Dissipative fractional standard maps: Riemann-Liouville and Caputo

Area of Science:

• Nonlinear Dynamics
• Fractional Calculus
• Chaos Theory

Background:

• Realistic dynamical systems exhibit inherent dissipation.
• Fractional dynamics offers a framework to model systems with memory.
• Fractional standard maps (fSMs) are nonlinear maps with memory.

Purpose of the Study:

• To investigate the effects of dissipation on fractional standard maps.
• To analyze the behavior of average action and average squared action in dissipative fSMs.
• To compare the influence of fractional order and dissipation strength on chaotic orbits.

Main Methods:

• Considered dissipative versions of Riemann-Liouville (RL) and Caputo (C) fractional standard maps.
• Focused on strongly chaotic orbits (K≫1).
• Analyzed average action (⟨In⟩) and average squared action (⟨In2⟩) as functions of nonlinearity (K), fractional order (α), and dissipation strength (γ).

Main Results:

• Demonstrated exponential decay of average action (⟨In⟩≈I0exp(-γn)) for |I0|>K in both dissipative fSMs.
• Showed that ⟨In2⟩RL-fSM is largely independent of α, except when α→1.
• Observed that any α<2 significantly influences the behavior of ⟨In2⟩C-fSM.
• Derived an analytical expression for ⟨In2⟩RL-fSM(K,α,γ).

Conclusions:

• Dissipation plays a crucial role in the long-term behavior of fractional dynamical systems.
• The fractional order α has a differential impact on the average squared action depending on the type of fractional derivative (RL vs. C).
• Analytical tools were developed to describe the dynamics of dissipative fractional maps.