Enhanced Symplectic Synchronization between Two Different Complex Chaotic Systems with Uncertain Parameters

and Applied Analysis 3 Equation (8) can be expressed as ̇ e 1 = − 2y 1 x 1 [a (x 2 − x 1 ) + x 4 ] − (1 + x 2 1 ) × [a 1 (y 2 − y 1 ) + y 4 ] − u 1 + 3x 2 4 (rx 4 + x 2 x 3 ) , ̇ e 2 = − 2y 2 x 2 (dx 1 + cx 2 − x 1 x 3 ) − (1 + x 2 2 ) × (b 1 y 1 − y 2 − y 1 y 3 ) − u 2 + 3x 2 1 [a (x 2 − x 1 ) + x 4 ] , ̇ e 3 = − 2y 3 x 3 (−bx 3 + x 1 x 2 ) − (1 + x 2 3 ) × (−c 1 y 3 + y 1 y 2 ) − u 3 + 3x 2 2 (dx 1 + cx 2 − x 1 x 3 ) , ̇ e 4 = − 2y 4 x 4 (rx 4 + x 2 x 3 ) − (1 + x 2 4 ) × (d 1 y 4 − y 1 y 3 ) − u 4 + 3x 2 3 (−bx 3 + x 1 x 2 ) , (16) where e 1 = −x 2 1 y 1 − y 1 + x 3 4 , e 2 = −x 2 2 y 2 − y 2 + x 3 1 , e 3 = −x 2 3 y 3 − y 3 + x 3 2 , and e 4 = −x 2 4 y 4 − y 4 + x 3 3 . Choose a positive definite Lyapunov function as V (e 1 , e 2 , e 3 , e 4 ) = 1 2 (e 2 1 + e 2 2 + e 2 3 + e 2 4 ) . (17) Its time derivative along any solution of (16) is ?̇? = e 1 {−2y 1 x 1 [a (x 2 − x 1 ) + x 4 ] − (1 + x 2 1 ) × [a 1 (y 2 − y 1 ) + y 4 ] − u 1 + 3x 2 4 (rx 4 + x 2 x 3 ) } + e 2 {−2y 2 x 2 (dx 1 + cx 2 − x 1 x 3 ) − (1 + x 2 2 ) × (b 1 y 1 − y 2 − y 1 y 3 ) − u 2 + 3x 2 1 [a (x 2 − x 1 ) + x 4 ] } + e 3 {−2y 3 x 3 (−bx 3 + x 1 x 2 ) − (1 + x 2 3 ) × (−c 1 y 3 + y 1 y 2 ) − u 3 + 3x 2 2 (dx 1 + cx 2 − x 1 x 3 ) } + e 4 {−2y 4 x 4 (rx 4 + x 2 x 3 ) − (1 + x 2 4 ) × (d 1 y 4 − y 1 y 3 ) − u 4 + 3x 2 3 (−bx 3 + x 1 x 2 ) } . (18) According to (10), we get the controller u 1 = − 2y 1 x 1 [a (x 2 − x 1 ) + x 4 ] + 3x 2 4 (rx 4 + x 2 x 3 ) − (1 + x 2 1 ) × [a 1 (y 2 − y 1 ) + y 4 ] + e 1 , u 2 = − 2y 2 x 2 (dx 1 + cx 2 − x 1 x 3 ) + 3x 2 1 [a (x 2 − x 1 ) + x 4 ] − (1 + x 2 2 ) × (b 1 y 1 − y 2 − y 1 y 3 ) + e 2 , u 3 = − 2y 3 x 3 (−bx 3 + x 1 x 2 ) + 3x 2 2 (dx 1 + cx 2 − x 1 x 3 ) + (1 + x 2 3 ) × (c 1 y 3 − y 1 y 2 ) + e 3 , u 4 = − 2y 4 x 4 (rx 4 + x 2 x 3 ) + 3x 2 3 (−bx 3 + x 1 x 2 ) − (1 + x 2 4 ) × (d 1 y 4 − y 1 y 3 ) + e 4 . (19) Equation (18) becomes ?̇? = − (e 2 1 + e 2 2 + e 2 3 + e 2 4 ) < 0, (20) which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization of the Chen system and the Lorenz system is achieved. The numerical results are shown in Figures 1, 2, and 3. After 1 second, the motion trajectories enter a chaotic attractor. Case 2 (a symplectic synchronization with uncertain parameters). The master Chen system with uncertain variable parameters is ?̇? 1 = a (t) (x 2 − x 1 ) + x 4 , ?̇? 2 = d (t) x 1 + c (t) x 2 − x 1 x 3 , ?̇? 3 = −b (t) x 3 + x 1 x 2 , ?̇? 4 = r (t) x 4 + x 2 x 3 , (21) where a(t), b(t), c(t), d(t), and r(t) are uncertain parameters. In simulation, we take a (t) = a (1 + k 1 sinω 1 t) , b (t) = b (1 + k 2 sinω 2 t) , c (t) = c (1 + k 3 sinω 3 t) , d (t) = d (1 + k 4 sinω 4 t) , r (t) = r (1 + k 5 sinω 5 t) , (22) where k 1 , k 2 , k 3 , k 4 , k 5 , ω 1 , ω 2 , ω 3 , ω 4 , andω 5 are constants. Take k 1 = 0.3, k 2 = 0.5, k 3 = 0.2, k 4 = 0.4, k 5 = 0.6, ω 1 = 13, ω 2 = 17, ω 3 = 19, ω 4 = 23, and ω 5 = 29. So, (21) is chaotic system, shown in Figure 4. We take F 1 (t) = x 3 4 (t), F 2 (t) = x 3 1 (t), F 3 (t) = x 3 2 (t), and F 4 (t) = x 3 3 (t).They are chaotic functions of time.H i (x, y, t) = −x 2 i y i (i = 1, 2, 3, 4) are given. By (6), we have lim t→∞ e i = lim t→∞ (−x 2 i y i − y i + x 3 j ) = 0, i = 1, 2, 3, 4; j = { 4, i = 1, i − 1, i ̸ = 1. (23) 4 Abstract and Applied Analysis