Variations in the dimension of chaotic systems

It is usually expected that the number of modes necessary to model turbulence increases with the appropriate control parameter. With the help of a concrete model, we show that this property may be shared by low-dimensional truncations of the Navier-Stokes equations.     

Fractal dimension of steady nonequilibrium flows

The Kaplan-Yorke information dimension of phase-space attractors for two kinds of steady nonequilibrium many-body flows is evaluated. In both cases a set of Newtonian particles is considered which interacts with boundary particles. Time-averaged boundary temperatures are imposed by Nose-Hoover thermostat forces. For both kinds of nonequilibrium systems, it is demonstrated numerically that external isothermal boundaries can drive the otherwise purely Newtonian flow onto a multifractal attractor with a phase-space information dimension significantly less than that of the corresponding equilibrium flow. Thus the Gibbs' entropy of such nonequilibrium flows can diverge.     

Reacting particles in open chaotic flows

We study the collision probability p of particles advected by open flows with chaotic advection. We show that p scales with the particle size (or, alternatively, reaction distance) δ as a power law whose coefficient is determined by the fractal dimensions of the invariant sets defined by the advection dynamics. We also argue that this same scaling also holds for the reaction rate of active particles in the low-density regime. These analytical results are compared to numerical simulations, and they are found to agree very well.     

Coexistence of inertial competitors in chaotic flows

We investigate the dynamics of inertial particles immersed in open chaotic flows. We consider the generic problem of competition between different species, e.g., phytoplankton populations in oceans. The strong influence from inertial effects is shown to result in the persistence of different species even in cases when the passively advected species cannot coexist. Multispecies coexistence in the ocean can be explained by the fact that the unstable manifold is different for each advected competitor of different size.     

Elementary quadratic chaotic flows with no equilibria

Three methods are used to produce a catalog of seventeen elementary three-dimensional chaotic flows with quadratic nonlinearities that have the unusual feature of lacking any equilibrium points. It is likely that most if not all the elementary examples of such systems have now been identified.     

Residence-time distributions for chaotic flows in pipes

In this paper we derive two rigorous properties of residence-time distributions for flows in pipes and mixers motivated by computational results of Khakhar et al. [Chem. Eng. Sci. 42, 2909 (1987)], using some concepts from ergodic theory. First, a curious similarity between the isoresidence-time plots and Poincare maps of the flow observed in Khakhar et al. is resolved. It is shown that in long pipes and mixers, Poincare maps can serve as a useful guide in the analysis of isoresidence-time plots, but the two are not equivalent. In particular, for long devices isoresidence-time sets are composed of orbits of the Poincare map, but each isoresidence-time set can be comprised of many orbits. Second, we explain the origin of multimodal residence-time distributions for nondiffusive motion of particles in pipes and mixers. It is shown that chaotic regions in the Poincare map contribute peaks to the appropriately defined and rescaled axial distribution functions. (c) 1999 American Institute of Physics.     

Alpha effect in a family of chaotic flows

We perform numerical experiments to calculate the kinematic alpha effect for a family of maximally helical, chaotic flows with a range of correlation times. We find that the value of depends on the structure of the flow, on its correlation time and on the magnetic Reynolds number in a nontrivial way. Furthermore, it seems that there is no clear relation between alpha and the helicity of the flow, contrary to what is often assumed for the parametrization of mean-field dynamo models.     

Design new chaotic maps based on dimension expansion

Based on the high-dimensional(HD) chaotic maps and the sine function, a new methodology of designing new chaotic maps using dimension expansion is proposed. This method accepts N dimensions of any existing HD chaotic map as inputs to generate new dimensions based on the combined results of those inputs. The main principle of the proposed method is to combine the results of the input dimensions, and then performs a sine-transformation on them to generate new dimensions.The characteristics of the generated dimensions are totally different compared to the input dimensions. Thus, both of the generated dimensions and the input dimensions are used to create a new HD chaotic map. An example is illustrated using one of the existing HD chaotic maps. Results show that the generated dimensions have better chaotic performance and higher complexity compared to the input dimensions. Results also show that, in the most cases, the generated dimensions can obtain robust chaos which makes them attractive to usage in a different practical application.     

Statistical description of chaotic attractors: The dimension function

A method for the investigation of fractal attractors is developed, based on statistical properties of the distributionP(δ, n) of nearest-neighbor distancesδ between points on the attractor. A continuous infinity of dimensions, called dimension function, is defined through the moments ofP(δ, n). In particular, for the case of self-similar sets, we prove that the dimension function DF yields, in suitable points, capacity, information dimension, and all other Renyi dimensions. An algorithm to compute DF is derived and applied to several attractors. As a consequence, an estimate of nonuniformity in dynamical systems can be performed, either by direct calculation of the uniformity factor, or by comparison among various dimensions. Finally, an analytical study of the distributionP(δ, n) is carried out in some simple, meaningful examples.     

Regular and chaotic transport of impurities in steady flows

This paper considers the properties of the transport of impurity particles in steady fluid flows and describes the principal modes of particle motion. An impurity consisting of particles with a lower density than that of the medium is localized at stationary points of the flow, whereas a heavy impurity can perform a spatially unbounded motion. The conditions for the transition from the bounded motion of a heavy impurity to the long-range transport mode, which occurs as a result of a loss of the stability of the heteroclinic trajectory, are obtained for a model two-dimensional flow having an eddy-cell structure. A mode is found in which a particle, after being transported over a long distance, is trapped forever within the confines of one cell. The transition from regular to chaotic particle transport is analyzed. The question of the effect of a small noise (for example, molecular diffusion) on the character of the motion of a heavy impurity is investigated. It is shown that this effect is important at high viscosity and leads to a transition from bounded motion of the impurity particle to diffusion-type chaotic motion. (c) 1994 American Institute of Physics.     

Chemical transients in closed chaotic flows: the role of effective dimensions

We investigate chemical activity in hydrodynamical flows in closed containers. In contrast to open flows, in closed flows the chemical field does not show a well-defined fractal property; nevertheless, there is a transient filamentary structure present. We show that the effect of the filamentary patterns on the chemical activity can be modeled by the use of time-dependent effective dimensions. We derive a new chemical rate equation, which turns out to be coupled to the dynamics of the effective dimension, and predicts an exponential convergence. Previous results concerning activity in open flows are special cases of this new rate equation.     

Small-scale structure of nonlinearly interacting species advected by chaotic flows

We study the spatial patterns formed by interacting biological populations or reacting chemicals under the influence of chaotic flows. Multiple species and nonlinear interactions are explicitly considered, as well as cases of smooth and nonsmooth forcing sources. The small-scale structure can be obtained in terms of characteristic Lyapunov exponents of the flow and of the chemical dynamics. Different kinds of morphological transitions are identified. Numerical results from a three-component plankton dynamics model support the theory, and they serve also to illustrate the influence of asymmetric couplings. (c) 2002 American Institute of Physics.     

Power spectrum of passive scalars in two dimensional chaotic flows

In this paper the power spectrum of passive scalars transported in two dimensional chaotic fluid flows is studied theoretically. Using a wave-packet method introduced by Antonsen et al., several model flows are investigated, and the fact that the power spectrum has the k(-1)-scaling predicted by Batchelor is confirmed. It is also observed that increased intermittency of the stretching tends to make the roll-off of the power spectrum at the high k end of the k(-1) scaling range more gradual. These results are discussed in light of recent experiments where a k(-1) scaling range was not observed. (c) 2000 American Institute of Physics.