Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks*

It is well known that neural networks have been applied in various areas such as imageprocessing, optimization, artificial intelligence, computer version, automatic control andso on [1, 17]. Thus the study on various dynamical behaviors of neural networks is aninteresting and important topic in today’s world. During the past decades, a great deal ofthe study achievements on various dynamics (including periodic solution, almost periodicsolution, Sp-almost periodic solution, pseudo almost periodic solution, piecewise pseudoalmost periodic solution, weight pseudo almost periodic solution, pseudo almost automorphicsolution, piecewise asymptotically almost automorphic solution, square-meanalmost autorphic solution, Stepanov-like weighted pseudo almost automorphic solution,weight pseudo almost automorphic solutions, anti-periodic solution, weighted pseudoanti-periodic solution, bifurcation, dissipativity, synchronization, global Mittag-Lefflerstability, fixed-time stabilization, etc.) have been reported. For example, Abdelaziz andChérif [1] discussed the piecewise asymptotic almost periodic solutions of fuzzy Cohen–Grossberg neural networks. Aouiti et al. [2] detailedly analyzed the piecewise pseudoalmost periodic solution to delayed neutral-type neural networks. Bohner et al. [5] investigatedthe almost periodic solutions of delayed Cohen–Grossberg neural networks. Huanget al. [12] studied the anti-periodic solutions for cellular neural networks with proportionaldelay. Zhao et al. [31] established the sufficient condition to ensure the existence, uniquenessand global exponential stability of weighted pseudo almost automorphic solutionsfor Hopfield neural networks with delays. Dhama1 and Abbas [7] made a systematicanalysis on the existence and stability of weighted pseudo almost automorphic solutionof dynamic equation. Zhou and Zhao [32] studied the synchronization issue of a class ofneural networks with proportional delays. For more detailed contents on these aspects, werefer the readers to [18 – 20].

However, the involved works above have been restricted to the integer-order delayeddifferential models. Nowadays many scholars find that fractional calculus has great application prospect in numerous fields such as electromagnetic waves, physics, viscoelasticity,biology, mechanics, neural networks, control science and so on [4, 21]. Lotsof researchers hold that fractional calculus can be regarded as a resultful tool to describethe actual problems of natural world since it possesses memory and hereditary propertiesduring the process of development and change of things [10, 14]. Recently, fractionalcalculus has attracted more and more attention from many scholars. Hopf bifurcationis a vital dynamical property of delayed differential systems. For a long time, a largenumber of research findings about Hopf bifurcation of integer-order delayed differentialmodels have been published. However, the study on Hopf bifurcation of fractional-orderdifferential systems is very rare. At present, there are some literatures that deal withthe Hopf bifurcation of fractional-order differential systems. For instance, Xu et al. [26] considered the effect of multiple time delays on bifurcation for fractional-order neuralnetworks. Huang et al. [16] investigated the stability and Hopf bifurcation for fractionalneural networks. Xiao et al. [24] dealt with the PD control technique of Hopf bifurcationsfor delayed fractional-order small-world networks. In details, one can see [8, 11, 22, 23,25, 27 – 30].

Here we would like to point out that few works are concerned with the comparison ofHopf bifurcation for integer-order and fractional-order cases. Time delay plays a vital rolein stabilizing system and changing the dynamical behavior of integer-order and fractionalordersystems. Fractional-order systems have one more parameter (fractional-order), thusthe impact of time delay on stability and Hopf bifurcation for integer-order and fractionalordersystems will display different style. How does the effect of time delay on thestability and Hopf bifurcation of integer-order and fractional-order differential systems?Motivated by this idea, in this work, we will focus on the comparative study on bifurcationbehavior for integer-order and fractional-order delayed BAM neural networks. In particular,in this work, we will handle the following problems (e.g., the main contribution ofthis paper): (i) reveal the effect of time delay on stability and Hopf bifurcation of integerorderand fractional-order delayed BAM neural networks; (ii) comparison between thebifurcation point of integer-order delayed BAM neural networks and the bifurcation pointof fractional-order delayed BAM neural networks is given.

In this article, we consider the following integer-order BAM neural networks with delay:

and the corresponding fractional-order BAM neural networks with delay:

where ${\gamma }_i(i=1,2,3,4)$ stands for the internal decay rate, $h_i(i=1,2,3,4)$ stands for thenonlinear feedback function, $k_i(i=1,2,3,4)$ stands for the connection function betweentwo neurons, $\eta \ge 0$ stands for the connection delay, $\textcolor{red}{}\sigma \in (0,1]$ is a constant. For moreconcrete information about system (1), one can see [9].

