1 Introduction

The inertial neural networks (INNs) were originally proposed by Babcock and Westervelt [1] in 1986 through adding inductors to the traditional neural circuit model. They found that the introduction of inertial term may lead to more complicated and unexpected behaviours, including chaos, oscillation and instability [2]. Furthermore, it should be emphasized that there are specific biological backgrounds for combining the inertial term with neural systems, see references [3,4,5] and cited therein. Attracted by these factors, more and more scholars are dedicated to the research on dynamic characteristics of INNs and considerable results have been acquired, such as synchronization, dissipativity, Lagrange stability, Hopf bifurcation and so on [6,7,8,9,10]. It is worth noting that early studies on INNs mainly adopt the reduced order approach, i.e. preprocessing the original systems via imposing adequate variable transformations. This will inevitably increase the dimensions of original systems, the diffculty of analyis as well as the complexity of derived criteria. In view of this deficiency, some researchers try to apply the non-reduced order approach to INNs and have obtained satisfactory conclusion. For example, several new results on stability and synchronization of delayed INNs were firstly presented in [11]; Duan and Li [12] investigated the fixed-time synchronization of fuzzy neutral-type BAM memristive INNs with proportional delays; Tu et al. [13] further generalized the approach to quaternion domain and gained several criteria such that the state estimation of IQVNNs could be realized with desired \(H_\infty \) performance.

Along with the accelerated development of digital computers, high dimensional neural network has attracted extensive attention in recent years considering its tremendous computing ability. In fact, there are some scenarios that cannot be handled well by one-dimensional data, such as high dimensional affine transformations, in this case, quaternion is a viable option. It is noteworthy that quaternions are much more intricate than real and complex numbers, because the commutative law is no longer applicable for quaternions. Consequently, quaternion-valued neural networks (QVNNs) have been explored extensively in [14,15,16,17,18,19,20,21,22,23,24]. Song et al. [14] investigated the stability problem for quaternion-valued system with stochastic terms. Considering that Markovian parameters have a significant effect on the dynamic performance of a system, the quaternion-valued model with Markovian jump was discussed in [15]. Liu and Lin [18] developed an event-triggered impulsive control scheme for the synchronization control problem. The dynamics of memristive quaternion-valued system have been developed in [20,21,22]. Early studies on the dynamics of quaternion memristive system mainly adopt the decomposing method, which may undermine the integrality of the system.

Recently, the synchronization issue has been a research hotspot since it is one of the most significant dynamic characteristics of neural networks. As is well known, there exist various kinds of synchronization, such as complete synchronization, anti-synchronization, lag synchronization and projective synchronization [25,26,27,28,29,30,31,32,33]. Among them, projective synchronization occupies an important position since it comprises several kinds of synchronization as special cases. It means that the slave system could be synchronized to the master system by a proportional factor. Due to the successful applications of projective synchronization in image processing and secure communication [28, 29], it has widely aroused scholars’ concern and plentiful results have been reported over the past few years. To name a few, Chen and Cao [30] investigated projective synchronization issue of delayed neural networks with parameter mismatch; Based on sliding mode control technique, the projective synchronization was achieved for a class of chaotic neural networks with mixed time delays [31]; Kumal et al. [32] further extended this framework to an impulsive neural system; Liu et al. [33] developed effective criteria on projective synchronization for fractional neural networks with discrete and distributed delays. It follows that the outcome of projective synchronization for IQVNNs is quite rare, which prompts us to enrich and improve the aforementioned content.

Motivated by the above discussions, this article concentrates on exploring the projective synchronization problem of delayed IQVNNs. The main highlights of this article are summarized as follows: (1) For the first time, we intend to study the projective synchronization issue of delayed IQVNNs; (2) Different from [6,7,8,9,10], the non-reduced order approach is employed to smoothly investigate the systems with inertial term, which could simplify the analysis processure and be more natural; (3) A novel quaternion-based Lyapunov functional is constructed to deal with quaternion-valued systems directly, which preserves the integrality of original systems.

The remainder of this article is structured as follows. In Sect. 2, we provide necessary preliminaries as well as inertial quaternion-valued master–slave systems. An appropriate controller is also designed in this part. In Sect. 3, several novel criteria are derived such that the aforementioned systems could achieve projective synchronization. In Sect. 4, numerical experiments are conducted to validate the effectiveness of main theorems. Finally, conclusions are drawn in Sect. 5.

