Dynamic analysis and synchronization of conformable fractional-order chaotic systems

Abstract.

In this paper, chaotic systems based on the conformable fractional-order derivative are investigated. First, some comments on the discrete Conformable Adomian decomposition method (discrete CADM) proposed recently are made. We demonstrate that the discrete CADM is not globally accurate for conformable fractional-order differential equations and some conclusions about conformable fractional-order chaotic systems in present papers based on the discrete CADM are inappropriate. Second, we restudy the conformable fractional-order simplified Lorenz system. And then, the conformable fractional-order Liu system is introduced and studied. The basic dynamical behaviors of the systems are analyzed in detail, such as dissipation, the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits and time series of states. Some interesting dynamic behaviors different from the Caputo or Riemann-Liouville fractional-order chaotic system are discovered. It is found that the system order of the commensurate case for chaos is $q \in (0,1]$. However, for the incommensurate case, dissipation and stability of equilibrium points depend on system order q_i and time t which may cause chaotic transients or period-doubling oscillation. In addition, the dynamic behaviors of conformable fractional-order chaotic systems will be static as $t \rightarrow + \infty$ which is named “dynamic death”. Finally, synchronization between conformable fractional-order simplified Lorenz system and conformable fractional-order Liu system is realized.