Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique

Abstract

This research deals with a comparative numerical analysis of chaos in two systems with non-integer derivatives. The one-scroll system and circle equilibrium system with different hidden attractors are simulated considering the fractal derivative, Khalil and Atangana conformable derivatives, and the truncated M-derivative considering a constant and variable-order. Phase portraits are shown, as well as bifurcation diagrams, and Lyapunov exponents are obtained. Later, 0–1 test, dynamic death analysis, and sensitivity to initial conditions are considered to choose which derivative produces richer chaotic behaviors. According to those mentioned above, we could observe that the M-derivative not only generalizes Khalil’s type conformable derivative but also its two non-integer orders produce interesting dynamic behaviors compared to the remaining derivatives. In the numerical results, we observe that the variable order makes the system more sensitive to the change in the initial conditions. The new chaotic behaviors with constant and variable order are used to develop a synchronization scheme of two identical one-scroll systems via the backstepping technique with the truncated M-derivative involved.