Notations: Let \(\mathbb {R} \) and \(\mathbb {H}\) separately denote the sets of real numbers and quaternions. \(\mathbb {R}_+\) represents the set of positive real numbers. For a quaternion \(h=h^{(R)}+ h^{(I)}i+ h^{(J)}j + h^{(K)}k\in \mathbb {H} \), where \(h^{(R)}, h^{(I)}, h^{(J)}, h^{(K)}\in \mathbb {R}\), i, j, k stand for imaginary units whose operations satisfy Hamilton rules, i.e. \(i^2=j^2=k^2=ijk=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j\). The conjugate of h is defined by \(\bar{h}=h^{(R)}- h^{(I)}i- h^{(J)}j - h^{(K)}k \). Besides, the modulus of h is denoted by \(|h|=(h\bar{h})^{\frac{1}{2}}=\sqrt{\left( h^{(R)}\right) ^2+\left( h^{(I)}\right) ^2+\left( h^{(J)}\right) ^2+\left( h^{(K)}\right) ^2}\). \(\Lambda \) stands for the set \(\{1,2,\cdots ,n\}\).

2 Preliminaries

In this article, the master system is described by the following IQVNNs model

$$\begin{aligned} \frac{d^2x_p(\iota )}{d\iota ^2}&=-a_p\frac{dx_p(\iota )}{d\iota }-b_px_p(\iota )+\sum _{q=1}^{n}c_{pq}g_q(x_q(\iota ))+\sum _{q=1}^{n}d_{pq} \nonumber \\&\quad \times g_q(x_q(\iota -\tau _q))+I_p, \quad p\in \Lambda , \end{aligned}$$
(1)

where \(x_p(\iota )\in \mathbb {H}\) denotes the state of the pth neuron at time \(\iota \); \(\frac{d^2x_p(\iota )}{d\iota ^2}\) represents the inertial term; \(a_p,b_p \in \mathbb {R}_+\) are constants; \(c_{pq}, d_{pq}\in \mathbb {H}\) are connection weight parameters; \(I_p\in \mathbb {H}\) is an external input; \(g_q(\cdot )\) denotes the neuron activation function; \(\tau _q\) signifies the transmission delay.

The initial conditions of (1) are presented by \(x_p(s)=\phi _p(s)\), \(\dot{x}_p(s)=\psi _p(s)\), \(-\tau \le s\le 0\) with \(\tau =\max \limits _{q\in \Lambda } \{\tau _q\}\). \(\phi _p(s)\) and \(\psi _p(s)\) are both bounded and continuous.

Correspondingly, the slave system for model (1) is constructed as

$$\begin{aligned} \frac{d^2y_p(\iota )}{d\iota ^2}&=-a_p\frac{dy_p(\iota )}{d\iota }-b_py_p(\iota )+\sum _{q=1}^{n}c_{pq}g_q(y_q(\iota ))+\sum _{q=1}^{n}d_{pq} \nonumber \\&\quad \times g_q(y_q(\iota -\tau _q))+I_p+u_p(\iota ), \quad p\in \Lambda , \end{aligned}$$
(2)

where \(u_p(\iota )\) represents the suitable controller that will be devised later.

Throughout this article, it is assumed that the activation function \(g_q(\cdot )\) possesses the following proposition.

Assumption 2.1

For any \(q\in \Lambda \), \(r,s\in \mathbb {H}\), \(r \ne s\), there exists \(l_q\in \mathbb {R}_+\), such that

$$\begin{aligned} |g_q(r)-g_q(s)|\le l_q|r-s|. \end{aligned}$$

Definition 2.1

For any two trajectories \(x_p(\iota )\) and \(y_p(\iota )\) of master–slave systems (1) and (2), if there exists a constant \(\alpha \in \mathbb {R}\) such that \(\lim \limits _{\iota \rightarrow +\infty } |y_p(\iota )-\alpha x_p(\iota )|=0\), then the projective synchronization is called to be realized between systems (1) and (2) with proportional factor \(\alpha \).

Let \(e_p(\iota )=y_p(\iota )-\alpha x_p(\iota )\) denotes the projective synchronization error, then \(\dot{e}_p(\iota )=\dot{y}_p(\iota )-\alpha \dot{x}_p(\iota )\), \(\ddot{e}_p(\iota )=\ddot{y}_p(\iota )-\alpha \ddot{x}_p(\iota )\).

