Fractal boundaries in open hydrodynamical flows: Signatures of chaotic saddles

Fractal structures in the chaotic advection of passive scalars in leaky planar hydrodynamical flows

The advection of passive scalars in time-independent two-dimensional incompressible fluid flows is an integrable Hamiltonian system. It becomes non-integrable if the corresponding stream function depends explicitly on time, allowing the possibility of chaotic advection of particles. We consider for a specific model (double gyre flow), a given number of exits through which advected particles can leak, without disturbing the flow itself. We investigate fractal escape basins in this problem and characterize fractality by computing the uncertainty exponent and basin entropy. Furthermore, we observe the presence of basin boundaries with points exhibiting the Wada property, i.e., boundary points that separate three or more escape basins.

Fractal generation in a two-dimensional active-nematic fluid

Active fluids, composed of individual self-propelled agents, can generate complex large-scale coherent flows. A particularly important laboratory realization of such an active fluid is a system composed of microtubules, aligned in a quasi-two-dimensional (2D) nematic phase and driven by adenosine-triphosphate-fueled kinesin motor proteins. This system exhibits robust chaotic advection and gives rise to a pronounced fractal structure in the nematic contours. We characterize such experimentally derived fractals using the power spectrum and discover that the power spectrum decays as k^-β for large wavenumbers k. The parameter β is measured for several experimental realizations. Though β is effectively constant in time, it does vary with experimental parameters, indicating differences in the scale-free behavior of the microtubule-based active nematic. Though the fractal patterns generated in this active system are reminiscent of passively advected dye in 2D chaotic flows, the underlying mechanism for fractal generation is more subtle. We provide a simple, physically inspired mathematical model of fractal generation in this system that relies on the material being locally compressible, though the total area of the material is conserved globally. The model also requires that large-scale density variations are injected into the material periodically. The model reproduces the power-spectrum decay k^-β seen in experiments. Linearizing the model of fractal generation about the equilibrium density, we derive an analytic relationship between β and a single dimensionless quantity r, which characterizes the compressibility.

Alteration of chaotic advection in blood flow around partial blockage zone: Role of hematocrit concentration

Spatial distributions of particles carried by blood exhibit complex filamentary pattern under the combined effects of geometrical irregularities of the blood vessels and pulsating pumping by the heart. This signifies the existence of so called chaotic advection. In the present article, we argue that the understanding of such pathologically triggered chaotic advection is incomplete without giving due consideration to a major constituent of blood: abundant presence of red blood cells quantified by the hematocrit (HCT) concentration. We show that the hematocrit concentration in blood cells can alter the filamentary structures of the spatial distribution of advected particles in an intriguing manner. Our results reveal that there primarily are two major impacts of HCT concentrations towards dictating the chaotic dynamics of blood flow: changing the zone of influence of chaotic mixing and determining the enhancement of residence time of the advected particles away from the wall. This, in turn, may alter the extent of activation of platelets or other reactive biological entities, bearing immense consequence towards dictating the biophysical mechanisms behind possible life-threatening diseases originating in the circulatory system.

New developments in classical chaotic scattering

Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described.

Are the fractal skeletons the explanation for the narrowing of arteries due to cell trapping in a disturbed blood flow?

We show that common circulatory diseases, such as stenoses and aneurysms, generate chaotic advection of blood particles. This phenomenon has major consequences on the way the biochemical particles behave. Chaotic advection leads to a peculiar filamentary particle distribution, which in turn creates a favorable environment for particle reactions. Furthermore, we argue that the enhanced stretching dynamics induced by chaos can lead to the activation of platelets, particles involved in the thrombus formation. In particular, we vary the size of both stenoses and aneurysms, and model them under resting and exercising conditions. We show that the filamentary particle distribution, governed by the fractal skeleton, depends on the size of the vessel wall irregularity, and investigate how it varies under resting or exercising conditions.

