Bifurcation Mechanisation of a Fractional-Order Neural Network with Unequal Delays

Abstract

The theme of bifurcation for a class of fractional-order neural networks (FONNs) with unique delay has been incalculably elucidated. It exhibits that multiple delays are capable of increasing the complicacy of realistic FONNs, but this has been insufficiently probed into. This paper attempts to conduct a research on the stability and bifurcation for a FONN with two unequal delays. By intercalating one delay and taking remnant delay as a bifurcation parameter, the incongruent critical values of diverse delays-induced bifurcations are exactly gained. Eventually, confirmation experiments are offered to endorse the procured theory.

1 Introduction

The explorations on the theory and applications of neural networks (NNs) have been garnered intense concerns thanks to their generally authentic applications [1,2,3,4,5,6,7]. Currently, fractional calculus has been favorably drawn into NNs in virtue of the pinpoint characterization for the dynamic response of the actual systems. It is exposed that fractional differentiation is emerged into NNs which can efficiently handle formation [8]. Some delightful results and applications have been founded for FONNs, such as image encryption [9], network approximation [10]. Therefore, it is requisite to explore the dynamics of FONNs. There has been an immensely ever-increasing attention in the explorations of FONNs, and some important and meritorious results were obtained [11,12,13,14,15]. In [11], the lag synchronization for delayed fractional-order memristive NNs was reached by the aid of switching jumps mismatch, and the lag quasi-synchronization conditions were further derived.

Researchers have the ability to develop prospective dynamical properties of nonlinear systems via impelling Hopf bifurcation methodology [16,17,18,19,20]. It is generally known that the bifurcations of integer-order systems have been excessively discussed. On account of high accurateness of describing NNs with the help of fractional calculus in comparison with accustomed ones, numerous scholars have been overwhelmingly enchanted by the bifurcations of fractional-order systems, and numerous valuable results have been reported [21,22,23,24,25,26,27,28]. In [21], the authors investigated a fractional delayed predator–prey model with Holling type II functional response including prey refuge and diffusion, and it indicated that the stability domain can be extended under the fractional order compared with integer-order one. In [24], the bifurcation control of a delayed fractional eco-epidemiological model with incommensurate orders was examined in terms of linear feedback strategy. It proclaimed that stability performance of the system can be varied by amplifying the control delay. In [25], the issue of bifurcation control for a delayed fractional-order predator–prey system was studied by using enhancing feedback method, and it divulged that the enhancing feedback can greatly diminish the control cost by comparison with dislocated feedback. In [26], the stability and bifurcation of a delayed fractional-order quaternion-valued neural network were considered. It manifested that the bifurcation phenomena transpire earlier as fractional order incrementally magnifies. Noticeably, the bulk of existing bifurcations results are focused on fractional-order systems including a single delay. Accordingly, it is reasonable and imperative to integrate multiple time delays into the fractional-order systems for accurately describing dynamical properties. Moreover, past findings has evidenced that the continuous state variables depending on different history, it is extremely essential to pierce the functions of diversified delays for NNs for well-implementing networks with electronic devices and enhancing the performance of optimizing networks in [29].

It is to be observed that some results available on bifurcations have been focused on investigating FONNs with a sole delay as a result of the more complicacy and difficulty for the academic inquiries involving multiple delays [30]. It is evident that systems with single delay can not well reflect the realistic dynamics of dynamical systems. Most currently, some scholars have tried to go a bit deeper and look at the effects of different delays with regard to the bifurcations for delayed FONNs [31, 32]. In [31], the authors considered the issue of bifurcation for a fractional-order predator–prey model comprising two discrepant delays, and the bifurcation points were deduced. The stability and bifurcation of a FONN with double delays, and the conditions of diverse delays-induced bifurcations were exactly derived in [32]. However, it should be pointed out that the investigations for the bifurcation of FONNs with multiple delays is insufficient with a vengeance.

Prompted by aforementioned discussions, we are dedicated to addressing a theoretical investigation of bifurcation for a FONN with two nonidentical delays in this paper. The essential benefits of this paper can be epitomized as follows: (1) It is a super-stretch challenging subject to identify the conditions of bifurcations for fractional order systems with multiple delays. This paper theoretically derives the exact bifurcation points based on disparate delays as the bifurcation parameters on account of the high complexity of stability analysis for dynamical systems with different delays. (2) Discriminating from [31, 32], this paper solves the bifurcation problems for a class of four-dimensional fractional-order system with two unequal delays. (3) This paper addresses an effectual scheme to theoretically investigate the bifurcation of higher dimensional FONNs with two different delays. It further provides an idea and framework to explore the issue of bifurcation for fractional-order systems with three or more delays.

The framework of the paper is portrayed in the following: In Sect. 2 addresses the fractional Caputo definition and basic stability results of fractional linear systems. In Sect. 3 formulates the mathematical model. Section 4 derives the outcomes of Hopf bifurcation by utilising different delays as bifurcation parameters. Section 5 gauges the efficiency of the proposed theory by exploiting simulation cases. Section 6 summaries the essential upshot.

2 Rudimentary Theoretical Tools

This section addresses the Caputo definition and lemma with respect to fractional calculus for the next theoretical analysis and simulations.

Definition 1

[33] The Caputo fractional-order derivative is defined by

$$\begin{aligned} D^{q}f(t)=\frac{1}{\varGamma (\iota -q)}\int _0^t (t-s)^{\iota -q-1}f^{(\iota )}(s)dt, \end{aligned}$$

where $\iota -1<q\le \iota \in Z^+$, $\varGamma (\cdot )$ is the Gamma function, $\varGamma (s)=\int _0^\infty t^{s-1}e^{-t}dt$.

