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In this Letter, an adaptive control scheme is developed to study the anti-synchronization behavior between two identical and different chaotic systems with unknown parameters. This adaptive anti-synchronization controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The adaptive anti-synchronization between two identical systems (Chen system) and different systems (Genesio and Lu systems) are taken as two illustrative examples to show the effectiveness of the proposed method. Theoretical analysis and numerical simulations are shown to verify the results.

In 1998, M.S. Baptista proposed a chaotic cryptosystem using the ergodicity property of the simple low-dimensional and chaotic logistic equation. Since then, many cryptosystems based on Baptista's work have been proposed. However, over the years research has shown that this cryptosystem is predictable and vulnerable to attacks and is widely discussed. Among the weaknesses are the non-uniform distribution of ciphertexts and succumbing to the one-time pad attack (a type of chosen plaintext attack). In this Letter, our objective is to modify the chaotic cryptographic scheme proposed previously. We use a matrix secret key such that the cryptosystem would no longer succumb to the one-time pad attack.

In this Letter, approximate solutions of singular two-point boundary value problems (BVPs) are obtained by the modified homotopy analysis methods (MHAM). MHAM provides a convenient way of controlling the convergence region and rate of the series solution. The numerical tests show the capability of MHAM for singular BVPs.

In this Letter, a general framework of the variational iteration method (VIM) is presented for solving systems of linear and nonlinear partial differential equations (PDEs). In VIM, a correction functional is constructed by a general Lagrange's multiplier which can be identified via a variational theory. VIM yields an approximate solution in the form of a rapid convergent series. Comparison with the exact solutions shows that VIM is a powerful method for the solution of linear and nonlinear systems of PDEs.

In this Letter, the homotopy analysis method (HAM) is employed to obtain a family of series solutions of the time-dependent reaction–diffusion problems. HAM provides a convenient way of controlling the convergence region and rate of the series solution.

In this Letter, variational iteration method (VIM) is applied to obtain approximate analytical solution of the sine-Gordon equation without any discretization. Comparisons with the exact solutions reveal that VIM is very effective and convenient.

In this Letter, variational iteration method (VIM) is employed to obtain analytical solution for the heat- and wave-like equations with singular behaviors. Comparisons with the exact solution show that the VIM is a powerful method for the solution of linear and nonlinear singular equations.