Fractal structures in nonlinear plasma physics

Fractal structures appear in many situations related to the dynamics of conservative as well as dissipative dynamical systems, being a manifestation of chaotic behaviour. In open area-preserving discrete dynamical systems we can find fractal structures in the form of fractal boundaries, associated to escape basins, and even possessing the more general property of Wada. Such systems appear in certain applications in plasma physics, like the magnetic field line behaviour in tokamaks with ergodic limiters. The main purpose of this paper is to show how such fractal structures have observable consequences in terms of the transport properties in the plasma edge of tokamaks, some of which have been experimentally verified. We emphasize the role of the fractal structures in the understanding of mesoscale phenomena in plasmas, such as electromagnetic turbulence.

Fractal structures in stenoses and aneurysms in blood vessels

Recent advances in the field of chaotic advection provide the impetus to revisit the dynamics of particles transported by blood flow in the presence of vessel wall irregularities. The irregularity, being either a narrowing or expansion of the vessel, mimicking stenoses or aneurysms, generates abnormal flow patterns that lead to a peculiar filamentary distribution of advected particles, which, in the blood, would include platelets. Using a simple model, we show how the filamentary distribution depends on the size of the vessel wall irregularity, and how it varies under resting or exercise conditions. The particles transported by blood flow that spend a long time around a disturbance either stick to the vessel wall or reside on fractal filaments. We show that the faster flow associated with exercise creates widespread filaments where particles can get trapped for a longer time, thus allowing for the possible activation of such particles. We argue, based on previous results in the field of active processes in flows, that the non-trivial long-time distribution of transported particles has the potential to have major effects on biochemical processes occurring in blood flow, including the activation and deposition of platelets. One aspect of the generality of our approach is that it also applies to other relevant biological processes, an example being the coexistence of plankton species investigated previously.

Chaotic advection in blood flow

In this paper we argue that the effects of irregular chaotic motion of particles transported by blood can play a major role in the development of serious circulatory diseases. Vessel wall irregularities modify the flow field, changing in a nontrivial way the transport and activation of biochemically active particles. We argue that blood particle transport is often chaotic in realistic physiological conditions. We also argue that this chaotic behavior of the flow has crucial consequences for the dynamics of important processes in the blood, such as the activation of platelets which are involved in the thrombus formation.

Transition to chaotic scattering: signatures in the differential cross section

We show that bifurcations in chaotic scattering manifest themselves through the appearance of an infinitely fine-scale structure of singularities in the cross section. These "rainbow singularities" are created in a cascade, which is closely related to the bifurcation cascade undergone by the set of trapped orbits (the chaotic saddle). This cascade provides a signature in the differential cross section of the complex pattern of bifurcations of orbits underlying the transition to chaotic scattering. We show that there is a power law with a universal coefficient governing the sequence of births of rainbow singularities and we verify this prediction by numerical simulations.

Onset of chaotic advection in open flows

In this paper we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time dependence in open flows with stagnant points, in contrast with flows with no stagnant points. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder.

Chaotic saddles at the onset of intermittent spatiotemporal chaos

In a recent study [Rempel and Chian, Phys. Rev. Lett. 98, 014101 (2007)], it has been shown that nonattracting chaotic sets (chaotic saddles) are responsible for intermittency in the regularized long-wave equation that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity and temporal chaos. In the present paper, it is demonstrated that a similar mechanism is present in the damped Kuramoto-Sivashinsky equation. Prior to the onset of STC, a spatiotemporally chaotic saddle coexists with a spatially regular attractor. After the transition to STC, the chaotic saddle merges with the attractor, generating intermittent bursts of STC that dominate the post-transition dynamics.

Fractal dimension in dissipative chaotic scattering

The effect of weak dissipation on chaotic scattering is relevant to situations of physical interest. We investigate how the fractal dimension of the set of singularities in a scattering function varies as the system becomes progressively more dissipative. A crossover phenomenon is uncovered where the dimension decreases relatively more rapidly as a dissipation parameter is increased from zero and then exhibits a much slower rate of decrease. We provide a heuristic theory and numerical support from both discrete-time and continuous-time scattering systems to establish the generality of this phenomenon. Our result is expected to be important for physical phenomena such as the advection of inertial particles in open chaotic flows, among others.

