In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented.

Due to finite lifespan of the particles or boundedness of the physical space, tempered fractional calculus seems to be a more reasonable physical choice. Stability is a central issue for the tempered fractional system. This paper focuses on the tempered Mittag–Leffler stability for tempered fractional systems, being much different from the ones for pure fractional case. Some new lemmas for tempered fractional Caputo or Riemann–Liouville derivatives are established. Besides, tempered fractional comparison principle and extended Lyapunov direct method are used to construct stability for tempered fractional system. Finally, two examples are presented to illustrate the effectiveness of theoretical results.

In this paper, we introduce fuzzy theory into the fractional cellular neural networks to dynamically enhance the coupling strength and propose a fractional fuzzy neural network model with interactions. Using the Lyapunov principle of fractional differential equations, we design the adaptive control schemes to realize the synchronization and obtain the synchronization criteria. Finally, we provide some numerical examples to show the effectiveness of our obtained results.

In this paper, we establish a reaction‐diffusion predator‐prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.