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

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Lorenz-type systems are dissipative dynamical systems that are described by three nonlinear equations with derivatives of the first order and are capable of exhibiting chaotic behavior. The generalization of Lorenz-type equations by using general fractional derivatives (GFDs) and periodical kicks is proposed. GFDs allow us to use the general form of memory functions as operator kernels to describe nonlinear dynamics with memory. The exact analytical solutions of Lorenz-type equations with GFDs are derived in the general case for the wide class of nonlinearity and memory functions. Using the exact solutions, we obtain discrete maps with memory (DMMs) that describe kicked GF Lorenz-type systems with general forms of memory and nonlinearity. The proposed maps describe the exact solution of nonlinear equations with GFDs at discrete time points as the function of all past discrete moments of time. The proposed multi-dimensional DMMs are derived from kicked GF Lorenz-type equations with GFDs without any approximations. The proposed results and the method to derive multi-dimensional DMMs are derived for arbitrary dimensions. The importance and unusualness of the proposed results lies in the fact that obtained solutions for equations of the Lorenz-type system are exact analytical solutions.