Influences of time delay and connection topology on a multi-delay inertial neural system

Abstract

Multiple delays and connection topology are the key parameters for the realistic modeling of networks. This paper discusses the influences of time delays and connection weight on multi-delay artificial neural models with inertial couplings. Firstly, sufficient conditions of some singularities involving static bifurcation, Hopf bifurcation, and pitchfork-Hopf bifurcation are presented by analyzing the transcendental characteristic equation. Secondly, taking self-connection weight and coupling delays as adjusting parameters and utilizing the parameter perturbation with the aid of the non-reduced order technique for the first time, rich dynamics near zero-Hopf interaction are obtained on the plane with self-connected weight and coupling delay as abscissa and ordinate. The multi-delay inertial neural system can exhibit coexisting attractors such as a pair of nontrivial equilibrium points and a periodic orbit with nontrivial equilibrium points. Self-connected weight can affect the number and dynamics of the system equilibrium points, while time delays can contribute to both trivial equilibrium and non-trivial equilibrium losing their stability and generating limit cycles. Simulation plots are displayed with computer software to support the established main results. Compared with the traditional reduced-order method, the used method here is simple and valid with less computation. The key research findings of this paper have significant theoretical guiding value in dominating and optimizing networks.

Appendix A

Proof

For simplification, let

$$P = \left( {\begin{array}{*{20}c} {p_{1} } \\ {p_{2} } \\ \ldots \\ {p_{n} } \\ \end{array} } \right),\;Q = \left( {\begin{array}{*{20}c} {q_{1} } \\ {q_{2} } \\ \ldots \\ {q_{n} } \\ \end{array} } \right),\;A_{0} = \left( {\begin{array}{*{20}c} {s_{1} } \\ {s_{2} } \\ \ldots \\ {s_{n} } \\ \end{array} } \right).$$

The solution (18) is rewritten as

$$Y\left( t \right) = P\cos \left( {\omega t} \right) + Q\sin \left( {\omega t} \right) + A_{0} .$$

(A.1)

One can obtain the following equations from (A.1)

$$\frac{dY}{{dt}} = - \omega P\sin \left( {\omega t} \right) + \omega Q\cos \left( {\omega t} \right).$$

(A.2)

$$\frac{{d^{2} Y}}{{dt^{2} }} = - \omega^{2} P\cos \left( {\omega t} \right) - \omega^{2} Q\sin \left( {\omega t} \right).$$

(A.3)

$$Y\left( {t + s} \right) = P\cos \left( {\omega t + \omega s} \right) + Q\sin \left( {\omega t + \omega s} \right) + A_{0} .$$

(A.4)

$$Y\left( {t + \tau_{0} } \right) = P\cos \left( {\omega t + \omega \tau_{0} } \right) + Q\sin \left( {\omega t + \omega \tau_{0} } \right) + A_{0} .$$

(A.5)

Substituting Eqs. (A.1) to (A.5) into Eq. (17), the following three equations are derived as

$$\begin{gathered} \left[ {\omega K^{T} - D_{10}^{T} \sin \left( {\omega s} \right) - D_{2}^{T} \sin \left( {\omega \tau_{0} } \right)} \right]P = \hfill \\ \left[ { - \omega^{2} I - B^{T} - D_{10}^{T} \cos \left( {\omega s} \right) - D_{2}^{T} \cos \left( {\omega \tau_{0} } \right)} \right]Q, \hfill \\ \end{gathered}$$

(A.6)

$$\begin{gathered} \left[ { - \omega K^{T} + D_{10}^{T} \sin \left( {\omega s} \right) + D_{2}^{T} \sin \left( {\omega \tau_{0} } \right)} \right]Q = \hfill \\ \left[ { - \omega^{2} I - B^{T} - D_{10}^{T} \cos \left( {\omega s} \right) - D_{2}^{T} \cos \left( {\omega \tau_{0} } \right)} \right]P, \hfill \\ \end{gathered}$$

(A.7)

$$\left( {B^{T} + D_{10}^{T} + D_{2}^{T} } \right)A_{0} = 0.$$

(A.8)

That is,

$$- F_{1}^{T} \left( {\begin{array}{*{20}c} {p_{1} } \\ {p_{2} } \\ \ldots \\ {p_{n} } \\ \end{array} } \right) = F_{2}^{T} \left( {\begin{array}{*{20}c} {q_{1} } \\ {q_{2} } \\ \ldots \\ {q_{n} } \\ \end{array} } \right),F_{1}^{T} \left( {\begin{array}{*{20}c} {q_{1} } \\ {q_{2} } \\ \ldots \\ {q_{n} } \\ \end{array} } \right) = F_{2}^{T} \left( {\begin{array}{*{20}c} {p_{1} } \\ {p_{2} } \\ \ldots \\ {p_{n} } \\ \end{array} } \right),F_{3}^{T} \left( {\begin{array}{*{20}c} {s_{1} } \\ {s_{2} } \\ \ldots \\ {s_{n} } \\ \end{array} } \right) = 0,$$

where the expressions of $F_{i} \left( {i = 1,2,3} \right)$ are consistent with those in Eq. (10).

This completes this proof of the lemma.