Refining Finite-Time Lyapunov Exponent Ridges and the Challenges of Classifying Them

While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic

qq-Breathers in finite lattices: nonlinearity and weak disorder

Nonlinearity and disorder are the recognized ingredients of the lattice vibrational dynamics, the factors that could be diminished, but never excluded. We generalize the concept of $q$-breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal linear mode space -- to the case of weak disorder, taking the Fermi-Pasta-Ulan chain as an example. We show, that these nonlinear vibrational modes remain exponentially localized near the central mode and stable, provided the disorder is sufficiently small. The instability threshold depends sensitively on a particular realization of disorder and can be modified by specifically designed impurities. Basing on it, an approach to controlling the energy flow between the modes is proposed. The relevance to other model lattices and experimental miniature arrays is discussed.Comment: 4 pages, 3 figure

q-Breathers and the Fermi-Pasta-Ulam Problem

The Fermi-Pasta-Ulam (FPU) paradox consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number $q$. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here $q$-Breathers (QB). They are characterized by time periodicity, exponential localization in the $q$-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.Comment: 4 pages, 4 figure

Null Energy Condition Violation and Classical Stability in the Bianchi I Metric

The stability of isotropic cosmological solutions in the Bianchi I model is considered. We prove that the stability of isotropic solutions in the Bianchi I metric for a positive Hubble parameter follows from their stability in the Friedmann-Robertson-Walker metric. This result is applied to models inspired by string field theory, which violate the null energy condition. Examples of stable isotropic solutions are presented. We also consider the k-essence model and analyse the stability of solutions of the form $\Phi(t)=t$.Comment: 27 pages, references added, accepted for publication in Phys. Rev.

Conceptual inconsistencies in finite-dimensional quantum and classical mechanics

Utilizing operational dynamic modeling [Phys. Rev. Lett. 109, 190403 (2012); arXiv:1105.4014], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechanics is a straightforward discretization of the corresponding infinite-dimensional formulation.Comment: 5 pages, 2 figure

Stability of Simple Periodic Orbits and Chaos in a Fermi -- Pasta -- Ulam Lattice

We investigate the connection between local and global dynamics in the Fermi -- Pasta -- Ulam (FPU) $\beta$ -- model from the point of view of stability of its simplest periodic orbits (SPOs). In particular, we show that there is a relatively high $q$ mode $(q=2(N+1)/{3})$ of the linear lattice, having one particle fixed every two oppositely moving ones (called SPO2 here), which can be exactly continued to the nonlinear case for $N=5+3m, m=0,1,2,...$ and whose first destabilization, $E_{2u}$, as the energy (or $\beta$) increases for {\it any} fixed $N$, practically {\it coincides} with the onset of a ``weak'' form of chaos preceding the break down of FPU recurrences, as predicted recently in a similar study of the continuation of a very low ($q=3$) mode of the corresponding linear chain. This energy threshold per particle behaves like $\frac{E_{2u}}{N}\propto N^{-2}$. We also follow exactly the properties of another SPO (with $q=(N+1)/{2}$) in which fixed and moving particles are interchanged (called SPO1 here) and which destabilizes at higher energies than SPO2, since $\frac{E_{1u}}{N}\propto N^{-1}$. We find that, immediately after their first destabilization, these SPOs have different (positive) Lyapunov spectra in their vicinity. However, as the energy increases further (at fixed $N$), these spectra converge to {\it the same} exponentially decreasing function, thus providing strong evidence that the chaotic regions around SPO1 and SPO2 have ``merged'' and large scale chaos has spread throughout the lattice.Comment: Physical Review E, 18 pages, 6 figure

Numerical integration of variational equations

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the \textit{`tangent map (TM) method'}, a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic map $S$, while the corresponding tangent map $TS$, is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.Comment: 27 pages, 11 figures, to appear in Phys. Rev.

Stability of Nonlinear Normal Modes in the FPU-β\beta Chain in the Thermodynamic Limit

All possible symmetry-determined nonlinear normal modes (also called by simple periodic orbits, one-mode solutions etc.) in both hard and soft Fermi-Pasta-Ulam-$\beta$ chains are discussed. A general method for studying their stability in the thermodynamic limit, as well as its application for each of the above nonlinear normal modes are presented

A saddle in a corner - a model of collinear triatomic chemical reactions

A geometrical model which captures the main ingredients governing atom-diatom collinear chemical reactions is proposed. This model is neither near-integrable nor hyperbolic, yet it is amenable to analysis using a combination of the recently developed tools for studying systems with steep potentials and the study of the phase space structure near a center-saddle equilibrium. The nontrivial dependence of the reaction rates on parameters, initial conditions and energy is thus qualitatively explained. Conditions under which the phase space transition state theory assumptions are satisfied and conditions under which these fail are derived