Based on the Laplace transform, we can express as

$$\begin{aligned} L\{D^{q}f(t);s\}=s^q F(s)-\sum _{k=0}^{\iota -1}s^{q-k-1}f^{(k)}(0),\quad \iota -1<q\le \iota \in Z^+. \end{aligned}$$

If $f^k(0)=0$, $k=1,2,\ldots ,n$, then $L\{D^{q}f(t);s\}=s^q F(s).$

Lemma 1

[34] Explore linear fractional-order systems with multiple variables

$$\begin{aligned} \left\{ \begin{aligned}&D^{q_1}z_{1}(t)=o_{11}z_{1}(t)+o_{12}z_{2}(t) +\cdots +o_{1n}z_{n}(t),\\&D^{q_2}z_{2}(t)=o_{21}z_{1}(t)+o_{22}z_{2}(t) +\cdots +o_{22}z_{n}(t),\\&\quad \vdots \\&D^{q_n}z_{n}(t)=o_{n1}z_{1}(t)+o_{n2}z_{2}(t) +\cdots +o_{nn}z_{n}(t), \end{aligned} \right. \end{aligned}$$

(1)

where $q_i\in (0,1](i=1,2,\ldots ,n)$. Suppose that q is the lowest common multiple of the denominators $\zeta _i$ of $q_i$, where $q_i=\frac{\sigma _i}{\zeta _i}$, $(\sigma _i,\zeta _i)=1$, $\sigma _i,\zeta _i\in Z^+$, for $i=1,2,\ldots ,n$. It is labeled as

$$\begin{aligned} \triangle (s)=\mathrm {}\left[ \begin{array}{cccc} s^{q_1}-o_{11} &{} \quad -o_{12} &{} \quad \cdots &{} \quad -o_{1n} \\ -o_{21} &{} \quad s^{q_2}-o_{22} &{} \quad \cdots &{} \quad -o_{2n}\\ \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots \\ -o_{n1} &{} \quad -o_{n2} &{} \quad \cdots &{} \quad s^{q_n}-o_{nn} \end{array} \right] . \end{aligned}$$

Then the zero solution of system (1) is globally asymptotically stable in the Lyapunov sense if all roots s of the equation $\det (\triangle (s))=0$ satisfy $|\arg (s)|>q_i\pi /2$.

3 Mathematical Modeling

Due to the high-precision description of fractional calculus for NNs. In this paper, we hence incorporate fractional calculus into the dynamical model [35]. The developed model can be addressed as

$$\begin{aligned} \left\{ \begin{aligned}&D^{q_1}z_1(t)=-\nu _1z_1(t)+a_1\varphi _1(z_1(t))+b_1\psi _1(z_4(t-\tau _2))+c_1\psi _1(z_2(t-\tau _2)),\\&D^{q_2}z_2(t)=-\nu _2z_2(t)+a_2\varphi _2(z_2(t))+b_2\psi _2(z_1(t-\tau _1))+c_2\psi _2(z_3(t-\tau _1)),\\&D^{q_3}z_3(t)=-\nu _3z_3(t)+a_3\varphi _3(z_3(t))+b_3\psi _3(z_2(t-\tau _2))+c_3\psi _3(z_4(t-\tau _2)),\\&D^{q_4}z_4(t)=-\nu _4z_4(t)+a_4\varphi _4(z_4(t))+b_4\psi _4(z_3(t-\tau _1))+c_4\psi _4(z_1(t-\tau _1)), \end{aligned} \right. \end{aligned}$$

(2)

where $q_i\in (0,1](i=1,2,3,4)$, $z_i(t)$ denote state variables, $\nu _i>0$ represent real values, $a_i$, $b_i$ and $c_i$ are connection weights, $\varphi _i(\cdot )$, $\psi _i(\cdot )$ denote activation functions, $\tau _1$ and $\tau _2$ are communication delays.

If setting up $q_i=1$, then FONN (2) can be clearly inverted into the classical model in [35].

The following assumptions are addressed for catching the main results:

$(\mathbf{H1} )$$\varphi _i, \psi _i\in C(R,R)$, $\varphi _i(0)=\psi _i(0)=0$, $z\varphi _i(z)>0$, $z\psi _i(z)>0$$(i=1,2,3,4)$ for $z\ne 0$.

4 Main Results

In this section, disparate delays are selected as a bifurcation parameter to study the stability and bifurcation for FONN (2), and the bifurcation points are exactly established.

4.1 The Existence of Bifurcation via $\tau _1$ of FONN (2)

In this subsection, we select $\tau _1$ as a bifurcation parameter to study the bifurcation of FONN (2). It is obvious that the origin is an equilibrium point of FONN (2) based on $(\mathbf{H1} )$. The linear equation of FONN (2) around the origin can be presented as

$$\begin{aligned} \left\{ \begin{aligned}&D^{q_1}z_1(t)=-k_1z_1(t)+m_1z_4(t-\tau _2)+n_1z_2(t-\tau _2),\\&D^{q_2}z_2(t)=-k_2z_2(t)+m_2z_1(t-\tau _1)+n_2z_3(t-\tau _1),\\&D^{q_3}z_3(t)=-k_3z_3(t)+m_3z_2(t-\tau _2)+n_3z_4(t-\tau _2),\\&D^{q_4}z_4(t)=-k_4z_4(t)+m_4z_3(t-\tau _1)+n_4z_1(t-\tau _1), \end{aligned} \right. \end{aligned}$$

(3)

where $k_i=\nu _i-a_i\varphi '_i(0)$, $m_i=b_i\psi '_i(0)$, $n_i=c_i\psi '_i(0)$$(i=1,2,3,4)$.