Consequently, the error system between (1) and (2) can be derived as follows

$$\begin{aligned} \frac{d^2e_p(\iota )}{d\iota ^2}&=-a_p\frac{dy_p(\iota )}{d\iota }-b_py_p(\iota )+\sum _{q=1}^{n}c_{pq}g_q(y_q(\iota ))+\sum _{q=1}^{n}d_{pq} \nonumber \\&\quad \times g_q(y_q(\iota -\tau _q))+I_p+u_p(\iota )-\alpha \Big (-a_p\frac{dx_p(\iota )}{d\iota }-b_px_p(\iota ) \nonumber \\&\quad +\sum _{q=1}^{n}c_{pq}g_q(x_q(\iota ))+\sum _{q=1}^{n}d_{pq}g_q(x_q(\iota -\tau _q))+I_p\Big ) \nonumber \\&=-a_p\big [\dot{y}_p(\iota )-\alpha \dot{x}_p(\iota )\big ]-b_p\big [y_p(\iota )-\alpha x_p(\iota )\big ]+\sum _{q=1}^{n}c_{pq} \nonumber \\&\quad \times \big [g_q(y_q(\iota ))-\alpha g_q(x_q(\iota ))\big ]+\sum _{q=1}^{n}d_{pq}\big [g_q(y_q(\iota -\tau _q)) \nonumber \\&\quad -\alpha g_q(x_q(\iota -\tau _q))\big ]+(1-\alpha )I_p+u_p(\iota ). \end{aligned}$$
(3)

An appropriate controller is introduced as

$$\begin{aligned} u_p(\iota )&=-\delta _p\big [\dot{y}_p(\iota )-\alpha \dot{x}_p(\iota )\big ]-k_p\big [y_p(\iota )-\alpha x_p(\iota )\big ]-(1-\alpha )I_p \nonumber \\&\quad -\sum _{q=1}^{n}c_{pq}\big [g_q(\alpha x_q(\iota ))-\alpha g_q(x_q(\iota ))\big ]-\sum _{q=1}^{n}d_{pq} \nonumber \\&\quad \times \big [g_q(\alpha x_q(\iota -\tau _q))-\alpha g_q(x_q(\iota -\tau _q))\big ], \quad p\in \Lambda , \end{aligned}$$
(4)

where \(\delta _p, k_p\in \mathbb {R}_+\) denote the control gains.

Remark 2.1

The design idea of controller (4) follows the similar line as that in [33]. The first two terms in the controller (4) stand for linear negative feedbacks, while the remaining terms could ensure \(e_p(\iota )=0\in \mathbb {H}\) is a trajectory of error system (3). The difference lies in that the controller in this article contains not only the term \(y_p(\iota )-\alpha x_p(\iota )\), but also its derivative \(\dot{y}_p(\iota )-\alpha \dot{x}_p(\iota )\). With this rational controller, the master–slave systems (1) and (2) could reach projective synchronization.

Therefore, the error system can be reduced as

$$\begin{aligned} \frac{d^2e_p(\iota )}{d\iota ^2}&=-\left( a_p+\delta _p\right) \dot{e}_p(\iota )-\left( b_p+k_p\right) e_p(\iota )+\sum _{q=1}^{n}c_{pq}f_q\left( e_q(\iota )\right) \nonumber \\&\quad +\sum _{q=1}^{n}d_{pq}f_q\left( e_q(\iota -\tau _q)\right) , \quad p\in \Lambda , \end{aligned}$$
(5)

where \(f_q\left( e_q(\iota )\right) =g_q\left( y_q(\iota )\right) -g_q\left( \alpha x_q(\iota )\right) \), \(f_q\left( e_q(\iota -\tau _q)\right) =g_q\left( y_q(\iota -\tau _q)\right) -g_q\left( \alpha x_q(\iota -\tau _q)\right) \).

In order to obtain the main results, several lemmas are presented as follows.

Lemma 2.1

[13] For \(u,v\in \mathbb {H}\), the following propositions can be obtained: (1) \(\overline{u+v}=\bar{u}+\bar{v}\); (2) \(\overline{uv}=\bar{v}\bar{u}\).

Lemma 2.2

[34] For \(u,v\in \mathbb {H}\), \(\mu \in \mathbb {R}_+\), the following inequality always holds: \(uv+\bar{v}\bar{u}\le \mu \bar{v}v+\frac{1}{\mu }u\bar{u}\).

Lemma 2.3

[35] Assuming that a function f(x) defined in \([0,+\infty )\) satisfies the following two conditions:

  1. (1)

    it is uniformly continuous;

  2. (2)

    \(\int _{0}^{+\infty }f(t)dt\) exists and is bounded. Then, one has \(\lim \limits _{x \rightarrow +\infty }f(x)=0\).

3 Main Results

In this section, we shall firstly establish several criteria such that the projective synchronization between master–slave systems could be realized. Whereafter, two corollaries are drawn directly by choosing specific proportional factor values.

