Fractal Structures of Normal and Anomalous Diffusion in Nonlinear Nonhyperbolic Dynamical Systems (original) (raw)

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Physica D-nonlinear Phenomena, 2002

The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green–Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.

Fractal properties of anomalous diffusion in intermittent maps

Physical Review E, 2007

An intermittent nonlinear map generating subdiffusion is investigated. Computer simulations show that the generalized diffusion coefficient of this map has a fractal, discontinuous dependence on control parameters. An amended continuous time random walk theory well approximates the coarse behavior of this quantity in terms of a continuous function. This theory also reproduces a full suppression of the strength of diffusion, which occurs at the dynamical transition from normal to anomalous diffusion. Similarly, the probability density function of this map exhibits a nontrivial fine structure while its coarse functional form is governed by a time fractional diffusion equation. A more detailed understanding of the irregular structure of the generalized diffusion coefficient is provided by an anomalous Taylor-Green-Kubo formula establishing a relation to de Rham-type fractal functions.

Deterministic diffusion in smooth nonlinear maps

From Quanta to Societies, Proceedings of EUROATTRACTOR, 2001

Deterministic diffusion under variation of a control parameter is studied both numerically and semi-analytically for one-dimensional smooth nonlinear maps on the line. The climbing sine map is chosen as a typical example. Numerical investigations of the climbing sine map reveal a fractal-like behavior of the diffusion coefficient as a function of a control parameter whenever diffusion is normal. Furthermore, there exist hierarchies of periodic windows on the parameter axis exhibiting sub-and superdiffusion. To understand this complicated ...

Simple deterministic dynamical systems with fractal diffusion coefficients

Physical Review E, 1998

We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional periodic array of scatterers in which point particles move from cell to cell as defined by a piecewise linear map. The microscopic chaotic scattering process of the map can be changed by a control parameter. This induces a parameter dependence for the macroscopic diffusion coefficient. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match to the ones of the simple phenomenological diffusion equation. Our main result is that the difffusion coefficient exhibits a fractal structure by varying the system parameter. To understand the origin of this fractal structure, we give qualitative and quantitative arguments. These arguments relate the sequence of oscillations in the strength of the parameter-dependent diffusion coefficient to the microscopic coupling of the single scatterers which changes by varying the control parameter.

Simple Maps with Fractal Diffusion Coefficients

Physical Review Letters, 1994

We consider chains of one-dimensional, piecewise linear, chaotic maps with uniform slope. We study the diffusive behaviour of an initially nonuniform distribution of points as a function of the slope of the map by solving the Frobenius-Perron equation. For Markov partition values of the slope, we relate the diffusion coefficient to eigenvalues of the topological transition matrix. The diffusion coefficient obtained shows a fractal structure as a function of the slope of the map. This result may be typical for a wide class of maps, such as two-dimensional sawtooth maps.

Chaotic and fractal properties of deterministic diffusion-reaction processes

Chaos, 1998

We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the reactive modes of the models as eigenstates of the Frobenius-Perron operator. The associated eigenvalues provide the dispersion relations of diffusion and reaction and, hence, they determine the reaction rate. For the simplest model we show explicitly that the reaction rate behaves as phenomenologically expected for one-dimensional diffusion-controlled reaction. Under parametric variation, we find that both the diffusion coefficient and the reaction rate have fractal-like dependences on the system parameter.

Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps

2011

Abstract We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation.

A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics

Chaos, 2015

Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process.

Fractal structures in nonlinear dynamics

Reviews of Modern Physics, 2009

In addition to the striking beauty inherent in their complex nature, fractals have become a fundamental ingredient of nonlinear dynamics and chaos theory since they were defined in the 1970s. Moreover, fractals have been detected in nature and in most fields of science, with even a certain influence in the arts. Fractal structures appear naturally in dynamical systems, in particular associated with the phase space. The analysis of these structures is especially useful for obtaining information about the future behavior of complex systems, since they provide fundamental knowledge about the relation between these systems and uncertainty and indeterminism. Dynamical systems are divided into two main groups: Hamiltonian and dissipative systems. The concepts of the attractor and basin of attraction are related to dissipative systems. In the case of open Hamiltonian systems, there are no attractors, but the analogous concepts of the exit and exit basin exist. Therefore basins formed by initial conditions can be computed in both Hamiltonian and dissipative systems, some of them being smooth and some fractal. This fact has fundamental consequences for predicting the future of the system. The existence of this deterministic unpredictability, usually known as final state sensitivity, is typical of chaotic systems, and makes deterministic systems become, in practice, random processes where only a probabilistic approach is possible. The main types of fractal basin, their nature, and the numerical and experimental techniques used to obtain them from both mathematical models and real phenomena are described here, with special attention to their ubiquity in different fields of physics.

The origin of diffusion: the case of non-chaotic systems

Physica D: Nonlinear Phenomena, 2003

We investigate the origin of diffusion in non-chaotic systems. As an example, we consider 1D map models whose slope is everywhere 1 (therefore the Lyapunov exponent is zero) but with random quenched discontinuities and quasi-periodic forcing. The models are constructed as non-chaotic approximations of chaotic maps showing deterministic diffusion, and represent one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the Ehrenfest wind-tree model). In particular, a simple construction shows that these maps define non-chaotic billiards in space-time. The models exhibit, in a wide range of the parameters, the same diffusive behavior of the corresponding chaotic versions. We present evidence of two sufficient ingredients for diffusive behavior in one-dimensional, non-chaotic systems: (i) a finite size, algebraic instability mechanism; (ii) a mechanism that suppresses periodic orbits.