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.