In view of FONN (3), the following form characteristic equation can be gained

$$\begin{aligned}&U_1(s)+U_2(s)e^{-s\tau _1}+U_3(s)e^{-2s\tau _1}=0 \end{aligned}$$

(4)

$$\begin{aligned}&U_1(s)=s^{q_1+q_2+q_3+q_4}+k_4s^{q_1+q_2+q_3}+k_3s^{q_1+q_2+q_4}+k_2s^{q_1+q_3+q_4}+k_1s^{q_2+q_3+q_4} \\&\quad +k_3k_4s^{q_1+q_2} +k_2k_4s^{q_1+q_3}+k_2k_3s^{q_1+q_4}+k_1k_4s^{q_2+q_3}+k_1k_3s^{q_2+q_4}+k_1k_2s^{q_3+q_4} \\&\quad +k_2k_3k_4s^{q_1}+k_1k_3k_4s^{q_2} +k_1k_2k_4s^{q_3}+k_1k_2k_3s^{q_4}+k_1k_2k_3k_4,\\&U_2(s)=-[n_3m_4s^{q_1+q_2}+n_4m_1s^{q_2+q_3}+m_2nn_1s^{q_3+q_4}+m_3n_2s^{q_1+q_4}+(k_2n_3m_4\\&\quad +k_4m_3n_2)s^{q_1} +(k_1n_3m_4+k_3n_4m_1)s^{q_2}+(k_2n_4m_1+k_4m_2n_1)s^{q_3}\\&\quad +(k_1m_3n_2+k_3m_2n_1)s^{q_4} +k_1k_2n_3m_4+k_3k_4m_3n_1+k_2k_3n_4m_1+k_1k_4m_3n_2]e^{-s\tau _2},\\ U_3(s)&=[m_1m_3n_2n_4+m_2m_4n_1n_3-m_1m_2m_3m_4-n_1n_2n_3n_4]e^{-2s\tau _2}. \end{aligned}$$

Multiplying $e^{s\tau _1}$ on both sides of Eq. (4), it obtains as

$$\begin{aligned} U_1(s)e^{s\tau _1}+U_2(s)+U_3(s)e^{-s\tau _1}=0. \end{aligned}$$

(5)

The real and imaginary parts of $U_l(s)(l=1,2,3)$ can be labeled by $U_l^r$, $U_l^i$, respectively. Assume that $s=w(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})$ is a purely imaginary root of Eq. (5), $w>0$. Substituting s into Eq. (5) and separating the real and imaginary parts of it, it results in

$$\begin{aligned} \left\{ \begin{aligned}&(U_1^r+U_3^r)\cos w\tau _1+(U_3^i-U_1^i)\sin w\tau _1=-U_2^r,\\&(U_1^i+U_3^i)\cos w\tau _1+(U_1^r-U_3^r)\sin w\tau _1=-U_2^i. \end{aligned} \right. \end{aligned}$$

(6)

Labeling

$$\begin{aligned}&\varUpsilon _1(w)=-U_2^r(U_1^r-U_3^r)+U_2^i(U_3^i-U_1^i),\\&\varUpsilon _2(w)=-U_2^i(U_1^r+U_3^r)+U_2^r(U_1^i+U_3^i),\\&\varUpsilon _3(w)=(U_1^r)^2+(U_1^i)^2-(U_3^r)^2-(U_3^i)^2.\\ \end{aligned}$$

As far as Eq. (6), it concludes that

$$\begin{aligned} \left\{ \begin{aligned}&\cos w\tau _1=\frac{\varUpsilon _1(w)}{\varUpsilon _3(w)},\\&\sin w\tau _1=\frac{\varUpsilon _2(w)}{\varUpsilon _3(w)}. \end{aligned} \right. \end{aligned}$$

(7)

By means of Eq. (7), it procures that

$$\begin{aligned} \varUpsilon _3^2(w)=\varUpsilon _1^2(w)+\varUpsilon _2^2(w). \end{aligned}$$

(8)

It defines from Eq. (8) that

$$\begin{aligned} \mathcal {H}(w)=\varUpsilon _3^2(w)-\varUpsilon _1^2(w)-\varUpsilon _2^2(w)=0. \end{aligned}$$

(9)

To establish the main results of this section, the following assumptions are useful and needed.

$(\mathbf{H2} )$ There exist positive real roots for Eq. (9).

Based on Eq. (7), it clearly derive as

$$\begin{aligned} \tau _{10}^{(k)}=\frac{1}{w}\Big [\frac{\varUpsilon _1(w)}{\varUpsilon _3(w)}+2k\pi \Big ],\quad k=0,1,2,\ldots . \end{aligned}$$

(10)

Label the bifurcation point of FONN (2) as

$$\begin{aligned} \tau _{10}=\min \{\tau _{10}^{(k)}\},\quad k=0,1,2,\ldots , \end{aligned}$$

where $\tau _{10}^{(k)}$ is defined by Eq. (10).

Equation (4) can be transformed into the following form when renouncing $\tau _1$

$$\begin{aligned}&\ell _1(s)+\ell _2(s)e^{-s\tau _2}+\ell _3e^{-2s\tau _2}=0, \end{aligned}$$

(11)

where

$$\begin{aligned} \ell _1(s)&=s^{q_1+q_2+q_3+q_4}+k_4s^{q_1+q_2+q_3}+k_3s^{q_1+q_2+q_4}+k_2s^{q_1+q_3+q_4}+k_1s^{q_2+q_3+q_4} \\&\quad +k_3k_4s^{q_1+q_2}+k_2k_4s^{q_1+q_3}+k_2k_3s^{q_1+q_4}+k_1k_4s^{q_2+q_3}+k_1k_3s^{q_2+q_4}+k_1k_2s^{q_3+q_4} \\&\quad +k_2k_3k_4s^{q_1}+k_1k_3k_4s^{q_2} +k_1k_2k_4s^{q_3}+k_1k_2k_3s^{q_4}+k_1k_2k_3k_4,\\ \ell _2(s)&=-[n_3m_4s^{q_1+q_2}+n_4m_1s^{q_2+q_3}+m_2n_1s^{q_3+q_4}+m_3n_2s^{q_1+q_4}\\&\quad +(k_2n_3m_4+k_4m_3n_2)s^{q_1} +(k_1n_3m_4+k_3n_4m_1)s^{q_2}+(k_2n_4m_1+k_4m_2n_1)s^{q_3}\\&\quad +(k_1m_3n_2+k_3m_2n_1)s^{q_4} +k_1k_2n_3m_4+k_3k_4m_3n_1+k_2k_3n_4m_1+k_1k_4m_3n_2],\\ \ell _3(s)&=m_1m_3n_2n_4+m_2m_4n_1n_3-m_1m_2m_3m_4-n_1n_2n_3n_4. \end{aligned}$$