In order to establish the key conclusions of this paper, we firstly make the following hypothesis:

The article is structured as follows. Section 2 presents some elementary knowledgeon fractional calculus and integer-order differential equations. Section 3 is concernedwith the discussion on the stability and the emergence of Hopf bifurcation of the integerorderdelayed BAM neural networks (1). Section 4 deals with the analysis on the stabilityand the emergence of Hopf bifurcation of the fractional-order delayed BAM neural networks (2). Section 5 carries out computer simulations and bifurcation diagrams to supportthe rationality of the derived key conclusions. Section 6 finishes our article.

In the part, we list some necessary knowledge on fractional-order dynamical system andinteger-order dynamical systems, which will be applied in the following proof.

Lemma 1. (See [16].) Consider the following exponential polynomial

where ${\eta }_j\ge 0(j=0,1,2,...,m)$ and $q_k^{\left(j\right)}(j=0,1,2,...,m;k=1,2,...,n)$ are constants. If $\textcolor{red}{}({\eta }_1{\eta }_2,...,m)$ change, the sum of the orders of $\textcolor{red}{}Q(\lambda ,e^{-\lambda n_1},...,e^{\lambda {\eta }_m})$ on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

Definition 1. (See [15].) Define Caputo fractional-order derivative as follows:

where $\textcolor{red}{}l\left(\upsilon \right)\in \left(\right[{\upsilon }_{0,}\infty ),ℝ),T\left(s\right)={\int }_0^{\infty }{\upsilon }^{s-1}{e}^{-\upsilon }d\upsilon ,\upsilon \ge {\upsilon }_0$, and$\textcolor{red}{}n\in {\mathbb{Z}}^{+},n-1\le \sigma <n.$.

The Laplace transform of Caputo fractional-order derivative is given by

where $G\left(s\right)=L\left\{g\right(t\left)\right\}$. Especially, if $g^{\left(j\right)}\left(0\right),j=1,2,...,m$, then $L\left\{D^{\sigma }g\right(t);s\}=s^{\sigma }G\left(s\right)$.

Definition 2. (See [3].) $({\omega }_{1*},{\omega }_{2*},{\omega }_{3*},{\omega }_{4*})$ is called an equilibrium point of system (1) (or system (2)), provided that the equations

are fulfilled.

Lemma 2. (See [13].) Suppose given the fractional-order system

where $\sigma \in (0,1]$ and $g(t,\omega (t\left)\right):{ℝ}^{+}\times {ℝ}^m\to {ℝ}^m$ . The equilibrium point of system (3) are locally asymptotically stable if all eigenvalues $\textcolor{red}{}\delta$ of the Jacobian matrix $\partial g(t,\omega )/\partial \omega$ evaluated near the equilibrium point satisfy $\left|arg\right(\delta \left)\right|>\delta \mathrm{\pi }/2$ .

Lemma 3. (See [27].) Consider the following fractional-order system:

where $0<{\sigma }_i<1(i=1,2,...,m)$ , the initial values $W_i\left(t\right)={\mathrm{\phi }}_i\left(t\right)\in C[-ma{x}_{\mathrm{i,l}},{\text{ϱ}}_{il},0]$ , $t\in [-ma{x}_{i,l},{\text{ϱ}}_{il},0],i,l=1,2,...,m$ . Denote

Then the zero solution of Eq. (4) is Lyapunov asymptotically stable provided that everyroot of $det(\Delta (\zeta \left)\right)=0$ possesses negative real parts.

[Hopf bifurcation exploration of system (1)] In this part, we are to explore the stabilityand the onset of Hopf bifurcation for system (1). In term of $\left(Q_1\right)$, one knows that Eq.(1) has a unique equilibrium $E(0,0,0,0)$. The linear system of Eq. (1) near $E(0,0,0,0)$ takes the form

where ${\alpha }_j={\gamma }_j-{h\text{'}}_j\left(0\right)(j=1,2,3,4)$.Then the associated characteristic equation of (5) takes the form

It follows from (6) that

Here $A_1\left(s\right)={\lambda }^4+a_3{\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0$,$A_2\left(s\right)=b_2{\lambda }^2+b_1\lambda +b_0$ where

Now we will discuss the distribution of the roots of Eq. (7). Assume that i is the root of Eq. (7), then one has

It follows from (9) that

By (10) one gets

which leads to

where

Let $y={\mathrm{\phi }}^2$, then Eq. (11) can be rewritten as

Let

Then

Denote

and let $x=y+r_3/4$, then Eq. (14) takes the form

where

Define

According to [6], we get the following conclusions.

