When chaotic systems are realized with finite precisions in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. In the literature, there does not seem to be much work on quantitative analysis of such degradation of digitized chaos and how to reduce its negative influence on chaos-based digital systems. Focusing on 1D piecewise linear chaotic maps (PWLCM), this paper reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on a digital PWLCM realized with a fixed-point finite precision. On top of that, the paper introduces a new method for studying digital chaos from an algorithmic point of view. In addition, the theoretical results obtained in this paper should be very helpful for the consideration of reducing negative influence of dynamical degradation in real design of various digital chaotic systems. As typical examples, the proposed dynamical indicators are applied to the performance comparison of different remedies for improving dynamical degradation, cryptanalysis of digital chaotic ciphers based on 1D PWLCM, and design of chaotic pseudo-random number generators with desired characteristics.

/pdf/on-the-dynamical-degradation-of-digital-piecewise-linear-4qax6vssk8.pdf

On the dynamical degradation of digital piecewise linear chaotic maps

We begin a systematic study of Euclidean piecewise isometric dynamical systems (p.i.d.s.) with a particular focus on the interplay between geometry, symbolic dynamics, and the group of isometries associated with p.i.d.s. We investigate various aspects of the dynamical information contained in the coding: symbolic growth and the periodic behavior of codings and cells. This theoretical investigation is motivated by the many examples of piecewise isometric dynamical systems found recently in the literature. Piecewise isometric dynamical systems are direct generalizations of interval exchange transformations to non-invertible, higher dimensional maps.

/pdf/dynamics-of-piecewise-isometries-1gsrk1ao5n.pdf

Dynamics of piecewise isometries

We discuss the dynamics of a class of non-ergodic piecewise affine maps of the torus. These maps exhibit highly complex and little understood behavior. We present computer graphics of some examples and analyses of some with a decreasing degree of completeness. For the best understood example, we show that the torus splits into three invariant sets on which the dynamics are quite different. These are: the orbit of the discontinuity set, the complement of this set in its closure, and the complement of the closure. There are still some unsolved problems concerning the orbit of the discontinuity set. However we do know that there are intervals of periodic orbits and at least one infinite orbit. The map on the second invariant set is minimal and uniquely ergodic. The third invariant set is one of full Lebesgue measure and consists of a countable number of open octagons whose points are periodic. Their orbits can be described in terms of a symbolism obtained from an equal length substitution rule or the triadic odometer.

Dynamics of non-ergodic piecewise affine maps of the torus

Elliptic behaviour in the sawtooth standard map

We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. For the case of eigenvalues of the associated linear map of the form eiθ with θ/π ∉ Q we present evidence that D has positive two dimensional Lebesgue measure. For θ/π ∈ Q we confirm that D has Lebesgue measure zero. As a by-product we get estimates of the exterior dimension of D. These results imply that if such filters are realized using finite-precision arithmetic then they will have a sizeable fraction of orbits that are periodic with high period overflows.

Lossless Digital Filter Overflow Oscillations; Approximation of Invariant Fractals

In this paper we present more analytical results about the complex behavior of a second order digital filter with overflow nonlinearity. We explore the parameter space to obtain a taxonomy of the different behaviors that occurs. In particular, we give a complete description of the chaotic behavior of the map F (which models the second order digital filter) in the parameter space (a,b). We prove that in the region R~/sub 5/ (the closure of R/sub 5/, where R/sub 5/={(a,b):b -1}) F is not chaotic; in the region |b| 1, if (a,b) are integers and b=-2(a-1), then F is an exact map. In addition, we obtain some results concerning the fractal behavior of the map F. We find an estimate of the Hausdorff dimension of the generalized hyperbolic attractor. We obtain results regarding the symbolic dynamics of F. For example, we prove that the set of points with aperiodic admissible sequences in the case |a|<2 and b=-1 is uncountable.

L.O. Choa

Papers

Complex behavior in digital filters with overflow nonlinearity: analytical results