Theorem 3.1

Under Assumption 1, the slave system (2) can be asymtotically projective synchronized to the master system (1) with controller (4) if there exist scalars \(\alpha _p\), \(\beta _p\) and \(\gamma _p\) such that

$$\begin{aligned} A_p<0, \quad A_pB_p>C_p^2, \quad p\in \Lambda , \end{aligned}$$
(6)

where

$$\begin{aligned} A_p&=2\alpha _p\gamma _p-2\left( a_p+\delta _p\right) \alpha _p^2+\sum _{q=1}^{n}\left( c_{pq}\bar{c}_{pq}+d_{pq}\bar{d}_{pq}\right) \alpha _p^2, \\ B_p&=-2\left( b_p+k_p\right) \alpha _p\gamma _p+\sum _{q=1}^{n}\left[ \left( \alpha _q^2+|\alpha _q\gamma _q|\right) l_p^2+\left( c_{pq}\bar{c}_{pq}\right. \right. \\&\quad \left. +\,d_{pq}\bar{d}_{pq}\right) |\alpha _p\gamma _p|\big ]+\sum _{q=1}^{n}\big (\alpha _q^2+|\alpha _q\gamma _q|\big )l_p^2, \\ C_p&=\beta _p+\gamma _p^2-\left( a_p+\delta _p\right) \alpha _p\gamma _p-\left( b_p+k_p\right) \alpha _p^2. \end{aligned}$$

Proof

Consider the following Lyapunov functional candidate

$$\begin{aligned} V(\iota )&=\sum _{p=1}^{n}\beta _p\bar{e}_p(\iota )e_p(\iota )+\sum _{p=1}^{n}\overline{\big (\alpha _p\dot{e}_p(\iota )+\gamma _pe_p(\iota )\big )}\big (\alpha _p\dot{e}_p(\iota )+\gamma _pe_p(\iota )\big ) \nonumber \\&\quad +\sum _{p=1}^{n}\sum _{q=1}^{n}\big (\alpha _q^2+|\alpha _q\gamma _q|\big )\int _{\iota -\tau _p}^{\iota }\bar{f}_p\left( e_p(s)\right) f_p\left( e_p(s)\right) ds. \end{aligned}$$
(7)

Calculating the derivative of \(V(\iota )\) along the trajectory of error system (5) leads to

$$\begin{aligned} \dot{V}(\iota )&=\sum _{p=1}^{n}\beta _p\big (\dot{\bar{e}}_p(\iota )e_p(\iota )+\bar{e}_p(\iota )\dot{e}_p(\iota )\big )+\sum _{p=1}^{n}\big (\alpha _p\ddot{\bar{e}}_p(\iota )+\gamma _p\dot{\bar{e}}_p(\iota )\big ) \nonumber \\&\quad \times \big (\alpha _p\dot{e}_p(\iota )+\gamma _pe_p(\iota )\big )+\sum _{p=1}^{n}\big (\alpha _p\dot{\bar{e}}_p(\iota )+\gamma _p\bar{e}_p(\iota )\big ) \nonumber \\&\quad \times \big (\alpha _p\ddot{e}_p(\iota )+\gamma _p\dot{e}_p(\iota )\big )+\sum _{p=1}^{n}\sum _{q=1}^{n}\left( \alpha _q^2+|\alpha _q\gamma _q|\right) \nonumber \\&\quad \times \big (\bar{f}_p(e_p(\iota ))f_p(e_p(\iota ))-\bar{f}_p(e_p(\iota -\tau _p))f_p(e_p(\iota -\tau _p))\big ) \nonumber \\&=\sum _{p=1}^{n}\left( \beta _p+\gamma _p^2\right) \big (\dot{\bar{e}}_p(\iota )e_p(\iota )+\bar{e}_p(\iota )\dot{e}_p(\iota )\big )+2\sum _{p=1}^{n}\alpha _p\gamma _p\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota ) \nonumber \\&\quad +\sum _{p=1}^{n}\alpha _p\ddot{\bar{e}}_p(\iota )\big (\alpha _p\dot{e}_p(\iota )+\gamma _pe_p(\iota )\big )+\sum _{p=1}^{n}\alpha _p\big (\alpha _p\dot{\bar{e}}_p(\iota )+\gamma _p\bar{e}_p(\iota )\big ) \nonumber \\&\quad \times \ddot{e}_p(\iota )+\sum _{p=1}^{n}\sum _{q=1}^{n}\left( \alpha _q^2+|\alpha _q\gamma _q|\right) \big (\bar{f}_p(e_p(\iota ))f_p(e_p(\iota )) \nonumber \\&\quad -\bar{f}_p(e_p(\iota -\tau _p))f_p(e_p(\iota -\tau _p))\big ) \nonumber \\&=\sum _{p=1}^{n}\left( \beta _p+\gamma _p^2\right) \big (\dot{\bar{e}}_p(\iota )e_p(\iota )+\bar{e}_p(\iota )\dot{e}_p(\iota )\big )+2\sum _{p=1}^{n}\alpha _p\gamma _p\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota ) \nonumber \\&\quad +\sum _{p=1}^{n}\alpha _p\Big (-(a_p+\delta _p)\dot{\bar{e}}_p(\iota )-(b_p+k_p)\bar{e}_p(\iota )+\sum _{q=1}^{n}\bar{f}_q(e_q(\iota ))\bar{c}_{pq} \nonumber \\&\quad +\sum _{q=1}^{n}\bar{f}_q(e_q(\iota -\tau _q))\bar{d}_{pq}\Big )\big (\alpha _p\dot{e}_p(\iota )+\gamma _pe_p(\iota )\big ) \nonumber \\&\quad +\sum _{p=1}^{n}\alpha _p\big (\alpha _p\dot{\bar{e}}_p(\iota )+\gamma _p\bar{e}_p(\iota )\big )\Big (-(a_p+\delta _p)\dot{e}_p(\iota )-(b_p+k_p)e_p(\iota ) \nonumber \\&\quad +\sum _{q=1}^{n}c_{pq}f_q(e_q(\iota ))+\sum _{q=1}^{n}d_{pq}f_q(e_q(\iota -\tau _q))\Big ) \nonumber \\&\quad +\sum _{p=1}^{n}\sum _{q=1}^{n}\left( \alpha _q^2+|\alpha _q\gamma _q|\right) \big (\bar{f}_p(e_p(\iota ))f_p(e_p(\iota ))-\bar{f}_p(e_p(\iota -\tau _p)) \nonumber \\&\quad \times f_p(e_p(\iota -\tau _p))\big ). \end{aligned}$$
(8)