Lemma 4. Equation (13) possesses at least one positive root if $r_0<0$ , where $r_0$ isdefined by (12).

Lemma 5. Let the condition $r_0\ge 0$ holds.

1. (i) I f$C\ge 0$, then Eq. (14) possesses positive roots, $\iff y_1>0$ and $f\left(y_1\right)<0$.

2. (ii) If $C<0$, then Eq. (14) possesses positive roots , there exists at least one $y_0∊\{y_1,y_2,y_3\}$ such that $y_{0>0}$ and $f\left(y_0\right)\le 0$.

Assume that Eq. (13) possesses positive roots. Here we suppose that Eq. (13) possesses four positive roots, say $y_j^{*}(j=1,2,3,4)$. Then Eq. (11) possesses thefollowing four positive roots:

By (10) one has

where $j=1,2,3,4;l=0,1,2,...$. Define

Now we make the following assumption:

Lemma 6. Assume that condition $\left(Q_2\right)$ holds.

1. (i) Let one of the three conditions is satisfied: (a) $r_0<0;$ (b) $r_0\ge 0,C\ge 0,y_1>0$ and$\textcolor{red}{}f\left(y_1\right)\le 0;$ (c) $r_0\ge 0,C<0$, and let there exists $y^{*}∊\{y_1,y_2,y_3\}$ such that$y^{*}>0$ and$f\left(y^{*}\right)\le 0$. Then all roots of Eq. (7) possesses negative real parts for$\eta \in [0,{\eta }_0)$.

2. (ii) If conditions (a)–(c) of (i) do not hold, then all roots of Eq. (7) possesses negativereal parts for$\eta \ge 0$.

The conclusions of Lemma 6 come from Lemmas 4, 5 and 1. The proof of Lemma 6 is similar to the proof of Lemma 2.4 in Hu and Huang [9]. Here we omit it.

Next, we check the transversality condition for the onset of Hopf bifurcation. Thefollowing assumption is made as follows:

Lemma 7. Assume that $s\left(\eta \right)=\alpha \left(\eta \right)+i\beta \left(\eta \right)$ is the root of (7) at $\eta ={\eta }_0$ and $\textcolor{red}{}\alpha \left({\eta }_0\right)=0,\beta \left({\eta }_0\right)={\mathrm{\phi }}_0$ , then $Redsd\eta |\eta ={\eta }_{0,\mathrm{\phi }={\mathrm{\phi }}_0}\ne 0.$ .

Proof. According to (7), one gets

Then

where

By $\left(Q_3\right)$ one has

Based on Lemmas 6 and 7, the following result can be established.

Theorem 1. For system (1), assume that conditions $\left(Q_1\right)$ and $\left(Q_2\right)$ hold. Then the followingconclusions hold true:

1. (i) Let one of the three conditions: (a)$r_0<0$; (b)$r_0\ge 0,C\ge 0,y_1>0$ and $f\left(y_1\right)\le 0;$ (c)$r_0\ge 0,C<0$ and let there exists $y^{*}\in \{y_1,y_2,y_3\}$ such that $y^{*}>0$ and$\left(y^{*}\right)\le 0$ do not hold. Then the zero equilibrium point$E(0,0,0,0)$is asymptotically stable for all $\eta \ge 0$;

2. (ii) If one of the conditions (a), (b) and (c) is fulfilled, then the zero equilibrium point $E(0,0,0,0)$ is asymptotically stable for$\eta \in [0,{\eta }_0).$

3. (iii) If conditions $\left(Q_3\right)$ and (ii) hold, then a Hopf bifurcation will happen around thezero equilibrium point $E(0,0,0,0)$.

