A new topological indicator for chaos in mechanical systems

Abstract

The main goal in this paper was to provide a novel chaos indicator based on a topological model which allows to calculate the fractal dimension of any curve. A fractal structure is a topological tool whose recursiveness becomes ideal to generalize the concept of fractal dimension. In this paper, we provide an algorithm to calculate a new fractal dimension specially designed for a parametrization of a curve or a random process, whose definition is made by means of fractal structures. As an application, we explore the use of this new concept of fractal dimension as a chaos indicator for dynamical systems, in a similar way to the classical maximal Lyapunov exponent. To illustrate it, we apply the new fractal dimension as an indicator to model the chaotic behavior of a satellite which is moving around a planet whose gravity field is approximated by the field of a point mass.

A survey on fractal dimension for fractal structures

“…[28,Theorem 3.12]) Let Γ Γ Γ be the natural fractal structure on R d and F be a compact subset of R d . Then the fractal dimensions IV, V, and VI of F equal the Hausdorff dimension of F, namely, dim H (F) = dim 6 Γ Γ Γ (F) = dim 5 Γ Γ Γ (F) = dim 4 Γ Γ Γ (F).…”

“…The main tool used in these areas to deal with fractals is the fractal dimension since it is their main invariant which throws some useful information about the complexity and irregularitires that a certain set presents once it has been explored with enough level of detail. It is worth mentioning that fractal dimension theory has also been applied in several scientific fields including the study of dynamical and mechanical systems [5,6], diagnosis of diseases (such as osteoporosis [7] or cancer [8]), ecology [9], earthquakes [10], detection of eyes in human face images [11], analysis of the human retina [12], and brain computer interface systems [13], to name a few.…”

“…The fractal dimension VI of F is given by the following critical point: dim 6 Γ Γ Γ (F) = inf{s : H s 6 (F) = 0} = sup{s : H s 6 (F) = ∞}.…”

Counterexamples for IFS-attractors

“…Recently, novel alternatives, with some desirable analytical properties as it happens with the Hausdorff dimension model, and being as easy to calculate as the box dimension, have been contributed. In particular, a new fractal dimension algorithm, specially appropriate to deal with time series, has allowed the authors to study the chaotic behavior of planar oscillations described by a satellite in the field of astrodynamics, based on a Beletsky model [11]. Moreover, also the Hurst exponent (resp.…”