Multiscale topology and dynamic internal vectorial geometry - Alpha group

This work sought to apply the Continuous Wavelet Transform (CWT) to the
multiscale time-frequency analysis of non-stationary signals associated with the
temporal evolution behavior of the complex eigenvalues of the matrix M(θ(t)), with
θ=2π radians. The results of CWT analysis have proven to be particularly effective
in identifying oscillatory patterns, transient regimes and resonant features that are
not evident through conventional spectral analysis, such as that which occurs
using Fourier Transform. The results show that the CWT scalogram clearly reveals
a persistent multiscale structure associated with an internal dynamic vector
geometry, which exhibits a complex foliated topology. Thus, the use of CWT
enabled a refined characterization of the system's spectral dynamics, serving as a
key tool for identifying topological phenomena, internal resonances, and recurrent
structures intrinsic to the vectorial geometry of the Alpha Group.