decays whereas a larger inoculum grows, and the threshold inoculum depends on the drug concentration. We characterized three distinct classes of drug-induced bistable growth dynamics and demonstrate that in rich medium, CAMPs correspond to the simplest class, bacteriostatic antibiotics to the second class, and all other traditional antibiotics to the third, more complex class. These findings provide a unifying universal framework for describing the dynamics of the inoculum effect induced by antimicrobials with inherently different killing mechanisms.

Novel strategies for granulocyte colony-stimulating factor treatment of severe prolonged neutropenia suggested by mathematical modeling.

PURPOSE: To improve the effectiveness of granulocyte colony-stimulating factor (G-CSF) treatment in high-risk neutropenic patients. EXPERIMENTAL DESIGN: We study G-CSF effects on chemotherapy-induced neutropenia by expanding a simple mathematical model of neutrophil dynamics in the blood. The final model is fitted and validated using published clinical data of neutrophil response to chemotherapy and standard s.c. G-CSF protocol (SG; filgrastim 5 microg/kg/d), single pegylated (pegG; pegfilgrastim 100 microg/kg), and continuous infusion (CG; filgrastim 10 microg/kg/d). The interpatient variability is studied by Monte-Carlo simulation of pegG compared with SG and placebo. RESULTS: The effect G-CSF support on neutropenia depends on the neutrophil count at the nadir. Three distinct neutropenia grades are identified: G1 (300 x 10(3)-500 x 10(3) cells/mL), G2 (50 x 10(3)-300 x 10(3) cells/mL), and G3 (< or =50 x 10(3) cells/mL). For many G2 patients, the G-CSF levels required for recovery are not attainable by the standard regimen, whereas the sustained pegG and CG seem to be significantly more effective. For G3 patients, G-CSF support alone is not sufficient and additional clinical approaches should be considered. The results presented here are robust and are only slightly affected by population variability. CONCLUSIONS: The model captures the G-CSF-neutrophil dynamics of severe chemotherapy-induced neutropenia. Our results clarify and complement the current American Society of Clinical Oncology recommendations for G-CSF administration in neutropenia: High sustained G-CSF levels are needed to treat severe neutropenia and may be achieved by either CG or pegG. The potential effect of sustained G-CSF on severe neutropenia should be studied within a framework of a prospective randomized clinical trial.

Leaky Fermi accelerators.

A Fermi accelerator is a billiard with oscillating walls. A leaky accelerator interacts with an environment of an ideal gas at equilibrium by exchange of particles through a small hole on its boundary. Such interaction may heat the gas: we estimate the net energy flow through the hole under the assumption that the particles inside the billiard do not collide with each other and remain in the accelerator for a sufficiently long time. The heat production is found to depend strongly on the type of Fermi accelerator. An ergodic accelerator, i.e., one that has a single ergodic component, produces a weaker energy flow than a multicomponent accelerator. Specifically, in the ergodic case the energy gain is independent of the hole size, whereas in the multicomponent case the energy flow may be significantly increased by shrinking the hole size.

Secondary homoclinic bifurcation theorems.

We develop criteria for detecting secondary intersections and tangencies of the stable and unstable manifolds of hyperbolic periodic orbits appearing in time-periodically perturbed one degree of freedom Hamiltonian systems. A function, called the "Secondary Melnikov Function" (SMF) is constructed, and it is proved that simple (resp. degenerate) zeros of this function correspond to transverse (resp. tangent) intersections of the manifolds. The theory identifies and predicts the rotary number of the intersection (the number of "humps" of the homoclinic orbit), the transition number of the homoclinic points (the number of periods between humps), the existence of tangencies, and the scaling of the intersection angles near tangent bifurcations perturbationally. The theory predicts the minimal transition number of the homoclinic points of a homoclinic tangle. This number determines the relevant time scale, the minimal stretching rate (which is related to the topological entropy) and the transport mechanism as described by the TAM, a transport theory for two-dimensional area-preserving chaotic maps. The implications of this theory on the study of dissipative systems have yet to be explored. (c) 1995 American Institute of Physics.

Parabolic resonances and instabilities.

A parabolic resonance is formed when an integrable two-degrees-of-freedom (d.o.f.) Hamiltonian system possessing a circle of parabolic fixed points is perturbed. It is proved that its occurrence is generic for one parameter families (co-dimension one phenomenon) of near-integrable, two d.o.f. Hamiltonian systems. Numerical experiments indicate that the motion near a parabolic resonance exhibits a new type of chaotic behavior which includes instabilities in some directions and long trapping times in others. Moreover, in a degenerate case, near a flat parabolic resonance, large scale instabilities appear. A model arising from an atmospherical study is shown to exhibit flat parabolic resonance. This supplies a simple mechanism for the transport of particles with small (i.e. atmospherically relevant) initial velocities from the vicinity of the equator to high latitudes. A modification of the model which allows the development of atmospherical jets unfolds the degeneracy, yet traces of the flat instabilities are clearly observed. (c) 1997 American Institute of Physics.

