Abstract
Currently, some works on studying complete synchronization of dynamical systems are usually restricted to its two special cases: 1) power-rate synchronization and 2) exponential synchronization. Therefore, how to give a generalization of these types of complete synchronization by the mathematical expression is an open question that needs to be urgently solved. To begin with, this article proposes multimode function synchronization by the mathematical expression for the first time, which is a generalization of exponential synchronization, power-rate synchronization, logarithmical synchronization, and so on. Moreover, two adaptive controllers are designed to achieve robust multimode function synchronization of memristive neural networks (MNNs) with mismatched parameters and uncertain parameters. Each adaptive controller includes function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r(t)$ </tex-math></inline-formula> and update gain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sigma $ </tex-math></inline-formula> . By choosing different types of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r(t)$ </tex-math></inline-formula> , multiple types of complete synchronization, including power-rate synchronization and exponential synchronization can be obtained. And update gain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sigma $ </tex-math></inline-formula> can be used to adjust the speed of synchronization. Therefore, our results enlarge and strengthen the existing results. Two examples are put forward to verify the validity of our results.
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