Stability and Hopf Bifurcation Analysis of a General Tri-diagonal BAM Neural Network with Delays

Abstract

In this article, a general tri-diagonal bidirectional associative memory (BAM) neural network model with 2n-neurons is proposed. Our investigates have some distinct superiorities, including forward transmission delay and feedback delay in the model are considered. Moreover, the neural network model considered with 2n-neurons is more general in application. We obtain the general expression of characteristic equation of BAM neural network model with 2n-neurons for investigating its stability and Hopf bifurcation. The distribution characteristics of roots of characteristic equation are discussed in detail to get the sufficient conditions for Hopf bifurcation of system caused by delays. Then the designed protocol is simpler and easier to implement compared with some previously investigates. Subsequently, by using the normal form method and central manifold theorem, the direction of Hopf bifurcation and the period and stability of bifurcation periodic solution are determined. Finally, several examples are also utilized to illustrate the validity of theoretical results.

Bifurcation analysis of motoneuronal excitability mechanisms under normal and ALS conditions

“…Neuronal firing is the outcome of many interacting non-linear membrane properties and ionic mechanisms, and bifurcation analysis provides a means to study and quantify the effect(s) each membrane property has on cell firing. As XPPAUT does not have tools to facilitate the development of high-fidelity neuronal models with detailed anatomy and multiple ion channels, bifurcation analysis has been traditionally conducted on reduced neuronal models of a single compartment with only a few ion channels ( White et al, 1995 ; Wang et al, 2019 , 2021 ; Xing et al, 2022 ). While bifurcation analysis of reduced neuronal models has been instrumental in shaping our understanding of the neuronal non-linear dynamics underlying membrane oscillations and cell firing under normal ( Jalics et al, 2010 ; Zhou et al, 2018 ), pathological ( Glass and Mackey, 1979 ; Kaper et al, 2013 ), and pharmacological ( V-Ghaffari et al, 2017 ) conditions, extending this analysis to higher-fidelity models with multiple compartments and dendritic channels, which mediate many non-linear firing behaviors in MNs ( Hounsgaard and Kiehn, 1993 ; Li and Bennett, 2007 ; Carlin et al, 2009 ) has been limited.…”

Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

“…Remark 2. It should be pointed out that this article adopts the linearization method [23,24,29,41] to deal with the dynamics analysis of systems, including the local stability, Turing instability and Hopf bifurcation. It is common knowledge that Lyapunov's second method is important to stability theory of dynamical systems and control theory.…”

“…has a pair of purely imaginary eigenvalues Λ 0 = {iω n 􏽥 τ, iω n 􏽥 τ}. α = 0 is a critical point of Hopf bifurcation in system (41). It follows from Riesz representation theorem that there is a 2 × 2 matrix function ε (κ, τ) satisfying…”

Spatiotemporal Evolution Characteristics of Time‐Delay Ecological Competition Systems with Food‐Limited and Diffusion

“…Remark 3. It should be pointed out that this paper adopts the linearization method [29,36,37,39,40] to deal with the dynamics analysis of system (9), including the local stability, Turing instability, and Hopf bifurcation. It is common knowledge that Lyapunov's second method is important to the stability theory of dynamical systems and control theory.…”

“…In particular, Hopf bifurcation is a dynamic bifurcation phenomenon, which shows that when the parameters change near the critical value, the stability of the equilibrium point will change and periodic solutions will be generated in its small neighborhood. Meanwhile, it is found that time delay as a bifurcation parameter can induce Hopf bifurcation [29,36,37]. erefore, in this paper, we consider the pregnancy delay of the predator population and analyse the dynamic characteristics of ecological competition systems.…”