On the chaotic systems of attraction with fractional operators in a novel Lorenz system

In this research, a novel variant model of the classical Lorenz system is proposed by reformulating both the classical and fractional-order Lorenz systems through a new framework based on piecewise fractional derivatives. The resulting piecewise Lorenz system is formulated as a nonlinear system of differential operators exhibiting rich and complex dynamical behaviors induced by the piecewise structure. The proposed model is developed within the frameworks of the Caputo fractional derivative, the Atangana–Baleanu fractional derivative, and the Caputo–Fabrizio fractional derivative. A rigorous qualitative analysis is carried out to establish the existence and uniqueness of solutions for the modified piecewise Lorenz system. Furthermore, the dynamical and chaotic properties of the system are investigated through numerical approximations based on the Newton interpolation formula. Numerical simulations and graphical illustrations demonstrate the effectiveness and accuracy of the proposed numerical scheme and reveal the influence of fractional orders and system parameters on the system dynamics. In particular, the system exhibits crossover behavior and chaotic dynamics characterized by the coexistence of two strange attractors, highlighting significant differences from the classical Lorenz system.