Numerical treatments for some stochastic–deterministic chaotic systems

We aim in this paper to propose classes of nonlinear ordinary differential equations based on the concept of piecewise differentiation. These classes are known as piecewise deterministic–stochastic differential equations or vice versa. Two deterministic–stochastic chaotic systems will be presented in this paper. These models are the hyperchaotic finance model and Bloch system with time delay, where the hybrid fractional derivatives are used. Existence, uniqueness and stability analysis are presented. In order to study the behavior of these models, three different cases for each model are presented. Moreover, two numerical methods are introduced to study these models. These methods are the nonstandard Milstein method to solve the stochastic models and Grünwald–Letnikov nonstandard finite difference to solving the hybrid fractional order deterministic models. Comparative study and numerical simulation are carried out.