On the Dynamics and Control of Fractional Chaotic Maps with Sine Terms

Abstract

This paper studies the dynamics of two fractional-order chaotic maps based on two standard chaotic maps with sine terms. The dynamic behavior of this map is analyzed using numerical tools such as phase plots, bifurcation diagrams, Lyapunov exponents and 0–1 test. With the change of fractional-order, it is shown that the proposed fractional maps exhibit a range of different dynamical behaviors including coexisting attractors. The existence of coexistence attractors is depicted by plotting bifurcation diagram for two symmetrical initial conditions. In addition, three control schemes are introduced. The first two controllers stabilize the states of the proposed maps and ensure their convergence to zero asymptotically whereas the last synchronizes a pair of non-identical fractional maps. Numerical results are used to verify the findings.

Dynamics Behaviours of a Discrete Memristor Map with Fractional Order

“…From equation (11), it is shows that the discrete memristor has memory effect characteristic similarly to the discrete fractional calculus, so the previous memristor can be extended to fractional order. Based on this equation (11), we changed the first difference operator ∆ with the Caputo like difference operator C ∆ γ a and got a new 2D discrete memristor as…”

“…Thanks to this unique characteristics, fractional order iterated map have been deeply studied in academic fields [6][7][8][9][10][11][12][13][14][15][16].…”