Adaptive impulsive synchronization for fractional-order quaternion-valued neural networks with different structures and unknown parameters

Abstract

The synchronization issue between fractional-order quaternion-valued neural networks with different structures is more common than that with the same structure in the real world. This paper addresses the Mittag-Leffler stability and adaptive impulsive synchronization between two fractional-order quaternion-valued neural networks with different structures. We also propose two assumptions to analyze the parameters of the nonlinear activation functions. Based on the Mittag-Leffler stability and fractional-order Lyapunov direct method, an impulsive controller scheme is designed to ensure the global stability of fractional-order time-delay quaternion-valued neural networks. We also analyzed the situation when the self-feedback matrix and connection matrix of quaternion-valued fractional-order neural networks with different structures are unknown and designed adaptive impulsive controllers to achieve synchronization. Finally, we verified the feasibility and effectiveness of the proposed theorem through numerical simulation experiments.

Keywords

Impulsive synchronization quaternion-valued fractional-order neural network unknown parameter different structure

1. Introduction

Fractional calculus has recently become a research hotspot in dynamic systems because of the modeling needs of many objects in the real world and its incomplete mathematical theory. Studies have shown that fractional-order description is more consistent with the true nature of reality, while the integer-order description is only an idealized and simplified representation (Podlubny, 1999; Zhang et al., 2019a). As is well known, artificial neural networks aim to simulate biological neural networks, so their unique structure and parallel processing ability, similar to biological neural networks, make them a very important class of nonlinear dynamic models and have a wide range of application scenarios. Hence, the dynamical behaviors of neural networks have been extensively investigated, such as the stability, chaos, synchronization, state estimation, and periodicity (Al-Sharman et al., 2019; Fan et al., 2019; Wang et al., 2019a; Cao et al., 2020; Cao and Cao, 2022). Factional derivatives have been brought into neural networks, in which fractional-order equations can describe their behaviors (Li et al., 2015a; Li and Zhang, 2016; (Lin et al., 2019)). Fractional-order real-valued neural networks and fractional-order complex-valued neural networks have been widely studied for their diffuse application in secure communication, engineering optimization, signal processing, and pattern recognition (Bao et al., 2016; Qi et al., 2019; Chen et al., 2020; You et al., 2020; Xu et al., 2021).

Many scholars have studied the stability analysis and synchronization of fractional-order neural networks using adaptive (Zhang et al., 2019a; Pratap et al., 2020a), impulsive (Zhang et al., 2019b; Pratap et al., 2020a), fuzzy (Liu et al., 2018a; Zhang et al., 2019a), projective (Udhayakumar et al., 2022), and other methods (Wang et al., 2019b; Mirrezapour et al., 2022; Gokyildirim et al., 2023). These methods are mainly based on Mittag-Leffler stability (Podlubny, 1999; Li et al., 2015b) and fractional-order Lyapunov direct methods (Aguila-Camacho et al., 2014). When solving the synchronization problem and stability analysis of fractional-order complex number domain neural networks, some scholars will divide the complex number based on the real part and the imaginary part (Luo and Wang, 2013). Although this method makes the problem simpler, it destroys the overall structure of the fractional-order complex number field system and ignores the nature of the complex number itself (Zhang et al., 2019b). Then the fractional-order Lyapunov direct method is extended to a complex number field (Zhang et al., 2019b).

Quaternion is a particular case of Clifford algebra invented by Hamilton in 1843 (Wei and Cao, 2019). Because of its characteristics, some general operation rules on real and complex numbers, such as the commutativity of multiplication, do not apply to quaternions. It consists of one real part and three imaginary parts, whose structure is particularly effective in processing multidimensional data (Lin et al., 2019). It has received extensive attention for its practical applications in space rotation, computer graphics, robotics, and image processing (Xia et al., 2015; Zou et al., 2016). Recently, some researchers have incorporated quaternion algebra into classical neural networks to establish quaternion-valued neural networks (QVNNs) (Liu et al., 2018b), whose connection weights, states, and activation functions of QVNNs take values in the quaternion field. Compared with real-valued neural networks (RVNNs) and complex-valued neural networks (CVNNs), a significant superiority of QVNNs is low dimension and high efficiency in handling multidimensional data (Wei and Cao, 2019).