Reactions in flows with nonhyperbolic dynamics

We study the reaction dynamics of active particles that are advected passively by 2D incompressible open flows, whose motion is nonhyperbolic. This nonhyperbolicity is associated with the presence of persistent vortices near the wake, wherein fluid is trapped. We show that the fractal equilibrium distribution of the reactants is described by an effective dimension d(eff) , which is a finite resolution approximation to the fractal dimension. Furthermore, d(eff) depends on the resolution epsilon and on the reaction rate 1/tau . As tau is increased, the equilibrium distribution goes through a series of transitions where the effective dimension increases abruptly. These transitions are determined by the complex structure of Cantori surrounding the Kolmogorov-Arnold-Moser (KAM) islands.

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.

On the estimate of the stochastic layer width for a model of tracer dynamics

An analytical estimate of the width of the generated chaotic layer in a time-periodically driven stream function model for the motion of passive tracers is discussed. It is based essentially on the method of the separatrix map and the use of the Melnikov theory. Energy-time variables are used to derive lower bounds for the half width of the layer. In order to perform a comparison with numerical simulations, the results are transformed into space variables. The analytic results of the layer thickness in both parallel and perpendicular directions to the shear flow are compared with numerical computations and some systematic deviations are discussed.

Countable and uncountable boundaries in chaotic scattering

We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.

Autocatalytic reactions of phase distributed active particles

We investigate the effect of asynchronism of autocatalytic reactions taking place in open hydrodynamical flows, by assigning a phase to each particle in the system to differentiate the timing of the reaction, while the reaction rate (periodicity) is kept unchanged. The chaotic saddle in the flow dynamics acts as a catalyst and enhances the reaction in the same fashion as in the case of a synchronous reaction that was studied previously, proving that the same type of nonlinear reaction kinetics is valid in the phase-distributed situation. More importantly, we show that, in a certain range of a parameter, the phenomenon of phase selection can occur, when a group of particles with a particular phase is favored over the others, thus occupying a larger fraction of the available space, or eventually leading to the extinction of the unfavored phases. We discuss the biological relevance of this latter phenomenon. (c) 2002 American Institute of Physics.

Metabolic network dynamics in open chaotic flow

We have analyzed the dynamics of metabolically coupled replicators in open chaotic flows. Replicators contribute to a common metabolism producing energy-rich monomers necessary for replication. The flow and the biological processes take place on a rectangular grid. There can be at most one molecule on each grid cell, and replication can occur only at localities where all the necessary replicators (metabolic enzymes) are present within a certain neighborhood distance. Due to this finite metabolic neighborhood size and imperfect mixing along the fractal filaments produced by the flow, replicators can coexist in this fluid system, even though coexistence is impossible in the mean-field approximation of the model. We have shown numerically that coexistence mainly depends on the metabolic neighborhood size, the kinetic parameters, and the number of replicators coupled through metabolism. Selfish parasite replicators cannot destroy the system of coexisting metabolic replicators, but they frequently remain persistent in the system. (c) 2002 American Institute of Physics.

Rainbow transition in chaotic scattering

We study the effects of classical chaotic scattering on the differential cross section, which is the measurable quantity in most scattering experiments. We show that the fractal set of singularities in the deflection function is not, in general, reflected on the differential cross section. We show that there are systems in which, as the energy (or some other parameter) crosses a critical value, the system's differential cross-section changes from a singular function having an infinite set of rainbow singularities with structure in all scales to a smooth function with no singularities, the scattering being chaotic on both sides of the transition. We call this metamorphosis the rainbow transition. We exemplify this transition with a physically relevant class of systems. These results have important consequences for the problem of inverse scattering in chaotic systems and for the experimental observation of chaotic scattering.

Driving trajectories in chaotic scattering

In this work we introduce a general approach for targeting in chaotic scattering that can be used to find a transfer trajectory between any two points located inside the scattering region. We show that this method can be used in association with a control of chaos strategy to drive around and keep a particle inside the scattering region. As an illustration of how powerful this approach is, we use it in a case of practical interest in celestial mechanics in which it is desired to control the evolution of two satellites that evolve around a large central body.