If $\tau _2=0$, it follows from Eq. (11) that

$$\begin{aligned}&s^{q_1+q_2+q_3+q_4}+l_1s^{q_1+q_2+q_3}+l_2s^{q_1+q_2+q_4}+l_3s^{q_1+q_3+q_4}+l_4s^{q_2+q_3+q_4} +l_5s^{q_1+q_2}+l_6s^{q_1+q_3}\nonumber \\&\quad +l_7s^{q_1+q_4}+l_8s^{q_2+q_3}+l_9s^{q_2+q_4}+l_{10}s^{q_3+q_4} +l_{11}s^{q_1}+l_{12}s^{q_2}+l_{13}s^{q_3}+l_{14}s^{q_4}+l_{15}=0, \end{aligned}$$

(12)

where

$$\begin{aligned} l_1&= k_4,\quad l_2=k_3,\quad l_3=k_2,\quad l_4=k_1, \quad l_5=k_3k_4-n_3m_4, \quad l_6=k_2k_4, \\ l_7&= k_2k_3-m_3n_2, \quad l_8=k_1k_4-n_4m_1,\quad l_9=k_1k_3, \quad l_{10}=k_1k_2-m_2n_1,\\ l_{11}&= k_2k_3k_4-(k_2n_3m_4+k_4m_3n_2),\quad l_{12}=k_1k_3k_4-(k_1n_3m_4+k_3n_4m_1),\\ l_{13}&= k_1k_2k_4-(k_2n_4m_1+k_4m_2n_1),\quad l_{14}=k_1k_2k_3-(k_1m_3n_2+k_3m_2n_1),\\ l_{15}&= k_1k_2k_3k_4-(k_1k_2n_3m_4+k_3k_4m_3n_1+k_2k_3n_4m_1+k_1k_4m_3n_2)\\&\quad +m_1m_3n_2n_4+m_2m_4n_1n_3-m_1m_2m_3m_4-n_1n_2n_3n_4. \end{aligned}$$

Assume that all roots s of the Eq. (12) obey Lemma 1, then we obtain that all the roots of Eq. (12) has negative real parts.

The real and imaginary parts of $\ell _i(s)(i=1,2,3)$ can be designated by $\ell _i^r$, $\ell _i^i$, respectively. Multiplying $e^{s\tau _2}$ on both sides of Eq. (11), it can be obtained as

$$\begin{aligned}&\ell _1(s)e^{s\tau _2}+\ell _2(s)+\ell _3(s)e^{-s\tau _2}=0. \end{aligned}$$

(13)

Presume that $s=\bar{w}(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})(\bar{w}>0)$ is a purely imaginary root of Eq. (13), then we can reason out

$$\begin{aligned} \left\{ \begin{aligned}&(\ell _1^r+\ell _3^r)\cos \bar{w}\tau _2+(\ell _3^i-\ell _1^i)\sin \bar{w}\tau _2=-\ell _2^r,\\&(\ell _1^i+\ell _3^i)\cos \bar{w}\tau _2+(\ell _1^r-\ell _3^r)\sin \bar{w}\tau _2=-\ell _2^i. \end{aligned} \right. \end{aligned}$$

(14)

It labels as

$$\begin{aligned}&G_1(\bar{w})=-\ell _2^r(\ell _1^r-\ell _3^r)+\ell _2^i(\ell _3^i-\ell _1^i),\\&G_2(\bar{w})=-\ell _2^i(\ell _1^r+\ell _3^r)+\ell _2^r(\ell _1^i+\ell _3^i),\\&G_3(\bar{w})=(\ell _1^r)^2+(\ell _1^i)^2-(\ell _3^r)^2-(\ell _3^i)^2.\\ \end{aligned}$$

It concludes from Eq. (14) that

$$\begin{aligned} \left\{ \begin{aligned}&\cos \bar{w}\tau _2=\frac{G_1(\bar{w})}{G_3(\bar{w})},\\&\sin \bar{w}\tau _2=\frac{G_2(\bar{w})}{G_3(\bar{w})}. \end{aligned} \right. \end{aligned}$$

(15)

By means of Eq. (15), it procures that

$$\begin{aligned} G^2_3(\bar{w})=G_1^2(\bar{w})+G_2^2(\bar{w}). \end{aligned}$$

(16)

It can be defined from Eq. (16) that

$$\begin{aligned} \rho (\bar{w})=G_3^2(\bar{w})-G_1^2(\bar{w})-G_2^2(\bar{w})=0. \end{aligned}$$

(17)

The following assumption is addressed.

$(\mathbf{H3} )$ There exists positive roots for Eq. (17).

By means of Eq. (17), the values of w can be obtained according to numerical software Maple 13, then the bifurcation point $\tau _{20}^*$ of FONN (2) with $\tau _1=0$ can be derived.

To throw out the bifurcation conditions, the following assumption is addressed

$(\mathbf{H4} )$$\frac{\varDelta _1\varLambda _1+\varDelta _2\varLambda _2}{\varLambda _1^2+\varLambda _2^2}\ne 0$, where $\varDelta _1$, $\varDelta _2$, $\varLambda _1$, $\varLambda _2$ are described by Eq. (20).

