Detecting variation in chaotic attractors

Identification of Colombian coffee price dynamics

The daily Colombian coffee price is a chaotic signal that has emerged from a complex economic system. This work proposes to identify its dynamics by means of two models: a single multiscroll Chua system and the coupling of two of these systems. Models are fine-tuned through an artificial bee colony optimization algorithm. Results show that this approach can reconstruct the price signal in terms of several statistics and points out a way for its long-term forecasting.

Grid-based partitioning for comparing attractors

Stationary dynamical systems have invariant measures (or densities) that are characteristic of the particular dynamical system. We develop a method to characterize this density by partitioning the attractor into the smallest regions in phase space that contain information about the structure of the attractor. To accomplish this, we develop a statistic that tells us if we get more information about our data by dividing a set of data points into partitions rather than just lumping all the points together. We use this method to show that not only can we detect small changes in an attractor from a circuit experiment, but we can also distinguish between a large set of numerically generated chaotic attractors designed by Sprott. These comparisons are not limited to chaotic attractors-they should work for signals from any finite-dimensional dynamical system.

Attractor comparisons based on density

Recognizing a chaotic attractor can be seen as a problem in pattern recognition. Some feature vector must be extracted from the attractor and used to compare to other attractors. The field of machine learning has many methods for extracting feature vectors, including clustering methods, decision trees, support vector machines, and many others. In this work, feature vectors are created by representing the attractor as a density in phase space and creating polynomials based on this density. Density is useful in itself because it is a one dimensional function of phase space position, but representing an attractor as a density is also a way to reduce the size of a large data set before analyzing it with graph theory methods, which can be computationally intensive. The density computation in this paper is also fast to execute. In this paper, as a demonstration of the usefulness of density, the density is used directly to construct phase space polynomials for comparing attractors. Comparisons between attractors could be useful for tracking changes in an experiment when the underlying equations are too complicated for vector field modeling.

Using filtering effects to identify objects

Reflecting signals off of targets is a method widely used to locate objects, but the reflected signal also contains information that can be used to identify the object. In radar or sonar, the signal amplitudes used are small enough that only linear effects are present, so we can consider the effect of the target on the signal as a linear filter. Using the known effects of linear filters on chaotic signals, we can create a reference that allows us to match a particular target to a particular reflected signal. Furthermore, if some parts of this "filter" vary only slowly as the aspect angle of the object changes, we can produce a reference that averages out the parts that are highly angle dependent so that one reference can be used to identify the target over a range of angles.

Nearest neighbors, phase tubes, and generalized synchronization

In this paper we report on the necessity of the refinement of the concept of generalized chaotic synchronization. We show that the state vectors of the interacting chaotic systems being in the generalized synchronization regime are related to each other by the functional, but not the functional relation as it was assumed until now. We propose the phase tube approach explaining the essence of generalized synchronization and allowing the detection and the study of this regime in many relevant physical circumstances. The finding discussed in this Brief Report provides great potential for different applications.