Islands of accelerator modes and homoclinic tangles.

Islands are divided according to their phase space structure-resonant islands and tangle islands are considered. It is proved that in the near-integrable limit these correspond to two distinct sets, hence that in general their definitions are not trivially equivalent. It is demonstrated and proved that accelerator modes of the standard map and of the web map are necessarily of the tangle island category. These islands have an important role in determining transport-indeed it has been demonstrated in various works that stickiness to these accelerator modes may cause anomalous transport even for initial conditions starting in the ergodic component. (c) 1999 American Institute of Physics.

Evidence for bistable bacteria-neutrophil interaction and its clinical implications.

Neutropenia, which may develop as a consequence of chemotherapy, increases the risk of bacterial infection. Similarly, increased risk of bacterial infection appears in disorders of phagocytic functions, such as the genetic disorder chronic granulomatous disease. To elucidate the organizing principles behind these distinct immunodeficiency conditions, we investigated the interaction between in vitro bacteria and human neutrophils by experiments and mathematical modeling. The model and the experiments showed that the in vitro bacterial dynamics exhibit bistability for a certain range of neutrophil concentration and function. Thus, there is a critical bacterial concentration above which infection develops, and below which neutrophils defeat the bacteria. Whereas with normal neutrophil concentration and function, an infection may develop when the initial bacterial concentration is very high, under neutropenic conditions or when there is neutrophil dysfunction, the critical bacterial concentration can be lower, within the clinically relevant range. We conclude that critical bacterial concentration has clinically relevant implications. The individual maximum bearable bacterial concentration depended on neutrophil concentration, phagocytic activity, and patient barrier integrity; thus, the resulting maximal bearable bacterial concentration may vary by orders of magnitude between patients. Understanding the interplay between neutrophils and bacteria may enhance the development of new therapeutic approaches to bacterial infections.

Bacteria-phagocyte dynamics, axiomatic modelling and mass-action kinetics.

Axiomatic modeling is ensued to provide a family of models that describe bacterial growth in the presence of phagocytes, or, more generally, prey dynamics in a large spatially homogenous eco-system. A classification of the possible bifurcation diagrams that arise in such models is presented. It is shown that other commonly used models that do not belong to this class may miss important features that are associated with the limited growth curve of the bacteria (prey) and the saturation associated with the phagocytosis (predator kill) term. Notably, these features appear at relatively low concentrations, much below the saturation range. Finally, combining this model with a model of neutrophil dynamics in the blood after chemotherapy treatments we obtain new insights regarding the development of infections under neutropenic conditions.

Exponential energy growth in a Fermi accelerator.

An unbounded energy growth of particles bouncing off two-dimensional (2D) smoothly oscillating polygons is observed. Notably, such billiards have zero Lyapunov exponents in the static case. For a special 2D polygon geometry--a rectangle with a vertically oscillating horizontal bar--we show that this energy growth is not only unbounded but also exponential in time. For the energy averaged over an ensemble of initial conditions, we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type. We demonstrate numerically that the ensemble averaged energy indeed grows exponentially, at a close to the analytically predicted rate-namely, the process is controllable.

Bistability and bacterial infections.

Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.

Parabolic resonance: a route to hamiltonian spatiotemporal chaos.

We show that initial data near an unperturbed stable plane wave can evolve into a regime of spatiotemporal chaos in the slightly forced conservative periodic one-dimensional nonlinear Schrödinger equation. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. The parameters and initial profiles that lead to such intermittency are predicted by utilizing a novel geometrical description of the integrable unforced equation.

Three types of chaos in the forced nonlinear Schrödinger equation.

Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance, and parabolic resonance) in Hamiltonian perturbations of the nonlinear Schrödinger (NLS) equation are described. The analysis is performed on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. It is demonstrated numerically that the forced NLS equation exhibits analogous types of chaotic phenomena. Thus, by adjusting the forcing frequency, the behavior near the plane wave solution may be set to any one of the three different types of chaos for any periodic box length.

Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials.

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.

Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model.

The truncated forced nonlinear Schrödinger (NLS) model is known to mimic well the forced NLS solutions in the regime at which only one linearly unstable mode exists. Using a novel framework in which a hierarchy of bifurcations is constructed, we analyze this truncated model and provide insights regarding its global structure and the type of instabilities which appear in it. In particular, the significant role of the forcing frequency is revealed and it is shown that a parabolic resonance mechanism of instability arises in the relevant parameter regime of this model. Numerical experiments demonstrating the different types of chaotic motion which appear in the model are provided.