Lemma 2

Let $s(\tau _1)=\xi (\tau _1)+i\rho (\tau _1)$ be the root of Eq. (4) near $\tau _1=(\tau _1)_j$ complying with $\xi ((\tau _1)_j)=0$, $w((\tau _1)_j)=w_0$, then the following transversality condition holds

$$\begin{aligned} \mathrm {Re}\Big [\frac{ds}{d\tau _1}\Big ] \Big |_{(w=w_0,\tau _1=\tau _{10})}\ne 0. \end{aligned}$$

Proof

The real and imaginary parts of $U'_p(s)p=1,2,3$ can be labeled by $U_p^{'r}$, $U_p^{'i}$. Using implicit function theorem to differentiate (4) with regard to $\tau _1$, then

$$\begin{aligned}&U'_1(s)\frac{ds}{d\tau _1}+\left[ U'_2(s)\frac{ds}{d\tau _1}e^{-s\tau _1} +U_2(s)e^{-s\tau _1}\left( -\tau _1\frac{ds}{d\tau _1}-s\right) \right] \nonumber \\&\quad +\left[ U'_3(s)\frac{ds}{d\tau _1}e^{-2s\tau _1}+U_3(s)e^{-2s\tau _1}\left( -2\tau _1\frac{ds}{d\tau _1}-2s\right) \right] =0. \end{aligned}$$

(18)

By mathematical operations from Eq. (18), we elicit

$$\begin{aligned} \frac{ds}{d\tau _1}=\frac{\varDelta (s)}{\varLambda (s)}, \end{aligned}$$

(19)

where

$$\begin{aligned} \varDelta (s)&=s[U_2(s)e^{-s\tau _1}+2U_3(s)e^{-2s\tau _1}],\\ \varLambda (s)&=U'_1(s)+[U'_2(s)-\tau _1 U_2(s)]e^{-s\tau _1}+[U'_3(s)-2\tau _1 U_3(s)]e^{-2s\tau _1}. \end{aligned}$$

It educes from Eq. (19) that

$$\begin{aligned} \mathrm {Re}\Big [\frac{ds}{d\tau _1}\Big ]\Big |_{(w=w_0,\tau =\tau _{10})} =\frac{\varDelta _1\varLambda _1+\varDelta _2\varLambda _2}{\varLambda _1^2+\varLambda _2^2}, \end{aligned}$$

(20)

where

$$\begin{aligned} \varDelta _1&=w_0(U_2^r\sin w_0\tau _{10}-U_2^i\cos w_0\tau _{10} +2U_3^r\sin 2w_0\tau _{10}-2U_3^i\cos 2w_0\tau _{10}),\\ \varDelta _2&=w_0(U_2^r\cos w_0\tau _{10}+U_2^i\sin w_0\tau _{10} +2U_3^r\cos 2w_0\tau _{10}+2U_3^i\sin 2w_0\tau _{10}),\\ \varLambda _1&=U_1^{'r}+(U_2^{'r}-\tau _{10}U_2^r)\cos w_0\tau _{10}+(U_2^{'i}-\tau _{10}U_2^i)\sin w_0\tau _{10}\\&\quad +(U_3^{'r}-2\tau _{10}U_3^r)\cos 2w_0\tau _{10}+(U_3^{'i}-2\tau _{10}U_3^i)\sin 2w_0\tau _{10},\\ \varLambda _2&=U_1^{'i}+(U_2^{'i}-\tau _{10}U_2^i)\cos w_0\tau _{10}-(U_2^{'r}-\tau _{10}U_2^r)\sin w_0\tau _{10}\\&\quad +(U_3^{'i}-2\tau _{10}U_3^i)\cos 2w_0\tau _{10}-(U_3^{'r}-2\tau _{10}U_3^r)\sin 2w_0\tau _{10}. \end{aligned}$$

$(\mathbf{H4} )$ indicates that transversality condition hold. It follows Lemma 2. $\square$

Based on the previous investigations, we can establish the following theorem.

Theorem 1

Under $(\mathbf{H1} )$–$(\mathbf{H4} )$, the following results are procurable.

• (1) If $\tau _2\in [0,\tau _{20}^*)$, then the origin of FONN (2) is asymptotically stable when $\tau _1=\tau _{10}$.

• (2) If $\tau _2\in [0,\tau _{20}^*)$, then FONN (2) undergoes a Hopf bifurcation at the origin when $\tau _1=\tau _{10}$.

4.2 Impact of $\tau _2$ on Bifurcation of FONN (2)

In this subsection, we turn our attention to choose $\tau _2$ as a bifurcation parameter, the bifurcation point is further acquired.

Equation (4) can be rewritten equally as

$$\begin{aligned}&J_1(s)+J_2(s)e^{-s\tau _2}+J_3(s)e^{-2s\tau _2}=0,\\&J_1(s)=s^{q_1+q_2+q_3+q_4}+k_4s^{q_1+q_2+q_3}+k_3s^{q_1+q_2+q_4}+k_2s^{q_1+q_3+q_4}+k_1s^{q_2+q_3+q_4}\nonumber \\&\quad +k_3k_4s^{q_1+q_2} +k_2k_4s^{q_1+q_3}+k_2k_3s^{q_1+q_4}+k_1k_4s^{q_2+q_3}+k_1k_3s^{q_2+q_4}+k_1k_2s^{q_3+q_4} \nonumber \\&\quad +k_2k_3k_4s^{q_1}+k_1k_3k_4s^{q_2} +k_1k_2k_4s^{q_3}+k_1k_2k_3s^{q_4}+k_1k_2k_3k_4,\nonumber \\&J_2(s)=-[n_3m_4s^{q_1+q_2}+n_4m_1s^{q_2+q_3}+m_2n_1s^{q_3+q_4}+m_3n_2s^{q_1+q_4}+(k_2n_3m_4+k_4m_3n_2)s^{q_1}\nonumber \\&\quad +(k_1n_3m_4+k_3n_4m_1)s^{q_2}+(k_2n_4m_1+k_4m_2n_1)s^{q_3}+(k_1m_3n_2+k_3m_2n_1)s^{q_4}\nonumber \\&\quad +k_1k_2n_3m_4+k_3k_4m_2n_1+k_2k_3n_4m_1+k_1k_4m_3n_2]e^{-s\tau _1},\nonumber \\&J_3(s)=[m_1m_3n_2n_4+m_2m_4n_1n_3-m_1m_2m_3m_4-n_1n_2n_3n_4]e^{-2s\tau _1}.\nonumber \end{aligned}$$

(21)

Multiplying $e^{s\tau _2}$ on both sides of Eq. (21), it obtains that

$$\begin{aligned} J_1(s)e^{s\tau _2}+J_2(s)+J_3(s)e^{-s\tau _2}=0. \end{aligned}$$

(22)

