Adaptive fixed-time robust control for function projective synchronization of hyperchaotic economic systems with external perturbations

Generalized Function Projective Synchronization of Two Different Chaotic Systems with Uncertain Parameters

“…Remark 2. The K matrix (34) is not antisymmetric, which does not satisfy the first condition in (24).…”

“…The numerical results are shown in Figures 1 and 2, in which the initial values of the systems ( 26) and ( 27) are taken as x 1 (0) = 3, x 2 (0) = 5, x 3 (0) = 7, x 4 (0) = 8, y 1 (0) = −12, y 2 (0) = 2, y 3 (0) = 3, y 4 (0) = −5 and the estimated parameters have initial conditions â(0) = 4, b(0) = −6, ĉ(0) = −2, d(0) = 1, r(0) = −5, â1 (0) = 1, b1 (0) = −3, ĉ1 (0) = 0.3 and d1 (0) = 0.1. Although K matrix (34) does not satisfy condition (24), Figure 1 shows that GFPS between systems ( 26) and ( 27) is achieved with the scaling function factors (28). Figure 2 illustrates that the estimated parameters â, b, ĉ, d, r, â1 , b1 , ĉ1 , d1 in systems ( 26) and ( 27 26) and ( 27) can be achieved with the scaling function factors (28) and matrix (34), and the parameter estimate errors converge to zero.…”

“…Although K matrix (34) does not satisfy condition (24), Figure 1 shows that GFPS between systems ( 26) and ( 27) is achieved with the scaling function factors (28). Figure 2 illustrates that the estimated parameters â, b, ĉ, d, r, â1 , b1 , ĉ1 , d1 in systems ( 26) and ( 27 26) and ( 27) can be achieved with the scaling function factors (28) and matrix (34), and the parameter estimate errors converge to zero. 26) and ( 27) is achieved with the scaling function matrix 26) and ( 27) for GFPS with the scaling function matrix h(x) = diag{2sin(x 1 + 1) + 1, 3sin(−2x 2 + 3) + 1, −sin(3x 3 − 1) + 1, −2sin(−x 4 + 4) + 1} and K matrix (34).…”

“…Chen et al [31,32] considered two chaotic systems synchronized up to a scaling function factor, which is called function projective synchronization (FPS). The FPS between two novel chaotic systems has been investigated by El Dessoky et al [33] and Bekiros et al [34] using the Lyapunov method of stability. Du et al [35] and Srisuntorn et al [36] discussed modified function projective synchronization (MFPS), in which two chaotic systems can be synchronized up to a desired scaling function matrix.…”

Fixed-Time Adaptive Chaotic Control for Permanent Magnet Synchronous Motor Subject to Unknown Parameters and Perturbations

“…Fixed-time control is primarily designed based on the bilimit homogeneous system theory [49] or the fixed-time Lyapunov stability theory [50,51]. Fixed-time controllers have been developed for the stabilization of chaotic systems in [52][53][54][55][56][57]. For instance, ref.…”