Further expand the expression (8), one arrives at that

$$\begin{aligned} \dot{V}(\iota )\le&\sum _{p=1}^{n}\Big (\beta _p+\gamma _p^2-(a_p+\delta _p)\alpha _p\gamma _p-(b_p+k_p)\alpha _p^2\Big )\big (\dot{\bar{e}}_p(\iota )e_p(\iota ) \nonumber \\&\quad +\bar{e}_p(\iota )\dot{e}_p(\iota )\big )+2\sum _{p=1}^{n}\big (\alpha _p\gamma _p-(a_p+\delta _p)\alpha _p^2\big )\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota ) \nonumber \\&\quad -2\sum _{p=1}^{n}\left( b_p+k_p\right) \alpha _p\gamma _p\bar{e}_p(\iota )e_p(\iota ) \nonumber \\&\quad +\sum _{p=1}^{n}\sum _{q=1}^{n}\alpha _p^2\big (\bar{f}_q(e_q(\iota ))\bar{c}_{pq}\dot{e}_p(\iota )+\dot{\bar{e}}_p(\iota )c_{pq}f_q(e_q(\iota ))\big ) \nonumber \\&\quad +\sum _{p=1}^{n}\sum _{q=1}^{n}|\alpha _p\gamma _p|\big (\bar{f}_q(e_q(\iota ))\bar{c}_{pq}e_p(\iota )+\bar{e}_p(\iota )c_{pq}f_q(e_q(\iota ))\big ) \nonumber \\&\quad +\sum _{p=1}^{n}\sum _{q=1}^{n}\alpha _p^2\big (\bar{f}_q(e_q(\iota -\tau _q))\bar{d}_{pq}\dot{e}_p(\iota )+\dot{\bar{e}}_p(\iota )d_{pq} \nonumber \\&\quad \times f_q(e_q(\iota -\tau _q))\big )+\sum _{p=1}^{n}\sum _{q=1}^{n}|\alpha _p\gamma _p|\big (\bar{f}_q(e_q(\iota -\tau _q))\bar{d}_{pq}e_p(\iota ) \nonumber \\&\quad +\bar{e}_p(\iota )d_{pq}f_q(e_q(\iota -\tau _q))\big )+\sum _{p=1}^{n}\sum _{q=1}^{n}\big (\alpha _q^2+|\alpha _q\gamma _q|\big )\nonumber \\&\quad \times \big (\bar{f}_p(e_p(\iota ))f_p(e_p(\iota ))-\bar{f}_p(e_p(\iota -\tau _p))f_p(e_p(\iota -\tau _p))\big ). \end{aligned}$$
(9)

Based on Assumption 2.1 and Lemma 2.2, one has

$$\begin{aligned}&\sum _{p=1}^{n}\sum _{q=1}^{n}\alpha _p^2\big (\bar{f}_q(e_q(\iota ))\bar{c}_{pq}\dot{e}_p(\iota )+\dot{\bar{e}}_p(\iota )c_{pq}f_q(e_q(\iota ))\big ) \nonumber \\&\quad \le \sum _{p=1}^{n}\sum _{q=1}^{n}\alpha _p^2\big (\bar{f}_q(e_q(\iota ))f_q(e_q(\iota ))+c_{pq}\bar{c}_{pq}\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota )\big ) \nonumber \\&\quad =\sum _{p=1}^{n}\sum _{q=1}^{n}\big (\alpha _q^2l_p^2\bar{e}_p(\iota )e_p(\iota )+\alpha _p^2c_{pq}\bar{c}_{pq}\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota )\big ). \end{aligned}$$
(10)