Marking the real and imaginary parts of $J_l(s)(l=1,2,3)$ as $J_l^r$, $J_l^i$, respectively. Assume that $s=\varpi (\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})$ is a purely imaginary root of Eq. (22), $\varpi >0$. Then it results in

$$\begin{aligned} \left\{ \begin{aligned}&(J_1^r+J_3^r)\cos \varpi \tau _2+(J_3^i-J_1^i)\sin \varpi \tau _2=-J_2^r,\\&(J_1^i+J_3^i)\cos \varpi \tau _2+(J_1^r-J_3^r)\sin \varpi \tau _2=-J_2^i. \end{aligned} \right. \end{aligned}$$

(23)

Labeling as

$$\begin{aligned}&\varPsi _1(\varpi )=-J_2^r(J_1^r-J_3^r)+J_2^i(J_3^i-J_1^i),\\&\varPsi _2(\varpi )=-J_2^i(J_1^r+J_3^r)+J_2^r(J_1^i+J_3^i),\\&\varPsi _3(\varpi )=(J_1^r)^2+(J_1^i)^2-(J_3^r)^2-(J_3^i)^2.\\ \end{aligned}$$

As far as Eq. (23), it concludes that

$$\begin{aligned} \left\{ \begin{aligned}&\cos \varpi \tau _2=\frac{\varPsi _1(\varpi )}{\varPsi _3(\varpi )},\\&\sin \varpi \tau _2=\frac{\varPsi _2(\varpi )}{\varPsi _3(\varpi )}. \end{aligned} \right. \end{aligned}$$

(24)

By means of Eq. (24), one reads

$$\begin{aligned} \varPsi _3^2(\varpi )=\varPsi _1^2(\varpi )+\varPsi _2^2(\varpi ). \end{aligned}$$

(25)

It labels from Eq. (25) that

$$\begin{aligned} \mathcal {G}(\varpi )=\varPsi _3^2(\varpi )-\varPsi _1^2(\varpi )-\varPsi _2^2(\varpi )=0. \end{aligned}$$

(26)

To establish the main results of this section, the following assumptions is useful and needed.

$(\mathbf{H5} )$ Eq. (26) has one positive real roots.

Based on Eq. (24), it actualizes that

$$\begin{aligned} \tau _{20}^{(k)}=\frac{1}{\varpi }\Big [\frac{\varPsi _1(\varpi )}{\varPsi _3(\varpi )}+2k\pi \Big ],\quad k=0,1,2,\ldots . \end{aligned}$$

(27)

Define the bifurcation point of system (5) as follows:

$$\begin{aligned} \tau _{20}=\min \{\tau _{20}^{(k)}\},\quad k=0,1,2,\ldots , \end{aligned}$$

where $\tau _{20}^{(k)}$ is defined by Eq. (27).

If $\tau _2=0$, then Eq. (21) can be inverted into

$$\begin{aligned}&\hbar _1(s)+\hbar _2(s)e^{-s\tau _1}+\hbar _3e^{-2s\tau _1}=0, \end{aligned}$$

(28)

where

$$\begin{aligned} \hbar _1(s)&=s^{q_1+q_2+q_3+q_4}+k_4s^{q_1+q_2+q_3}+k_3s^{q_1+q_2+q_4}+k_2s^{q_1+q_3+q_4}+k_1s^{q_2+q_3+q_4} \\&\quad +k_3k_4s^{q_1+q_2} +k_2k_4s^{q_1+q_3}+k_2k_3s^{q_1+q_4}+k_1k_4s^{q_2+q_3}+k_1k_3s^{q_2+q_4}+k_1k_2s^{q_3+q_4} \\&\quad +k_2k_3k_4s^{q_1}+k_1k_3k_4s^{q_2} +k_1k_2k_4s^{q_3}+k_1k_2k_3s^{q_4}+k_1k_2k_3k_4,\\ \hbar _2(s)&=-[n_3m_4s^{q_1+q_2}+n_4m_1s^{q_2+q_3}+m_2n_1s^{q_3+q_4}+m_3n_2s^{q_1+q_4}+(k_2n_3m_4+k_4m_3n_2)s^{q_1}\\&\quad +(k_1n_3m_4+k_3n_4m_1)s^{q_2}+(k_2n_4m_1+k_4m_2n_1)s^{q_3}+(k_1m_3n_2+k_3m_2n_1)s^{q_4}\\&\quad +k_1k_2n_3m_4+k_3k_4m_2n_1+k_2k_3n_4m_1+k_1k_4m_3n_2],\\ \hbar _3(s)&=m_1m_3n_2n_4+m_2m_4n_1n_3-m_1m_2m_3m_4-n_1n_2n_3n_4. \end{aligned}$$

The real and imaginary parts of $\hbar _i(s)(i=1,2,3)$ can be denoted by $\hbar _i^r$, $\hbar _i^i$, respectively. Multiplying $e^{s\tau _1}$ on both sides of Eq. (28), it can be obtained as

$$\begin{aligned} \hbar _1(s)e^{s\tau _1}+\hbar _2(s)+\hbar _3(s)e^{-s\tau _1}=0. \end{aligned}$$

(29)

Suppose that $s=\bar{\varpi }(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})(\bar{\varpi }>0)$ is a purely imaginary root of Eq. (29), then we arrive at

$$\begin{aligned} \left\{ \begin{aligned}&(\hbar _1^r+\hbar _3^r)\cos \bar{\varpi }\tau _1+(\hbar _3^i-\hbar _1^i)\sin \bar{\varpi }\tau _1=-\hbar _2^r,\\&(\hbar _1^i+\hbar _3^i)\cos \bar{\varpi }\tau _1+(\hbar _1^r-\hbar _3^r)\sin \bar{\varpi }\tau _1=-\hbar _2^i, \end{aligned} \right. \end{aligned}$$

(30)

It is further labeled as

$$\begin{aligned}&\phi _1(\bar{\varpi })=-\hbar _2^r(\hbar _1^r-\hbar _3^r)+\hbar _2^i(\hbar _3^i-\hbar _1^i),\\&\phi _2(\bar{\varpi })=-\hbar _2^i(\hbar _1^r+\hbar _3^r)+\hbar _2^r(\hbar _1^i+\hbar _3^i),\\&\phi _3(\bar{\varpi })=(\hbar _1^r)^2+(\hbar _1^i)^2-(\hbar _3^r)^2-(\hbar _3^i)^2.\\ \end{aligned}$$

