Áron Péntek

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.

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.

Competing populations in flows with chaotic mixing

We investigate the effects of spatial heterogeneity on the coexistence of competing species in the case when the heterogeneity is dynamically generated by environmental flows with chaotic mixing properties. We show that one effect of chaotic advection on the passively advected species (such as phytoplankton, or self-replicating macro-molecules) is the possibility of coexistence of more species than that limited by the number of niches they occupy. We derive a novel set of dynamical equations for competing populations.

Fractality, chaos, and reactions in imperfectly mixed open hydrodynamical flows

We investigate the dynamics of tracer particles in time-dependent open flows. If the advection is passive the tracer dynamics is shown to be typically transiently chaotic. This implies the appearance of stable fractal patterns, so-called unstable manifolds, traced out by ensembles of particles. Next, the advection of chemically or biologically active tracers is investigated. Since the tracers spend a long time in the vicinity of a fractal curve, the unstable manifold, this fractal structure serves as a catalyst for the active process. The permanent competition between the enhanced activity along the unstable manifold and the escape due to advection results in a steady state of constant production rate. This observation provides a possible solution for the so-called “paradox of plankton”, that several competing plankton species are able to coexists in spite of the competitive exclusion predicted by classical studies. We point out that the derivation of the reaction (or population dynamics) equations is analog to that of the macroscopic transport equations based on a microscopic kinetic theory whose support is a fractal subset of the full phase space.

Autocatalytic reactions in systems with hyperbolic mixing: Exact results for the active Baker map

We investigate the effects of hyperbolic hydrodynamical mixing on the reaction kinetics of autocatalytic systems. Exact results are derived for the two-dimensional open Baker map as an underlying mixing dynamics for a two-component autocatalytic system, A + B→2B. We prove that chaotic advection modelled by the Baker map enhances the productivity of the reaction which is due to the fact that the reaction kinetics is catalysed by the fractal unstable manifold of the chaotic set of the reaction-free dynamics. The results are compared with phenomenological theories of active advection.

Stabilizing chaotic vortex trajectories: an example of high-dimensional control

Abstract A chaos control algorithm is developed to actively stabilize unstable periodic orbits of higher-dimensional systems. The method assumes knowledge of the model equations and a small number of experimentally accessible parameters. General conditions for controllability are discussed. The algorithm is applied to the Hamiltonian problem of point vortices inside a circular cylinder with applications to an experimental plasma system.

TRANSIENT CHAOTIC MIXING IN OPEN HYDRODYNAMICAL FLOWS

We investigate particle motion and mixing in time-dependent open flows with uniform inflow and outflow velocities. The dynamics is typically chaotic, of the transient type that can be observed on finite time scales only. There exists an underlying chaotic set consisting of an infinity of unstable tracer orbits, restricted to a finite domain of the flow. Transient dynamics is accompanied by persistent fractality. The latter is reflected by tracer patterns and by the singularity distribution of the particles’ time delay function. As an illustrative example, we consider a planar incompressible flow modeling the leapfrogging motion of two vortex rings.

Controlled capture of a continuous vorticity distribution

We develop a chaotic control and targeting scheme to capture and stabilize a concentrated vortex around a cylindrical body embedded in an open fluid flow. We demonstrate that the point vortex-based control model is also valid in a continuum fluid framework, by simulating the system using a Navier-Stokes-like dynamics.

Dynamical classification of hydroacoustic data from a Baltic Sea experiment

The authors analyze the acoustic signatures of a small boat using nonlinear dynamical signal models. Specifically, they discuss the estimation of the parameters of a nonlinear delay differential equation from this data. Using the model coefficients as detection features they implement a Mahalanobis distance-based decision criteria to perform rigorous hypothesis testing. By analyzing acoustic data recorded in shallow water in the Baltic Sea, they compare the performance of the dynamical detector with a frequency band-matched energy detector and show that the former provides increased detection performance. In addition, they demonstrate that the delay differential equation model can distinguish details of the boats signature, such as the engine RPM, hence it could be used for classification purposes as well.

Acoustic discrimination between aircraft and land vehicles using nonlinear dynamical signal models

One of the most important applications of nonlinear dynamics is the estimation of empirical dynamical models from data, in order to explain time series derived from physical processes. Such derived models can then be used for a variety of data processing applications, in particular for detection and classification problems. Previously, we presented a theory and numerical approach for the estimation of such nonlinear dynamical models in the detection and classification of acoustic data. Here, we apply these ideas to perform discrimination between aircraft and land vehicles based on acoustic signature recordings, derived from a field test of the USA Army.