Analogously, one can derive that

$$\begin{aligned} \sum _{p=1}^{n}\sum _{q=1}^{n}&|\alpha _p\gamma _p|\big (\bar{f}_q(e_q(\iota ))\bar{c}_{pq}e_p(\iota )+\bar{e}_p(\iota )c_{pq}f_q(e_q(\iota ))\big ) \nonumber \\&\quad \le \sum _{p=1}^{n}\sum _{q=1}^{n}\big (|\alpha _q\gamma _q|l_p^2+|\alpha _p\gamma _p|c_{pq}\bar{c}_{pq}\big )\bar{e}_p(\iota )e_p(\iota ), \end{aligned}$$
(11)
$$\begin{aligned}&\sum _{p=1}^{n}\sum _{q=1}^{n}\alpha _p^2\big (\bar{f}_q(e_q(\iota -\tau _q))\bar{d}_{pq}\dot{e}_p(\iota )+\dot{\bar{e}}_p(\iota )d_{pq}f_q(e_q(\iota -\tau _q))\big ) \nonumber \\&\quad \le \sum _{p=1}^{n}\sum _{q=1}^{n}\big (\alpha _q^2\bar{f}_p(e_p(\iota -\tau _p))f_p(e_p(\iota -\tau _p))+\alpha _p^2d_{pq}\bar{d}_{pq}\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota )\big ), \end{aligned}$$
(12)

and

$$\begin{aligned}&\sum _{p=1}^{n}\sum _{q=1}^{n}|\alpha _p\gamma _p|\big (\bar{f}_q(e_q(\iota -\tau _q))\bar{d}_{pq}e_p(\iota )+\bar{e}_p(\iota )d_{pq}f_q(e_q(\iota -\tau _q))\big ) \nonumber \\&\quad \le \sum _{p=1}^{n}\sum _{q=1}^{n}\big (|\alpha _q\gamma _q|\bar{f}_p(e_p(\iota -\tau _p))f_p(e_p(\iota -\tau _p))+|\alpha _p\gamma _p|d_{pq}\bar{d}_{pq}\bar{e}_p(\iota )e_p(\iota )\big ). \end{aligned}$$
(13)

Substituting (10)–(13) into (9) yields that

$$\begin{aligned} \dot{V}(\iota )&\le \sum _{p=1}^{n}\Big (\beta _p+\gamma _p^2-(a_p+\delta _p)\alpha _p\gamma _p-(b_p+k_p)\alpha _p^2\Big )\big (\dot{\bar{e}}_p(\iota )e_p(\iota ) \nonumber \\&\quad +\bar{e}_p(\iota )\dot{e}_p(\iota )\big )+\sum _{p=1}^{n}\Big (2\alpha _p\gamma _p-2(a_p+\delta _p)\alpha _p^2+\sum _{q=1}^{n}(c_{pq}\bar{c}_{pq} \nonumber \\&\quad +d_{pq}\bar{d}_{pq})\alpha _p^2\Big )\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota )+\sum _{p=1}^{n}\Big (-2(b_p+k_p)\alpha _p\gamma _p \nonumber \\&\quad +\sum _{q=1}^{n}\left[ \left( \alpha _q^2+|\alpha _q\gamma _q|\right) l_p^2+(c_{pq}\bar{c}_{pq}+d_{pq}\bar{d}_{pq})|\alpha _p\gamma _p|\right] \nonumber \\&\quad +\sum _{q=1}^{n}\big (\alpha _q^2+|\alpha _q\gamma _q|\big )l_p^2\Big )\bar{e}_p(\iota )e_p(\iota ) \nonumber \\&=\sum _{p=1}^{n}\Big (C_p\big (\dot{\bar{e}}_p(\iota )e_p(\iota )+\bar{e}_p(\iota )\dot{e}_p(\iota )\big )+A_p\dot{\bar{e}}_p(\iota )\dot{e}_p(\iota ) \nonumber \\&\quad +B_p\bar{e}_p(\iota )e_p(\iota )\Big ) \nonumber \\&=\sum _{p=1}^{n}A_p\overline{\left( \dot{e}_p(\iota )+\frac{C_p}{A_p}e_p(\iota )\right) }\left( \dot{e}_p(\iota )+\frac{C_p}{A_p}e_p(\iota )\right) \nonumber \\&\quad +\sum _{p=1}^{n}\left( B_p-\frac{C_p^2}{A_p}\right) \bar{e}_p(\iota )e_p(\iota ). \end{aligned}$$
(14)

