Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular...

Area of Science:

• Nonlinear dynamics
• Fractional calculus
• Mathematical physics

Background:

• Fractional calculus (FC) effectively models complex phenomena across scientific disciplines.
• Investigating nonlinear dynamics in fractional systems is crucial for understanding intricate behaviors.

Purpose of the Study:

• Introduce a new physical model utilizing a nonlocal fractional differential operator with a singular kernel.
• Derive new Routh-Hurwitz stability conditions for fractional orders within [0,2).
• Analyze nonlinear dynamics, including chaotic behavior and stability, in the proposed fractional model.

Main Methods:

• Development of a new physical model with a nonlocal fractional differential operator.
• Derivation of novel Routh-Hurwitz stability criteria for fractional-order systems.
• Application of Hopf bifurcation theory to investigate periodic solutions.
• Analysis of chaotic dynamics using Lyapunov exponents and spectrum.
• Implementation of chaos suppression techniques.

Main Results:

• New Routh-Hurwitz stability conditions derived for fractional orders [0,2).
• Existence and uniqueness of solutions proven for the proposed system.
• Chaotic dynamics identified in the commensurate fractional system across various orders.
• Lyapunov exponents and spectrum calculated for the model.
• Successful chaos suppression demonstrated using two distinct methods.

Conclusions:

• The proposed fractional model exhibits rich nonlinear dynamics, including chaos.
• Novel stability conditions enhance the analysis of fractional-order systems.
• Methods for chaos control are effective in the studied fractional system.