Characterizing and quantifying weak chaos in fractional dynamics (2504.03007v1)

Abstract: A particularly intriguing and unique feature of fractional dynamical systems is the cascade of bifurcations type trajectories (CBTT). We examine the CBTTs in a generalized version of the standard map that incorporates the Riemann-Liouville fractional derivative, known as the Riemann-Liouville Fractional Standard Map (RLFSM). Due to the memory effects inherent in fractional maps, traditional techniques for characterizing the dynamics, such as Lyapunov exponents, are not applicable. Instead, we propose a methodology using two quantifiers based only on the system's time series: the Hurst exponent and the recurrence time entropy. This approach allows us to effectively characterize the dynamics of the RLFSM, including regions of CBTT and chaotic behavior. Our analysis demonstrates that regions of CBTT are associated with trajectories that exhibit lower values of these quantifiers compared to strong chaotic regions, indicating weakly chaotic dynamics during the CBTTs.

Characterizing Weak Chaos in Fractional Dynamical Systems

The paper "Characterizing and quantifying weak chaos in fractional dynamics" explores a complex yet significant area of paper in fractional dynamical systems (FDS), exploring the phenomenon known as cascade of bifurcations type trajectories (CBTT). This research provides a structured examination of fractional systems using novel methodologies aimed at overcoming traditional limitations encountered in analyzing their complex behaviors.

Fractional dynamical systems are governed by fractional differential equations (FDEs) that incorporate fractional time derivatives, reflecting nonlinear behaviors with memory effects. These integrate computational models existing in biological systems, electromagnetism, and quantum mechanics. Within this context, the authors focus on the Riemann-Liouville Fractional Standard Map (RLFSM), a generalization of the standard map extended to accommodate fractional dynamics using the Riemann-Liouville fractional derivative.

The paper's analysis highlights a particular regime unique to fractional systems known as CBTT, which constitutes sequences of bifurcations along a single trajectory, occurring not through parameter variations but through temporal evolution. This behavior aligns with characteristics observed in sticky orbits of two-dimensional, area-preserving maps where chaotic trajectories intermittently exhibit weaker chaos. CBTT emerges within regions where traditional dynamical measures such as Lyapunov exponents fail due to fractional system memory effects, prompting the investigation of alternative quantifiers.

Two quantifiers, the Hurst exponent and recurrence time entropy (RTE), are proposed to characterize weak chaos during CBTTs. Utilizing finite-time analysis, these quantifiers are applied to time series derived from the RLFSM. The results reveal chaotic regimes as those with high quantifier values, periodic regimes with low values, and intermediate CBTT regions exhibiting lower values than chaotic ones but exceeding periodic levels. Finite-time analysis proved crucial in discerning these regimes, highlighting transitions in dynamical behavior.

The exposition of probability distributions for these quantifiers further enriches the understanding of CBTT, illustrating transitions between chaotic, CBTT, and periodic regimes through multiple peaks. Such distributions not only anchor the quantification of weak chaos but also reveal more intricate structures potentially inherent within CBTT regions. Methodologically, this approach circumvents computational complexities linked to fractional system dynamics, rendering significant contributions without reliance on traditional metrics compromised by system memory effects.

As fractional dynamical systems continue to underpin diverse scientific applications, the paper’s methodologies and findings carry practical implications for characterizing complex systems in computational models. The insights gained from analyzing fractional maps propel theoretical discussions on memory effects within dynamical systems, suggesting broader applicability and the need for continual adaptation in quantifying such phenomena.

This research opens avenues for future investigations into fractional systems, especially concerning the development and refinement of quantifiers tailored to capture dynamical behaviors affected by system memory. It invites exploration beyond traditional chaotic dynamics into fractional domains, marking critical intersections between mathematical theory and real-world applications.