As far as Eq. (30), it concludes that

$$\begin{aligned} \left\{ \begin{aligned}&\cos \bar{\varpi }\tau _1=\frac{\phi _1(\bar{\varpi })}{\phi _3(\bar{\varpi })},\\&\sin \bar{\varpi }\tau _1=\frac{\phi _2(\bar{\varpi })}{\phi _3(\bar{\varpi })}. \end{aligned} \right. \end{aligned}$$

(31)

It procures from Eq. (31) that

$$\begin{aligned} \phi ^2_3(\bar{\varpi })=\phi _1^2(\bar{\varpi })+\phi _2^2(\bar{\varpi }). \end{aligned}$$

(32)

In terms of Eq. (32), it is clear that

$$\begin{aligned} \varrho (w)=\phi _3^2(\bar{\varpi })-\phi _1^2(\bar{\varpi })-\phi _2^2(\bar{\varpi })=0. \end{aligned}$$

(33)

The following assumption is addressed.

$(\mathbf{H6} )$ Eq. (33) has at least positive roots.

Based on Eq. (33), the values of $\bar{\varpi }$ can be obtained using numerical software Maple 13, then the bifurcation point $\tau _{10}^*$ of FONN (2) with $\tau _2=0$ can be derived.

To throw out the bifurcation conditions, the following assumption is addressed

$(\mathbf{H7} )$$\frac{\varPi _1\varTheta _1+\varPi _2\varTheta _2}{\varTheta _1^2+\varTheta _2^2}\ne 0$,

where $\varPi _1$, $\varPi _2$, $\varTheta _1$, $\varTheta _2$ are described by Eq. (36).

Lemma 3

Let $s(\tau _2)=\eta (\tau _2)+i\varpi (\tau _2)$ be the root of Eq. (21) near $\tau _2=(\tau _2)_j$ complying with $\eta ((\tau _2)_j)=0$, $\varpi ((\tau _2)_j)=\varpi _0$, then the following transversality condition holds

$$\begin{aligned} \mathrm {Re}\Big [\frac{ds}{d\tau _2}\Big ] \Big |_{(\varpi =\varpi _0,\tau _2=\tau _{20})}\ne 0. \end{aligned}$$

Proof

The real and imaginary parts of $J'_p(s)(p=1,2,3)$ can be labeled by $J_p^{'r}$, $J_p^{'i}$. Employing implicit function theorem to differentiate Eq. (21) concerning $\tau _2$, then

$$\begin{aligned}&J'_1(s)\frac{ds}{d\tau _2}+\Big [J'_2(s)\frac{ds}{d\tau _2}e^{-s\tau _2} +J_2(s)e^{-s\tau _2}\Big (-\tau _2\frac{ds}{d\tau _2}-s\Big )\Big ]\nonumber \\&\quad +\Big [J'_3(s)\frac{ds}{d\tau _2}e^{-2s\tau _2} +J_3(s)e^{-2s\tau _2}\Big (-2\tau _2\frac{ds}{d\tau _2}-2s\Big )\Big ]=0. \end{aligned}$$

(34)

Direct deduction from Eq. (34) yields

$$\begin{aligned} \frac{ds}{d\tau _2}=\frac{\varPi (s)}{\varTheta (s)}, \end{aligned}$$

(35)

where

$$\begin{aligned} \varPi (s)&=s[J_2(s)e^{-s\tau _2}+2J_3(s)e^{-2s\tau _2}],\\ \varTheta (s)&=J'_1(s)+[J'_2(s)-\tau _2 J_2(s)]e^{-s\tau _2}+[J'_3(s)-2\tau _2 J_3(s)]e^{-2s\tau _2}. \end{aligned}$$

It educes from Eq. (35) that

$$\begin{aligned} \begin{aligned} \mathrm {Re}\Big [\frac{ds}{d\tau _2}\Big ]\Big |_{(\varpi =\varpi _0,\tau =\tau _{20})} =\frac{\varPi _1\varTheta _1+\varPi _2\varTheta _2}{\varTheta _1^2+\varTheta _2^2}, \end{aligned} \end{aligned}$$

(36)

where

$$\begin{aligned} \varPi _1&=\varpi _0(J_2^r\sin \varpi _0\tau _{20}-J_2^i\cos \varpi _0\tau _{20} +2J_3^r\sin 2\varpi _0\tau _{20}-2J_3^i\cos 2\varpi _0\tau _{20}),\\ \varPi _2&=\varpi _0(J_2^r\cos \varpi _0\tau _{20}+J_2^i\sin \varpi _0\tau _{20} +2J_3^r\cos 2\varpi _0\tau _{20}+2J_3^i\sin 2\varpi _0\tau _{20}),\\ \varTheta _1&=J_1^{'r}+(J_2^{'r}-\tau _{20}J_2^r)\cos \varpi _0\tau _{20}+(J_2^{'i}-\tau _{20}J_2^i)\sin \varpi _0\tau _{20}\\&\quad +(J_3^{'r}-2\tau _{20}J_3^r)\cos 2\varpi _0\tau _{20}+(J_3^{'i}-2\tau _{20}J_3^i)\sin 2\varpi _0\tau _{20},\\ \varTheta _2&=J_1^{'i}+(J_2^{'i}-\tau _{20}J_2^i)\cos \varpi _0\tau _{20}-(J_2^{'r}-\tau _{20}J_2^r)\sin \varpi _0\tau _{20}\\&\quad +(J_3^{'i}-2\tau _{20}J_3^i)\cos 2\varpi _0\tau _{20}-(J_3^{'r}-2\tau _{20}J_3^r)\sin 2\varpi _0\tau _{20}. \end{aligned}$$

$(\mathbf{H7} )$ indicates that transversality condition hold. We accomplish the proof of Lemma 3.

