Theoretical and numerical studies of chaotic mixing

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity

The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current

This article reviews several recently developed Lagrangian tools and shows how their combined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the climate impact of ocean transport processes, we illustrate a 2D application on altimeter data sets over the area of the Kuroshio Current, although the proposed techniques are general and applicable to arbitrary time dependent aperiodic flows. The first challenge for describing transport in aperiodical time dependent flows is obtaining a representation of the phase portrait where the most relevant dynamical features may be identified. This representation is accomplished by using global Lagrangian descriptors that when applied for instance to the altimeter data sets retrieve over the ocean surface a phase portrait where the geometry of interconnected dynamical systems is visible. The phase portrait picture is essential because it evinces which transport routes are acting on the whole flow. Once these routes are roughly recognised it is possible to complete a detailed description by the direct computation of the finite time stable and unstable manifolds of special hyperbolic trajectories that act as organising centres of the flow.Comment: 40 pages, 24 figure

Complex Dynamics in Dedicated / Multifunctional Neural Networks and Chaotic Nonlinear Systems

We study complex behaviors arising in neuroscience and other nonlinear systems by combining dynamical systems analysis with modern computational approaches including GPU parallelization and unsupervised machine learning. To gain insights into the behaviors of brain networks and complex central pattern generators (CPGs), it is important to understand the dynamical principles regulating individual neurons as well as the basic structural and functional building blocks of neural networks. In the first section, we discuss how symbolic methods can help us analyze neural dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations in various models of individual neurons, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits, such as network bursting from non-intrinsic bursters. The second section is focused on the origin and coexistence of multistable rhythms in oscillatory neural networks of inhibitory coupled cells. We discuss how network connectivity and intrinsic properties of the cells affect the dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. Our analyses can help generate verifiable hypotheses for neurophysiological experiments on central pattern generators. In the last section, we demonstrate the inter-disciplinary nature of this research through the applications of these techniques to identify the universal principles governing both simple and complex dynamics, and chaotic structure in diverse nonlinear systems. Using a classical example from nonlinear laser optics, we elaborate on the multiplicity and self-similarity of key organizing structures in 2D parameter space such as homoclinic and heteroclinic bifurcation curves, Bykov T-point spirals, and inclination flips. This is followed by detailed computational reconstructions of the spatial organization and 3D embedding of bifurcation surfaces, parametric saddles, and isolated closed curves (isolas). The generality of our modeling approaches could lead to novel methodologies and nonlinear science applications in biological, medical and engineering systems

Vortex-induced vibrations of a non-linearly supported rigid cylinder

Vortex-Induced Vibrations (VIV) are a complex fluid-structure interaction problem.VIV are particularly strong for low-mass structures subject to a low damping, asencountered in the offshore industry, in which structures experiencing VIV can alsobe subject to strong structural non-linearities.In this project, investigation of the VIV of a low-mass low-damping rigid cylindersubject to structural non-linearities is carried out first experimentally. Non-linearitiesconsidered are the symmetric or asymmetric limitation of the amplitude of thecylinder with soft or stiff stops placed at different offsets from the cylinder andimplying a non-smooth non-linearity of the system. Experimental results show that astrong perturbation of the dynamics of the cylinder occurs when amplitude limitationis strong, and flow visualisations displaying a modification of the vortex wake suggesta change in the fluid-structure interaction affecting the vortex formation process.Attention is also given to the impact velocities in the different cases of amplitudelimitation with stiff stops, as they are an important factor in the design of structures.Two different wake oscillator models are then used to simulate the VIV of the samerigid circular cylinder in the same conditions of non-linear structural restraints.Results show that these simple models exhibit some features observed experimentally,giving in some cases a good estimation of the experimental data

Critical Market Crashes

This review is a partial synthesis of the book ``Why stock market crash'' (Princeton University Press, January 2003), which presents a general theory of financial crashes and of stock market instabilities that his co-workers and the author have developed over the past seven years. The study of the frequency distribution of drawdowns, or runs of successive losses shows that large financial crashes are ``outliers'': they form a class of their own as can be seen from their statistical signatures. If large financial crashes are ``outliers'', they are special and thus require a special explanation, a specific model, a theory of their own. In addition, their special properties may perhaps be used for their prediction. The main mechanisms leading to positive feedbacks, i.e., self-reinforcement, such as imitative behavior and herding between investors are reviewed with many references provided to the relevant literature outside the confine of Physics. Positive feedbacks provide the fuel for the development of speculative bubbles, preparing the instability for a major crash. We demonstrate several detailed mathematical models of speculative bubbles and crashes. The most important message is the discovery of robust and universal signatures of the approach to crashes. These precursory patterns have been documented for essentially all crashes on developed as well as emergent stock markets, on currency markets, on company stocks, and so on. The concept of an ``anti-bubble'' is also summarized, with two forward predictions on the Japanese stock market starting in 1999 and on the USA stock market still running. We conclude by presenting our view of the organization of financial markets.Comment: Latex 89 pages and 38 figures, in press in Physics Report

Comparative evaluation of approaches in T.4.1-4.3 and working definition of adaptive module

The goal of this deliverable is two-fold: (1) to present and compare different approaches towards learning and encoding movements us- ing dynamical systems that have been developed by the AMARSi partners (in the past during the first 6 months of the project), and (2) to analyze their suitability to be used as adaptive modules, i.e. as building blocks for the complete architecture that will be devel- oped in the project. The document presents a total of eight approaches, in two groups: modules for discrete movements (i.e. with a clear goal where the movement stops) and for rhythmic movements (i.e. which exhibit periodicity). The basic formulation of each approach is presented together with some illustrative simulation results. Key character- istics such as the type of dynamical behavior, learning algorithm, generalization properties, stability analysis are then discussed for each approach. We then make a comparative analysis of the different approaches by comparing these characteristics and discussing their suitability for the AMARSi project

Theoretical and numerical studies of chaotic mixing

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity

Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics

The research presented in this PhD thesis within the framework of nonlinear deterministic dynamical systems depending on parameters. The work is divided into four Chapters, where the first is a general introduction to the other three. Chapter two deals with the investigation of a time-periodic three-dimensional system of ordinary differential equations depending on three parameters, the Lorenz-84 model with seasonal forcing. The model is a variation on an autonomous system proposed in 1984 by the meteorologist E. Lorenz to describe general atmospheric circulation at mid latitude of the northern hemisphere. ... Zie: Summary