Quantitative chaos analysis for discrete fractional systems

Develop a consistent, rigorous quantitative framework to characterize chaos in discrete fractional systems defined by Volterra difference equations of convolution type with power-law memory kernels, including the fractional logistic map and the fractional difference logistic map. The framework should provide reproducible chaos indicators compatible with non-Markovian memory (for example, Lyapunov spectra, entropy rates, and invariant measures) and enable clear discrimination between chaotic and non-chaotic regimes across parameter ranges.

Background

The paper analyzes chaotic behavior in fractional difference logistic maps using Poincaré (return) plots and notes quasi-fractal, multi-scroll structures resembling dissipative attractors. While qualitative features are evident, the authors emphasize that discrete fractional systems possess power-law memory and non-Markovian dynamics, complicating the use of standard chaos quantifiers developed for memoryless maps.

Against this backdrop, the authors explicitly state that a consistent quantitative analysis of chaos in discrete fractional systems remains unresolved, motivating the need for a robust methodology adapted to the fractional setting (e.g., appropriate definitions and computation of Lyapunov exponents or invariant measures that remain valid under memory effects).