[Hopf bifurcation exploration of system (2)] In this section, we are to analyze the stabilityand the existence of Hopf bifurcation of model (2). In term of $\left(Q_1\right)$, it is easy to seethat Eq. (2) has a unique equilibrium $E(0,0,0,0)$. The linear system of Eq. (2) around $E(0,0,0,0)$ is given by

where ${\alpha }_j={\gamma }_j-{h\text{'}}_j\left(0\right)(j=1,2,3,4)$. The associated characteristic equation of (15) takes the form

It follows from (16) that

Here

where $a_i(i=0,1,2,3)$ and are defined by (8).

Suppose that $s=\mathrm{i\varrho }=\varrho \left(\cos (\mathrm{\pi }/2)+isin(\mathrm{\pi }/2)\right)$ is a root of (17) and $B_{iR}\left(s\right)$ and $B_{il}\left(s\right)(i=1,2)$ stand for the real parts and imaginary parts of$B_i\left(s\right)(i=1,2)$,respectively. Then we have

where

By (18) one gets

Let

Then (19) can be rewritten as

In view of (20) and (21), one gets

where

and

By (23) it follows from (22) that

where

Set

and

Lemma 8.

1. (i) In addition to the condition $a_0+b_0\ne 0,if{\text{ϑ}}_l>0(l=1,2,...,13)$, then Eq. (17) possesses no the root with zero real parts.

2. (ii) If ${\text{ϑ}}_{13}>0$ and there exists ${\xi }_0>0$, which satisfies $N\left({\xi }_0\right)<0$, then Eq. (17)possesses at least two pairs of purely imaginary roots.

Proof. (i) In term of (25), one has

By ${\text{ϑ}}_l>0(l=1,2,...,12)$ one gets $dM(\varrho )/\mathrm{d\varrho }>0\text{ϱ}>0$. Noting that $M\left(0\right)={\text{ϑ}}_{13}>0$, one obtains that Eq. (24) has no positive real root. In addition, by $a_0+b_0\ne 0$ one knows that $s=0$ is not the root of (17). This completes the proof of (i).

(ii) By $N\left(0\right)={\text{ϑ}}_{13}>0,N\left({\xi }_0\right)<0({\xi }_0>0)$ and $\underset{∊\to 0+\infty }{\lim }N\left(\epsilon \right)/d\epsilon =+\infty$ one can find${\xi }_1\in (0,{\xi }_0)and{\xi }_2\in ({\xi }_{0,}+\infty )$, which satisfy $N\left({\xi }_1\right)=N\left({\xi }_2\right)=0$. Then possesses at least two positive real roots. Therefore possesses at least twopairs of purely imaginary roots. The proof of (ii) is finished.

Assume that Eq. (24) has twelve positive real roots ${\text{ϱ}}_{j(j=1,2,...,12)}$. It followsfrom (20) that

where $l=0,1,2,...,j=1,2,...,12$. Denote

In the sequel, we make the following hypothesis:

$\left(Q_4\right)W_{11}{W}_{21}+W_{12}{W}_{22}>0$ ,where

Lemma 9. Suppose that $s\left(\eta \right)=\rho \left(\eta \right)+ik\left(\eta \right)$ is the root of Eq. (17) at $\eta ={\eta }_0$ , whichsatisfies $\rho \left({\eta }_0\right)=0,k\left({\eta }_0\right)={\text{ϱ}}_0$ . Then one has $Re[ds/d\eta ]{|}_{\eta ={\eta }_{0,}\text{ϱ}={\text{ϱ}}_0}>0$ .

Proof. It follows from Eq. (17) that

Then

where

Thus

In view of $\left(Q_4\right)$, we have

The proof of Lemma 9 finishes.

Lemma 10. If $\eta =0$ and $\left(Q_2\right)$ holds true, then system (2) is locally asymptotically stable.

Proof. When $\eta =0$ , then (17) takes the form

By $\left(Q_2\right)$ one knows that all roots ${\lambda }_i$ of (26) satisfy $\left|arg\right({\lambda }_i\left)\right|>\mathrm{\sigma }\mathrm{\pi }/2(i=1,2,3,4)$. Sowe know that Lemma 10 is correct. The proof ends.

Based on the investigation above, we have the following conclusion.

Theorem 2. Let hypotheses $\left(Q_1\right)-\left(Q_4\right)$ hold. Then the zero equilibrium point $E(0,0,0,0)$ of system (2) is locally asymptotically stable if $\textcolor{red}{}\eta$ fall into the interval $[0,{\eta }_0)$ and a Hopfbifurcation will take place in the vicinity of $E(0,0,0,0)$ when $\eta ={\eta }_0$ .