Projective synchronization between different fractional-order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique (Q2922247)

Consider the systems of differential equations with fractional derivatives NEWLINE\[NEWLINE \displaystyle{D^q_tx = f(x) + F(x)\alpha} NEWLINE\]NEWLINE called the drive system and NEWLINE\[NEWLINE \displaystyle{D^q_ty = g(y) +G(y)\beta + U(x,y,\alpha,\beta)} NEWLINE\]NEWLINE called the response system. Achieving NEWLINE\[NEWLINE \displaystyle{\lim_{t\rightarrow\infty}|e(t,x_0,y_0)| = \lim_{t\rightarrow\infty}|y(t,y_0)-Ax(t,x_0)| = 0} NEWLINE\]NEWLINE is called projective synchronization; here \(\alpha\) and \(\beta\) are vector parameters. It is shown that the adaptive nonlinear control law defined by NEWLINE\[NEWLINEU = A(f(x)+F(x)\alpha) - g(y) - G(y)\beta + D_t^{q-1}[AF(x)e_\alpha - G(y)e_\beta-ek]NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot{\tilde{\alpha}} = -\left([AF(x)]^Te+e_\alpha\right)\,\dot{\tilde{\beta}} = [G(y)]^Te - e_\beta, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE e_\alpha = \tilde{\alpha} - \alpha\;, e_\beta = \tilde{\beta} - \beta ,NEWLINE\]NEWLINE ensures adaptive projective synchronization and also NEWLINE\[NEWLINE{\lim_{t\rightarrow\infty} |e_\alpha(t)| = \lim_{t\rightarrow\infty} |e_\beta(t)| = 0}. NEWLINE\]NEWLINE Applications to the synchronization of the Lorenz and Lü chaotic systems are given.