Abstract: The selection of periodical or chaotic attractors becomes initial-dependent in that setting different initial values can trigger a different profile of attractors in a dynamical system with memory by adding a nonlinear term such as z²y in the Rössler system. The memory effect means that the outputs are very dependent on the initial value for variable z, e.g. magnetic flux for a memristor. In this study, standard nonlinear analyses, including phase portrait, bifurcation analysis, and Lyapunov exponent analysis were carried out. Synchronization between two coupled oscillators and a network was investigated by resetting initial states. A statistical synchronization factor was calculated to find the dependence of synchronization on the coupling intensity when different initial values were selected. Our results show that the dynamics of the attractor depends on the selection of the initial value for one variable z. In the case of coupling between two oscillators, appropriate initial values are selected to trigger two different nonlinear oscillators (periodical and chaotic). Results show that complete synchronization between periodical oscillators, chaotic oscillators, and periodical and chaotic oscillators can be realized by applying an appropriate unidirectional coupling intensity. In particular, two periodical oscillators can be coupled bidirectionally to reach chaotic synchronization so that periodical oscillation is modulated to become chaotic. When the memory effect is considered on some nodes of a chain network, enhancement of memory function can decrease the synchronization, while a small region for intensity of memory function can contribute to the synchronization of the network. Finally, dependence of attractor formation on the initial setting was verified on the field programmable gate array (FPGA) circuit in digital signal processing (DSP) builder block under Matlab/Simulink.

Key words: Synchronization; Bifurcation; Synchronization factor; Field programmable gate array (FPGA)

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