Applying delayed dynamical models to characterize the dynamics of neural networks has attracted great interest from scientific community. In this current manuscript, a kind of new tri‐neuron bidirectional associative memory (BAM) neural networks including delay are formulated. The properties of solution and Hopf bifurcation issue of the established tri‐neuron BAM neural networks including delay are investigated. First, we check the existence and uniqueness of the solution to the formulated delayed tri‐neuron BAM neural networks by virtue of fixed point theorem. Second, we seek the boundedness of the solution of the formulated delayed tri‐neuron BAM neural networks in view of a suitable function and inequality skills. Third, the delay‐independent criteria on stability and bifurcation of the formulated delayed tri‐neuron BAM neural networks are acquired. Fourth, Hopf bifurcation control aspect of the formulated delayed tri‐neuron BAM neural networks is explored via designing two proper hybrid controllers. To prove the accuracy of gained primary assertions, software simulation experiments are put into practice.

Fractional-order differential models plays a pivotal role in depicting the relationship among concentration changes of various chemical substances in chemistry. In this current study, we will explore the dynamics of a delayed chemostat model. First of all, we prove that the solution of the delayed chemostat model exists and is unique by virtue of fixed point theorem. Secondly, we demonstrate that the solution of the delayed chemostat model is non-negative by applying some suitable inequality strategies. Thirdly, the boundedness of the solution to the delayed chemostat model is explored via constructing a reasonable function. Fourthly, the Hopf bifurcation and stability of the delayed chemostat model are dealt with by exploiting the stability criterion and bifurcation theory on fractional dynamical system. Fifthly, the stability domain and Hopf bifurcation of the delayed chemostat model are resoundingly controlled by making use of an extended hybrid controller. Sixthly, the stability domain and Hopf bifurcation of the delayed chemostat model are effectively adjusted by making use of an another extended hybrid controller. The role of delay in this chemostat model is revealed. Seventhly, software experiments are given to illustrate the rightness of the gained key conclusions. The acquired outcomes of this work are perfectly innovative and have crucial theoretical value in controlling the concentrations of various chemical substances.

Establishing financial models or economic models to describe economic phenomena in real life has become a heated discussion in society at present. From a mathematical point of view, the exploration on dynamics of financial models or economic models is a valuable work. In this study, we build a new delayed finance model and explore the dynamical behavior containing existence and uniqueness, boundedness of solution, Hopf bifurcation, and Hopf bifurcation control of the considered delayed finance model. By virtue of fixed point theorem, we prove the existence and uniqueness of the solution to the considered delayed finance model. Applying a suitable function, we obtain the boundedness of the solutions for the considered delayed finance model. Taking advantage of the stability criterion and bifurcation argument of delayed differential equation, we establish a delay-independent condition ensuring the stability and generation of Hopf bifurcation of the involved delayed finance model. Exploiting hybrid controller including state feedback and parameter perturbation, we efficaciously adjust the stability region and the time of occurrence of Hopf bifurcation of the involved delayed finance model. The study manifests that time delay is a fundamental parameter in controlling stability region and the time of onset of Hopf bifurcation of the involved delayed finance model. To examine the soundness of established key results, computer simulation figures are concretely displayed. The derived conclusions of this study are perfectly new and has momentous theoretical value in economical operation.