Detection and Characterization of Cusp Singularities

Abstract

Studies on nonlinear analysis of system dynamics have increased in recent years. Since most systems that exist in nature have complex dynamics and therefore exhibit nonlinear behavior; there are various methods and theories developed in this context. Self-similar functions are mathematical functions exhibiting self-similar and scale-invariant behaviors performing fractal structures. Singularities are the basis for producing the self-similar behavior of these functions. Singularity analysis is mainly carried out by using wavelet transform (WT). The cusp singularities are non-oscillating singularities which are characterized by their singularity strength. However, the representation and behavior of this type of singularity differs depending on the sign of the exponent, known as the singularity strength. The cusp singularity functions with negative exponent show irregular behavior progressively different than positively valued functions since the value of the function is undefined at that particular singular point. It is commonly accepted that the singularity strength is studied as Holder exponent of the cusp function, but by definition, the value of this exponent cannot take negative values. We present a new method to estimate the singularity strength of cusp singularities with negative exponent. The developed method is based on analyzing and redistributing the amplitudes of a cusp function with negative exponent by taking the WT. The redistribution of amplitudes over time is achieved by applying curve fitting process to frequency values of the analyzing function.