Abstract
In this manuscript, we principally probe into a class of fractional-order tri-neuron neural networks incorporating delays. Making use of fixed point theorem, we prove the existence and uniqueness of solution to the fractional-order tri-neuron neural networks incorporating delays. By virtue of a suitable function, we prove the uniformly boundedness of the solution to the fractional-order tri-neuron neural networks incorporating delays. With the aid of the stability theory and bifurcation knowledge of fractional-order differential equation, a new delay-independent condition to guarantee the stability and creation of Hopf bifurcation of the fractional-order tri-neuron neural networks incorporating delays is established. Taking advantage of the mixed controller that contains state feedback and parameter perturbation, the stability region and the time of onset of Hopf bifurcation of the fractional-order trineuron neural networks incorporating delays are successfully controlled. Software simulation plots are displayed to illustrate the established key results. The obtained conclusions in this article have important theoretical significance in designing and controlling neural networks.
Highlights
After the classical work of Hopfield [1] on neural networks, many scholars pay much attention to the study of the dynamics of different types of neural networks since neural networks have displayed extraordinary application value in various fields such as associate memory, automatic control, image processing, biological engineering, pattern recognition and so on [2–5]
A natural problem arises: What is the impact of time delay on the dynamical behavior of neural networks? During the past decades, in order to reveal the influence of time delay on various dynamics of neural networks, a great deal of researchers from mathematics and engineering have made great efforts to explore the dynamics of delayed neural networks and lots of valuable fruits have been achieved
Stimulated by the idea, in this present work, we are to deal with the following three aspects: (i) Explore the existence and uniqueness, the boundedness of solution of the involved neural networks with single delay; (ii) Set up a sufficient criterion ensuring the stability and and the creation of Hopf bifurcation of the involved neural networks with single delay; (iii) Make use of a suitable mixed controller that contains state feedback and parameter perturbation to control the time of onset of Hopf bifurcation for the involved neural networks with single delay
Summary
After the classical work of Hopfield [1] on neural networks, many scholars pay much attention to the study of the dynamics of different types of neural networks since neural networks have displayed extraordinary application value in various fields such as associate memory, automatic control, image processing, biological engineering, pattern recognition and so on [2–5]. Set up a novel sufficient condition to ensure the global Mittag–Leffler stability of impulsive fractional-order delayed complex-valued BAM neural networks. Xu et al [38] established a sufficient condition to ensure the stability and the onset of Hopf bifurcation of fractional-order six-neuron BAM neural networks with multi-delays. Stimulated by the idea, in this present work, we are to deal with the following three aspects: (i) Explore the existence and uniqueness, the boundedness of solution of the involved neural networks with single delay; (ii) Set up a sufficient criterion ensuring the stability and and the creation of Hopf bifurcation of the involved neural networks with single delay; (iii) Make use of a suitable mixed controller that contains state feedback and parameter perturbation to control the time of onset of Hopf bifurcation for the involved neural networks with single delay.
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