Dissipative chaotic scattering

We show that weak dissipation, typical in realistic situations, can have a metamorphic consequence on nonhyperbolic chaotic scattering in the sense that the physically important particle-decay law is altered, no matter how small the amount of dissipation. As a result, the previous conclusion about the unity of the fractal dimension of the set of singularities in scattering functions, a major claim about nonhyperbolic chaotic scattering, may not be observable.

Analyses of transient chaotic time series

We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits, from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

Stagger-and-step method: detecting and computing chaotic saddles in higher dimensions

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.

Chaotic flow: the physics of species coexistence

Hydrodynamical phenomena play a keystone role in the population dynamics of passively advected species such as phytoplankton and replicating macromolecules. Recent developments in the field of chaotic advection in hydrodynamical flows encourage us to revisit the population dynamics of species competing for the same resource in an open aquatic system. If this aquatic environment is homogeneous and well-mixed then classical studies predict competitive exclusion of all but the most perfectly adapted species. In fact, this homogeneity is very rare, and the species of the community (at least on an ecological observation time scale) are in nonequilibrium coexistence. We argue that a peculiar small-scale, spatial heterogeneity generated by chaotic advection can lead to coexistence. In open flows this imperfect mixing lets the populations accumulate along fractal filaments, where competition is governed by an "advantage of rarity" principle. The possibility of this generic coexistence sheds light on the enrichment of phytoplankton and the information integration in early macromolecule evolution.

Detecting unstable periodic orbits from transient chaotic time series

We address the detection of unstable periodic orbits from experimentally measured transient chaotic time series. In particular, we examine recurrence times of trajectories in the vector space reconstructed from an ensemble of such time series. Numerical experiments demonstrate that this strategy can yield periodic orbits of low periods even when noise is present. We analyze the probability of finding periodic orbits from transient chaotic time series and derive a scaling law for this probability. The scaling law implies that unstable periodic orbits of high periods are practically undetectable from transient chaos.

Chaotic advection, diffusion, and reactions in open flows

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.

Unstable periodic orbits and the natural measure of nonhyperbolic chaotic saddles

Chaotic saddles are nonattracting dynamical invariant sets that physically lead to transient chaos. We examine the characterization of the natural measure by unstable periodic orbits for nonhyperbolic chaotic saddles in dissipative dynamical systems. In particular, we compare the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in it. Our systematic computations indicate that the periodic-orbit theory of the natural measure, previously shown to be valid only for hyperbolic chaotic sets, is applicable to nonhyperbolic chaotic saddles as well.

Chemical or biological activity in open chaotic flows

We investigate the evolution of particle ensembles in open chaotic hydrodynamical flows. Active processes of the type A+B-->2B and A+B-->2C are considered in the limit of weak diffusion. As an illustrative advection dynamics we consider a model of the von Kármán vortex street, a time-periodic two-dimensional flow of a viscous fluid around a cylinder. We show that a fractal unstable manifold acts as a catalyst for the process, and the products cover fattened-up copies of this manifold. This may account for the observed filamental intensification of activity in environmental flows. The reaction equations valid in the wake are derived either in the form of dissipative maps or differential equations depending on the regime under consideration. They contain terms that are not present in the traditional reaction equations of the same active process: the decay of the products is slower while the productivity is much faster than in homogeneous flows. Both effects appear as a consequence of underlying fractal structures. In the long time limit, the system locks itself in a dynamic equilibrium state synchronized to the flow for both types of reactions. For particles of finite size an emptying transition might also occur leading to no products left in the wake.

Universal behavior in the parametric evolution of chaotic saddles

Chaotic saddles are nonattracting dynamical invariant sets that physically lead to transient chaos. As a system parameter changes, chaotic saddles can evolve via an infinite number of homoclinic or heteroclinic tangencies of their stable and unstable manifolds. Based on previous numerical evidence and a rigorous analysis of a class of representative models, we show that dynamical invariants such as the topological entropy and the fractal dimension of chaotic saddles obey a universal behavior: they exhibit a devil-staircase characteristic as a function of the system parameter.

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (c) 1997 American Institute of Physics.