Let \(\xi =\min \limits _{p\in \Lambda }\{\frac{C_p^2}{A_p}-B_p\}\), then \(\xi >0\) according to Theorem 3.1. Moreover, one has

$$\begin{aligned} \dot{V}(\iota )\le \sum _{p=1}^{n}\left( B_p-\frac{C_p^2}{A_p}\right) \bar{e}_p(\iota )e_p(\iota ) \le -\xi \sum _{p=1}^{n}\bar{e}_p(\iota )e_p(\iota ). \end{aligned}$$
(15)

It follows from (15) that \(V(\iota )\le V(0), \iota \ge 0\), which implies that \(e_p(\iota )\) and \(\dot{e}_p(\iota )\) are both bounded for \(p\in \Lambda , \iota \ge 0\). Therefore, the derivative of \(\sum _{p=1}^{n}\bar{e}_p(\iota )e_p(\iota )\) is also bounded, which results in \(\sum _{p=1}^{n}\bar{e}_p(\iota )e_p(\iota )\) is uniformly continuous.

On the other hand, integrating both sides of (15) and taking the limitation as \(\iota \rightarrow +\infty \), one has

$$\begin{aligned} \lim \limits _{\iota \rightarrow +\infty }\int _0^\iota \sum _{p=1}^{n}\bar{e}_p(s)e_p(s)ds\le \frac{V(0)}{\xi }< +\infty . \end{aligned}$$

Combining above discussions, it can be obtained that \(\lim \limits _{\iota \rightarrow +\infty }\sum _{p=1}^{n}\bar{e}_p(\iota )e_p(\iota )=0\) on account of Lemma 2.3. Thus, the error system (5) is asymptotically stable, which completes our proof. \(\square \)

Remark 3.1

It could be found from above proof procedure that we do not implement any real or plural decomposition for the quaternion-valued system. On the contrary, we derive a component-wise criterion via adequately utilizing the proposition of quaternions (Lemmas 2.1–2.2), which could be regarded as a natural extension of [23, 24] and simple to verify.

Remark 3.2

The traditional approach adopted in handling inertial neural networks is reduced order approach, which needs to conduct a transformation for the original system and will inevitably increase the dimensions and make analysis more complicated. The non-reduced order approach employed in this article could provide a new way in dealing with systems with inertial term, and a relatively laconic conclusion is drawn in Theorem 3.1.

Remark 3.3

By employing the non-reduced order approach, a variety of results of IRVNNs have sprung up over the past few years [11, 12]. However, the approach cannot be simply applied to IQVNNs due to the non-commutativity of quaternions. To tackle this difficulty, a novel quaternion-based Lyapunov functional (7) is firstly established. It consists of both \(e_p(\iota )\) and \(\dot{e}_p(\iota )\), which is the key point of non-reduced order approach in dealing with second-order systems.

Remark 3.4

There are three parameters \(\alpha _p\), \(\beta _p\) and \(\gamma _p\) appeared in Theorem 3.1. We could reduce the number of parameters by the following line. Let \(C_p=0\), which implies \(\beta _p=-\gamma _p^2+(a_p+\delta _p)\alpha _p\gamma _p+(b_p+k_p)\alpha _p^2\)>0. Hence, the equivalent form of Theorem 3.1 could be presented as below without detailed proof.

Theorem 3.2

Under Assumption 1, the slave system (2) can be asymtotically projective synchronized to the master system (1) with controller (4) if there exist scalars \(\alpha _p\) and \(\gamma _p\) such that

$$\begin{aligned}&-\gamma _p^2+\left( a_p+\delta _p\right) \alpha _p\gamma _p+\left( b_p+k_p\right) \alpha _p^2>0, \\&\quad A_p<0, \quad B_p<0, \quad p\in \Lambda , \end{aligned}$$

where \(A_p\) and \(B_p\) are the same as those defined in Theorem 3.1.

It is worth noting that choosing different values of proportional factor \(\alpha \) will result in various kinds of synchronization between (1) and (2). In what follows, two corollaries are presented to display various kinds of synchronization.

When \(\alpha =1\), the projective synchronization is equivalent to complete synchronization. In this case, the controller (4) becomes

$$\begin{aligned} u_p(\iota )=-\delta _p\dot{e}_p(\iota )-k_pe_p(\iota ). \end{aligned}$$
(16)

Then, the first corollary can be drawn directly.

Corollary 3.1

The master–slave systems (1) and (2) can achieve asymptotical complete synchronization with controller (16) if (6) holds.

