Kicked General Fractional Lorenz-Type Equations: Exact Solutions and Multi-Dimensional Discrete Maps

Area of Science:

• Nonlinear Dynamics and Chaos Theory
• Fractional Calculus
• Mathematical Physics

Background:

• Lorenz-type systems are fundamental models of chaotic behavior in dissipative dynamical systems.
• Existing models often lack the capacity to describe systems with memory effects.
• General Fractional Derivatives (GFDs) offer a framework to incorporate memory functions into dynamical systems.

Purpose of the Study:

• To generalize Lorenz-type equations by incorporating general fractional derivatives (GFDs) and periodical kicks.
• To derive exact analytical solutions for these generalized systems, accommodating nonlinearity and memory.
• To develop discrete maps with memory (DMMs) that precisely represent the dynamics of kicked GF Lorenz-type systems.

Main Methods:

• Application of General Fractional Derivatives (GFDs) to Lorenz-type equations.
• Derivation of exact analytical solutions for nonlinear equations with GFDs and memory functions.
• Construction of multi-dimensional Discrete Maps with Memory (DMMs) from the exact solutions without approximations.

Main Results:

• Exact analytical solutions were obtained for a wide class of nonlinear Lorenz-type equations with GFDs.
• Novel Discrete Maps with Memory (DMMs) were derived, accurately describing the dynamics at discrete time points.
• The methodology allows for the derivation of DMMs for arbitrary dimensions and complex nonlinearities.

Conclusions:

• The study successfully generalized Lorenz-type systems using GFDs, enabling the modeling of systems with memory.
• The derived exact analytical solutions and DMMs provide powerful tools for analyzing complex chaotic dynamics.
• This work offers a significant advancement in understanding and solving fractional-order chaotic systems.