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.

Richardson equations are widely studied models of simplified multi-player arms races. In this paper, we derive a novel and more theoretically-consistent formulation, which implicitly includes a mapping from observed economic data to the model variables. We define new variables which are time-integrated versions of scaled `defense' expenditures, and show that they may be robustly estimated from historical data. We illustrate the model's usefulness with historical examples and a simplified risk analysis of simulated data.

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, $A+B \to 2B$. We prove that the hyperboliticity exponentially enhances the productivity of the reaction which is due the fact that the reaction kinetics is catalyzed by the fractal unstable manifold of the chaotic set of the reaction-free dynamics. The results are compared with phenomenological theories of active advection.

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.

We analyze sonar recordings of various boats as well as ambient sea noise using nonlinear dynamical signal models. Specifically, we discuss the estimation of the parameters of nonlinear delay differential equations from data. Using the model parameters as classification features we implement a three class Bayesian minimum-error-rate classifier and demonstrate almost perfect classification of the data set considered. This indicates that, classifiers based on nonlinear dynamical models can be useful in sonar applications.

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.

1. Recent developments in the field of chaotic advection in hydrodynamical/environmental flows encourage us to revisit the population dynamics of competing species in open aquatic systems. 2. We assume that these species are in competition for a common limiting resource in open flows with chaotic advection dynamics. As an illustrative example, we consider a time periodic two‐dimensional flow of viscous fluid (water) around a cylindrical obstacle. 3. Individuals accumulate along a fractal set in the wake of the cylinder, which acts as a catalyst for the biological reproduction process. While in homogeneous, well mixed environments only one species could survive this competition, coexistence of competitors is typical in our hydrodynamical system. 4. It is shown that a steady state sets in after sufficiently long times. In this state, the relative density of competitors is determined rather by the fractal nature of the spatial distribution of the advected species, and by their initial conditions, than by their competitive abilities. We argue that two factors, the strong chaotic mixing along a fractal set and the boundary layer around the obstacle, are responsible for the coexistence.

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. Typically, the parameters of such dynamical models are estimated directly from the time series by minimizing a cost function with least squares. In this paper we discuss the theory and applications of an alternate approach for estimation of such nonlinear dynamical models and the use of these models for detection and classification of seismic and acoustic data. We apply these ideas to real data derived from seismic station recordings in the region of the Panama Canal. Finally we compare our results with that previously achieved by the method of master-event correlations, and find improved performance. This indicates that a dynamical model approach incorporates additional signal information in this example

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.

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 boat's signature, such as the engine RPM, hence it could be used for classification purposes as well

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.

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.

Abstract We discuss a method for estimating nonlinear dynamical models from data, to construct a closed-form, few-parameter representation. The model coefficients are estimated using a Yule-Walker type equation involving higher-order correlations, which provides computational speed, numerical stability and noise robustness. We implement a multi-feature detector based on Mahalanobis distance and compute ROC curves for several signal classes. We show that the dynamical detectors’ performance can be improved significantly by increasing the sampling rate.

We investigate the evolution of active particle ensembles in open chaotic flows. The active processes of the type $A+B\ensuremath{\rightarrow}2B$ and $A+B\ensuremath{\rightarrow}2C$ are considered in the limit of weak diffusion. As an illustrative advection dynamics, we choose a model of the von K\'arm\'an vortex street, and show that the backbone of the active processes is the fractal structure associated with the passive dynamics' chaotic saddle. This fractal dynamics leads to singularly enhanced concentrations, resulting in a distribution of products that differs entirely from the one in conventional active processes. This may account for the observed filamental intensification of activity in environmental flows.

We investigate the vortex dynamics near a translating and rotating circular cylinder in a two-dimensional uniform viscous flow. In analogy with the point-vortex and Eulerian dynamics, there is an interesting scattering effect of vortices approaching the cylinder from far upstream. The vortex-boundary-layer interaction plays an important role in the scattering processes. We implement a modified Ott, Grebogi, and Yorke chaos control scheme, based on a low-dimensional Hamiltonian model of the flow, to capture and stabilize a concentrated vortex around the cylinder. This point-vortex-based control model can successfully be applied in a viscous flow when control is actuated by uniformly rotating the cylinder and actively changing the background flow velocity far from the body. We demonstrate that such a control mechanism can simultaneously control the vortex dynamics, and also suppress the vortex shedding. An analysis of the vortex-boundary-layer interaction is presented to explain the absence of vortex shedding during control simulations.

A high-dimensional chaos control algorithm is applied to stabilize unstable orbits of symmetric point-vortex configurations. We discuss possible applications to an experimental plasma system.

Dyes of different colors advected by two-dimensional flows which are asymptotically simple can form a fractal boundary that coincides with a chaotic saddle's unstable manifold. We show that such dye boundaries can have the Wada property: every boundary point of a given color on this fractal set is on the boundary of at least two other colors. The condition for this is the nonempty intersection of the saddle's stable manifold with at least three differently colored domains in the asymptotic inflow region.