On the Use of Interval Extensions to Estimate the Largest Lyapunov Exponent from Chaotic Data

A method to estimate the (positive) largest Lyapunov exponent (LLE) from data using interval extensions is proposed. The method differs from the ones available in the literature in its simplicity since it is only based on three rather simple steps. Firstly, a polynomial NARMAX is used to identify a model from the data under investigation. Secondly, interval extensions, which can be easily extracted from the identified model, are used to calculate the lower bound error. Finally, a simple linear fit to the logarithm of lower bound error is obtained and then the LLE is retrieved from it as the third step. To illustrate the proposed method, the LLE is calculated for the following well-known benchmarks: sine map, Rössler Equations, and Mackey-Glass Equations from identified models given in the literature and also from two identified NARMAX models: a chaotic jerk circuit and the tent map. In the latter, a Gaussian noise has been added to show the robustness of the proposed method.

A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Hénon, Lorenz and Rössler equations and the Mackey–Glass system.

Interval Extensions of Signed Distance Functions: iSDF-reps and Reliable Membership Classification

This paper considers the problem of inferring the geometry of an object from values of the signed distance sampled on a uniform grid. The problem is motivated by the desire to effectively and efficiently model objects obtained by 3D imaging technology such as magnetic resonance, computed tomography, and positron emission tomography. Techniques recently developed for automated segmentation convert intensity to signed distance, and the voxel structure imposes the uniform sampling grid. The specification of the signed distance function (SDF) throughout the ambient space would provide an implicit and function-based representation (f-rep) model that uniquely specifies the object, and we refer to this particular f-rep as the signed distance function representation (SDF-rep). However, a set of uniformly sampled signed distance values may uniquely determine neither the distance function nor the shape of the object. Here, we employ essential properties of the signed distance to construct the upper and lower bounds on the allowed variation in signed distance, which combine to produce interval-valued extensions of the signed distance function. We employ an interval extension of the signed distance function as an interval SDF-rep that defines the range of object geometries that are consistent with the sampled SDF data. The particular interval extensions considered include a tight global extension and more computationally efficient local extensions that provide useful criteria for root exclusion/isolation. To illustrate a useful application of the interval bounds, we present a reliable approach to top-down octree membership classification for uniform samplings of signed distance functions.