Study of low-dimensional nonlinear fractional difference equations of complex order

Area of Science:

• Nonlinear Dynamics and Chaos Theory
• Fractional Calculus
• Complex Systems Analysis

Background:

• Fractional calculus extends traditional calculus to non-integer orders.
• Fractional maps are generalizations of discrete dynamical systems to fractional orders.
• Understanding the behavior of fractional maps is crucial for modeling complex phenomena.

Purpose of the Study:

• To investigate the dynamics of complex fractional-order maps in one and two dimensions.
• To explore the impact of map properties (smoothness, differentiability) on chaotic behavior.
• To analyze the occurrence of chaos and multistability in various fractional maps.

Main Methods:

• Studied fractional maps of complex order (αe) for 0 < α < 1 and 0 ≤ r < 1.
• Analyzed one-dimensional maps: logistic, tent, Gauss, circle, and Bernoulli.
• Investigated two-dimensional maps: Hénon, Duffing, and Lozi.
• Examined two distinct generalization methods for 2D fractional maps.

Main Results:

• Smooth maps (logistic, Gauss, Duffing, Hénon) did not exhibit chaos.
• Discontinuous maps (Bernoulli, circle) and non-differentiable maps (tent, Lozi) showed chaotic behavior.
• Two-dimensional complex fractional-order maps displaying chaos also exhibited multistability.
• Different generalization methods for 2D fractional maps yielded distinct bifurcation diagrams.

Conclusions:

• Smooth complex fractional-order maps tend towards regular behavior.
• Discontinuous or non-differentiable maps are more prone to chaos and multistability.
• The generalization approach significantly influences the dynamics of 2D fractional maps.