Complex modified function projective synchronization of complex chaotic systems with known and unknown complex parameters

Much progress has been made in the research of modified function projective synchronization (MFPS) for real (complex) chaotic systems with real parameters. The scaling functions are always chosen as real-valued ones in previous MFPS schemes for chaotic systems evolving in the same or inverse directions simultaneously. However, MFPS with different complex-valued scaling functions (CMFPS) has not been previously reported, where complex-variable chaotic (hyperchaotic) systems (CVCSs) evolve in different directions with a time-dependent intersection angle. Therefore, CMFPS is discussed for CVCSs with known and unknown complex parameters in this paper. By constructing appropriate Lyapunov functions defined on complex field, and employing adaptive control technique, a set of simple and practical sufficient conditions for achieving CMFPS are derived, and complex update laws for estimating unknown parameters are also given. The corresponding theoretical proofs and computer simulations are worked out to demonstrate the effectiveness and feasibility of the proposed schemes.