Chaos in the incommensurate fractional order system and circuit simulations

In this paper, the dynamics of an autonomous three dimensional system with three nonlinearity quadratic terms is investigated. With the help of stability analysis of equilibrium points, the Lyapunov exponent, and the phase portraits we study the dynamical behavior of the fractional order system. We find that system can display double scroll and double-four wing chaotic attractor. The commensurate order has been investigated and we find that the necessary condition for chaos to appear is 0.84<α≤ 1 . For the incommensurate order, we show that the instability measure of the equilibrium points for all the saddle points of index 2 must be non-negative for a system with five equilibrium points to be chaotic. Numerical simulations and the analog simulations are carried out in Multisim.