When \(\alpha =-1\), the projective synchronization turns into anti-synchronization. With the extra assumptions that \(g_q\) is odd and \(I_p=0\), the controller (4) is with the following form

$$\begin{aligned} u_p(\iota )=-\delta _p\big [\dot{y}_p(\iota )+\dot{x}_p(\iota )\big ]-k_p\big [y_p(\iota )+x_p(\iota )\big ]. \end{aligned}$$
(17)

Then, the second corollary are concluded as follows.

Corollary 3.2

Suppose the activation function \(g_q\) is odd and the external input \(I_p=0\), the master–slave systems (1) and (2) can achieve asymptotical anti-synchronization with controller (17) if (6) holds.

Remark 3.5

The aforementioned corollaries show that the model considered in this article is rather general, it contains some existing works as special cases. For example, [11] explored the complete synchronization problem for IRVNNs, whose results could be included in Corollary 3.1; [36] investigated the anti-synchronization issue for QVNNs, which could be deemed as a special case of Corollary 3.2.

Fig. 1
figure 1

Transient behaviors of master–slave systems (1) and (2) with controller (16)

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Fig. 2
figure 2

Transient behaviors of synchronization errors with controller (17)

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4 Numerical Simulation

This section will demonstrate the effectiveness and superiority of the scheme presented above.

Example 4.1

Consider the 2-dimensional systems (1) and (2) whose parameters are given by \(a_1=8.8, a_2=8.3, b_1=6.6, b_2=5.9, g_1(\iota )=g_2(\iota )=0.3\tanh (\iota ), \tau _1=\tau _2=1, I_1=0.8-0.5i+0.4j+0.5k, I_2=0.6+0.4i-0.5j-0.5k\) and

$$\begin{aligned} c_{11}&=0.26-0.44i-0.80j+0.09k, \ \ \ \ c_{12}=0.91+0.60i-0.03j-0.72k, \\ c_{21}&=-\,0.16+0.58i+0.83j+0.92k, \ \ c_{22}=0.36+0.49i+0.52j-0.22k, \\ d_{11}&=0.31+0.41i-0.66j-0.94k, \ \ \ \ d_{12}=-\,0.12+0.53i-0.24j+0.59k, \\ d_{21}&=-\,0.63-0.11i-0.02j+0.29k, \ \ d_{22}=0.42-0.45i+0.51j+0.36k. \end{aligned}$$

When \(\alpha =1\), the projective synchronization is equivalent to complete synchronization. Besides, we set the control gains values in (4) as \(\delta _1=2, \delta _2=2.1, k_1=1, k_2=1.5\). The scalars \(\alpha _p, \beta _p, \gamma _p (p=1,2)\) in Theorem 3.1 are chosen as \(\alpha _1=\alpha _2=0.5, \beta _1=\beta _2=10, \gamma _1=\gamma _2=1\). With simple calculation, we can derive that \(A_1=-3.1747, A_2=-3.2380, B_1=-4.8794, B_2=-5.2061, C_1=3.7000, C_2=3.9500\), which indicates that the criterion (6) is fulfilled. Therefore, the master–slave systems (1) and (2) can achieve complete synchronization with controller (4) (or (16)) in accordance with Theorem 3.1 (or Corollary 3.1). Figure 1 dipicts the transient behaviors of master–slave systems (1) and (2) with initial conditions \((2.7+1.9i-1.7j-0.8k, -\,1.5+1.2i+2.5j+2.6k)\) and \((-\,0.8-2.4i+1.5j+1.9k, 1.6-1.7i-2.3j-1.8k)\). Since \(x_p(\iota )\) and \(y_p(\iota )\) are both quaternion-valued, Fig. 1 is composed of 4 subfigures, each of which dipicts the behaviors of a component of \(x_p(\iota )\) and \(y_p(\iota )\). Through comparing and analyzing, it can be found that the obtained theory results coincide well with the simulation results.

When \(\alpha =-1\), the projective synchronization turns into anti-synchronization. If \(I_1=I_2=0\), and other parameters remain unchanged, one can easily derive that conditions in Corollary 3.2 are all satisfied, which indicates the inertial quaternion-valued neural network (2) could be asymptotically anti-synchronized to (1) with controller (17). Transient behaviors of anti-synchronization errors are illustrated in Fig. 2.

5 Conclusion

In this article, we study the projective synchronization problem of IQVNNs by employing the non-reduced order approach. To realize the synchronization goal, an appropriate controller is firstly proposed. Then, by virtue of Lyapunov functional method, some sufficient conditions ensuring the asymptotical stability of error system are presented in component form. Subsequently, two corollaries are obtained to display various kinds of synchronization with different values of proportional factor. Finally, the validity of our approach is proved through a numerical example. In view of the convenience and effectiveness of non-reduced order approach, it would be meaningful to branch out into more general inertial models, such as inertial Cohen-Grossberg QVNNs and fractional-order INNs, with this useful tool in the future.