It follows from the prevenient analysis that the next theorem is obtainable. $\square$

Theorem 2

Under $(\mathbf{H1} )$, $(\mathbf{H5} )$–$(\mathbf{H7} )$, the following results are available.

• (1) If $\tau _1\in [0,\tau _{10}^*)$, then the origin of FONN (2) is asymptotically stable when $\tau _2\in [0,\tau _{20})$.

• (2) If $\tau _1\in [0,\tau _{10}^*)$, then FONN (2) undergoes a Hopf bifurcation at the origin when $\tau _2\in [\tau _{20},+\infty )$.

Fig. 1

Time responses of FONN (37) with $\tau _2=0.18$, $\tau _1=1.2<\tau _{10}=1.4537$

Fig. 2

Phase diagrams of FONN (37) with $\tau _2=0.18$, $\tau _1=1.2<\tau _{10}=1.4537$

Fig. 3

Time responses of FONN (37) with $\tau _2=0.18$, $\tau _1=1.6>\tau _{10}=1.4537$

Fig. 4

Phase diagrams of FONN (37) with $\tau _2=0.18$, $\tau _1=1.6>\tau _{10}=1.4537$

5 Simulation Examples

In this section, numerical results exemplify the effectiveness and practicability of our theoretical achievements.

Fig. 5

Time responses of FONN (38) with $\tau _1=0.24$, $\tau _2=0.6<\tau _{20}=0.8459$

Fig. 6

Phase portraits of FONN (38) with $\tau _1=0.24$, $\tau _2=0.6<\tau _{20}=0.8459$

5.1 Example 1

Consider the following FONN model

$$\begin{aligned} \left\{ \begin{aligned}&D^{0.94}z_1(t)=-0.5z_1(t)-0.5\tanh (z_1(t))+0.4\tanh (z_4(t-\tau _2))+0.4\tanh (z_2(t-\tau _2)),\\&D^{0.96}z_2(t)=-1.5z_2(t)+0.5\tanh (z_2(t))-1.2\tanh (z_1(t-\tau _1))-1.2\tanh (z_3(t-\tau _1)),\\&D^{0.97}z_3(t)=-0.6z_3(t)-0.4\tanh (z_3(t))+0.6\tanh (z_2(t-\tau _2))+0.6\tanh (z_4(t-\tau _2)),\\&D^{0.99}z_4(t)=-1.8z_4(t)+0.8\tanh (z_4(t))-0.8\tanh (z_3(t-\tau _1))-0.8\tanh (z_1(t-\tau _1)). \end{aligned} \right. \end{aligned}$$

(37)

Based on computation, we can first determine that $\tau _{20}^*=1.6337$ in Theorem 1. Then the initial values are selected as $(z_1(0),z_2(0),z_3(0),z_4(0))=(0.02,0.03,0.05,0.04)$. If choosing $\tau _2=0.18\in (0,\tau _{20}^*)$, we further procure $w_0=0.9549$, $\tau _{10}=1.4537$. It simply authenticates that the conditions in Theorem 1 are met. Figures 1 and 2 simulate that locally asymptotical steadiness of the zero equilibrium point of FONN (37) when $\tau _1=1.2<\tau _{10}$, while Figs. 3 and 4 reflect that the instability of the zero equilibrium point FONN (37), Hopf bifurcation takes place when $\tau _1=1.6>\tau _{10}$.

5.2 Example 2

Investigate the following FONN model

$$\begin{aligned} \left\{ \begin{aligned}&D^{0.91}z_1(t)=-0.4z_1(t)-0.5\tanh (z_1(t))+0.4\tanh (z_4(t-\tau _2))+0.4\tanh (z_2(t-\tau _2)),\\&D^{0.93}z_2(t)=-0.4z_2(t)-0.5\tanh (z_2(t))-1.5\tanh (z_1(t-\tau _1))-1.5\tanh (z_3(t-\tau _1)),\\&D^{0.95}z_3(t)=-0.4z_3(t)-0.5\tanh (z_3(t))+0.5\tanh (z_2(t-\tau _2))+0.5\tanh (z_4(t-\tau _2)),\\&D^{0.98}z_4(t)=-0.4z_4(t)-0.5\tanh (z_4(t))-1.2\tanh (z_3(t-\tau _1))-1.2\tanh (z_1(t-\tau _1)). \end{aligned} \right. \end{aligned}$$

(38)

By calculation, it first derives that $\tau _{10}^*=1.0859$ in Theorem 2. Next the initial values are taken as $(z_1(0),z_2(0),z_3(0),z_4(0))=(0.02,0.03,0.02,0.03)$. If setting up $\tau _1=0.24\in (0,\tau _{10}^*)$, then we further procure that $\varpi _0=1.1798$, $\tau _{20}=0.8459$. It easily justifies that the conditions of in Theorem 2 hold. Figures 5 and 6 depict that the zero equilibrium point of FONN (38) is locally asymptotically steady when $\tau _2=0.6<\tau _{20}$, while Figs. 7 and 8 describe that the zero equilibrium point of FONN (38) is precarious, Hopf bifurcation occurs when $\tau _2=1>\tau _{20}$.

Fig. 7

Time responses of FONN (38) with $\tau _1=0.24$, $\tau _2=1>\tau _{20}=0.8459$

Fig. 8

Phase portraits of FONN (38) with $\tau _1=0.24$, $\tau _2=1>\tau _{20}=0.8459$

6 Conclusion

In this paper, one delay has been established and another has been delay selected as a bifurcation parameter, the critical values of bifurcation for a FONN with two different delays have been derived. It has been discovered that the bifurcation occurs when the selected delay passes through the critical value, and FONN turns unstable. To certificate the effectiveness of our analytic results, two simulation examples have been ultimately underpinned. There are some enchanting and essential issues that deserve further investigation in the future work, such as (i) how to analyze the problem of bifurcation of the NNs with different delays by using system parameter or fraction order as a bifurcation parameter. (ii) how to generalize the derived bifurcation results to more high-order FONNs with numerous delays.