Дослідження емульсій м/в методами ротаційної віскозиметрії та спінових зондів

The study of o/w emulsions using the rotating viscometer method and the method of spin probesThe rheological properties of o/w emulsions used as vehicles for semi-solid preparations (SSP) depend on the composition of o/w and w/o emulsifiers to a large extent, and it is related to the structure of their aggregates in emulsions. In order to control the rheological parameters of emulsions it is necessary to know the regularities of the influence of the composition and properties of emulsifiers on these parameters, as well as the mechanism of emulsion stabilization, which depends on the structure of the aggregates formed by emulsifiers.Aim. To study the relationship between the structure of the aggregates formed by emulsifiers and the rheological properties of o/w emulsions.Materials and methods. The objects of the study were o/w emulsions stabilized with macrogol-40 stearate (M40S), glyceryl monostearate 40-55 (type II) (GMS) and cetostearyl alcohol (CSA). The rheological studies were performed by the rotational viscometer method. The structure of aggregates of emulsifiers was studied by the spin probes method. Probes simulating o/w and w/o emulsifiers with different localization of the radical were used.Results and discussion. The apparent viscosity of emulsions stabilized with the o/w emulsifier (M40S) and a mixture of w/o emulsifiers (GMS and CSA) is maximal at a certain ratio between o/w and w/o emulsifiers. The increase in the rheological parameters correlates with the increase in the packing density of emulsifier molecules in the polar part of their aggregates; it in this case practically does not change at the level of the 5 and 16 carbon atoms of alkyl chains. This indicates formation of non-spherical aggregates which form a coagulation structure in the emulsion.Conclusions. The change in the ratio between o/w and w/o emulsifiers results in the change in the structure of their aggregates, and it affects the rheological properties of o/w emulsions. Based on the results of the studies it is possible to reasonably control the rheological parameters of preparations with emulsions as a vehicle.Реологические свойства эмульсий м/в, применяемых в качестве основ для мягких лекарственных средств (МЛС), во многом зависят от состава эмульгаторов м/в и в/м, что связано со структурой их агрегатов в эмульсиях. Чтобы управлять реопараметрами эмульсий, необходимо знать закономерности влияния на них состава и свойств эмульгаторов, а также механизм стабилизации эмульсий, зависящий от структуры агрегатов, образованных эмульгаторами. Цель. Исследовать связь между структурой агрегатов, образованных эмульгаторами, и реологическими свойствами эмульсий м/в. Материалы и методы. Объекты исследований – эмульсии м/в, стабилизированные макрогол-40 стеаратом (М40С), глицерилмоностеаратом 40-55 (типа II) (ГМС) и цетостеариловым спиртом (ЦСС). Реологические исследования проводили методом ротационной вискозиметрии. Структуру агрегатов эмульгаторов исследовали методом спиновых зондов. Использовали зонды, моделирующие эмульгаторы м/в и в/м, с разной локализацией радикала.Результаты и их обсуждение. Структурная вязкость эмульсий, стабилизированных эмульгатором м/в М40С и смесью эмульгаторов в/м (ГМС и ЦСС), максимальна при определённом соотношении между эмульгаторами м/в и в/м. Увеличение реопараметров коррелирует с повышением плотности упаковки молекул эмульгаторов в полярной части их агрегатов, которая при этом не изменяется на уровне 5 и 16 атомов углерода алкильных цепей. Это свидетельствует об образовании агрегатов несферической формы, из которых в эмульсии формируется коагуляционная структура. Выводы. Изменение соотношения между эмульгаторами м/в и в/м изменяет структуру их агрегатов, что влияет на реологические свойства эмульсий м/в. На основании результатов исследований можно обоснованно управлять реологическими параметрами препаратов на основе эмульсий.Реологічні властивості емульсій м/в, що застосовуються як основи для м’яких лікарських засобів (МЛЗ), багато в чому залежать від складу емульгаторів м/в та в/м, що пов’язано зі структурою їх агрегатів в емульсіях. Щоб управляти реопараметрами емульсій необхідно знати закономірності впливу на них складу та властивостей емульгаторів, а також механізм стабілізації емульсій, який залежить від структури агрегатів, утворених емульгаторами. Мета роботи. Дослідити зв’язок між структурою агрегатів, утворених емульгаторами, і реологічними властивостями емульсій м/в. Матеріали та методи. Об’єкти досліджень – емульсії м/в, стабілізовані макрогол-40 стеаратом (М40С), гліцерилмоностеаратом 40-55 (типу II) (ГМС) і цетостеариловим спиртом (ЦСС). Реологічні дослідження проводили методом ротаційної віскозиметрії. Структуру агрегатів емульгаторів досліджували методом спінових зондів. Використовували зонди, що моделюють емульгатори м/в та в/м, з різною локалізацією радикалу.Результати та їх обговорення. Структурна в’язкість емульсій, стабілізованих емульгатором м/в М40С і сумішшю емульгаторів в/м (ГМС і ЦСС), максимальна при певному співвідношенні між емульгаторами м/в та в/м. Збільшення реопараметрів корелює з підвищенням щільності упаковки молекул емульгаторів у полярній частині їх агрегатів, яка при цьому не змінюється на рівні 5 і 16 атомів вуглецю алкільних ланцюгів. Це свідчить про утворенням агрегатів несферичної форми, з яких в емульсії формується коагуляційна структура. Висновки. Зміна співвідношення між емульгаторами м/в та в/м змінює структуру їх агрегатів, що впливає на реологічні властивості емульсій м/в. За результатами досліджень можна обґрунтовано управляти реологічними параметрами препаратів на основі емульсій