To analyze the dynamic, stability, and synchronization of integer-order QVNNs, researchers have proposed the real decomposition approach (Wei and Cao, 2019; Qi et al., 2019), plural decomposition approach (Liu et al., 2016; Chen et al., 2018), and direct quaternion approach (Chen et al., 2017; Tu et al., 2018; Liu et al., 2018b; Singh et al., 2022). Based on Mittag-Leffler’s stability, scholars try to realize the stability and synchronization of fractional-order quaternion-valued neural networks (abbreviated as FQVNNs). The fractional-order Lyapunov direct method, suitable for real-valued and complex-valued, is extended to quaternions (Lin et al., 2019). The Mittag-Leffler stability and synchronization of FQVNNs with linear threshold neurons are analyzed (Yang et al., 2018). Classical synchronization methods, such as projection (Zhang et al., 2021), adaptive (Pratap et al., 2020a), impulsive (Pratap et al., 2020a), and finite-time (Pratap et al., 2020b; Zhang et al., 2022; Mo and Bo, 2022), are also implemented on FQVNNs.

Unknown parameters in neural networks are a more complex problem than parameter uncertainty. Adaptive methods are highly effective for integer-order (Zhang and Zeng, 2019; Zhou et al., 2021; Cao et al., 2022) and fractional-order (Gu et al., 2020; Sun and Liu, 2020, 2021) neural networks to address this issue. However, most of these methods do not discuss the nonlinear part.

It is worth noting that the above methods are carried out between FQVNNs with the same structure. The synchronization between networks with different structures is a more complex and easily encountered phenomenon in reality. Therefore, we address the adaptive impulsive synchronization of FQVNNs with time delay and different structures based on the Mittag-Leffler stability and fractional-order Lyapunov direct method. We propose two assumptions to analyze the parameters of the nonlinear activation functions and design an efficient impulsive control scheme to ensure the global Mittag-Leffler stability of fractional-order time-delay quaternion-valued neural networks. We also analyzed the situation when the self-feedback matrix and connection matrix of quaternion-valued fractional-order neural networks with different structures are unknown and designed adaptive impulsive controllers to achieve synchronization. The main contributions are concluded as follows:

• We proposed two necessary assumptions to analyze the parameters of the nonlinear activation function of FQVNNs and provided a proof process.

• Impulsive synchronization between FQVNNs with different structures was achieved based on Mittag-Leffler stability theory.

• Furthermore, we designed an adaptive impulsive method to synchronize the self-feedback matrix and connection matrix of FQVNNs with different structures in unknown situations.

2. Preliminaries and model description

Let $R, C, Q$ denote the real-valued, complex-valued, and quaternion-valued, respectively. Then a quaternion-valued function z_p(t) can be described as

z_{p} (t) = z_{p}^{R} (t) + i z_{p}^{I} (t) + j z_{p}^{J} (t) + k z_{p}^{K} (t),

(1)

where

z_{p}^{R} (t), z_{p}^{I} (t), z_{p}^{J} (t), z_{p}^{K} (t) \in R

, and i, j, k are imaginary units, which obey the Hamilton rules:

\begin{array}{l} i^{2} = j^{2} = k^{2} = i j k = - 1, \\ i j = - j i = k, \\ j k = - k j = i, \\ k i = - i k = j . \end{array}

Then $\bar{z_{p} (t)} = z_{p}^{R} (t) - i z_{p}^{I} (t) - j z_{p}^{J} (t) - k z_{p}^{K} (t)$ is the conjugate of z_p(t). And the module of z_p(t) can be calculated by $‖ z_{p} (t) ‖ = \sqrt{\bar{z_{p} (t)} z_{p} (t)} = \sqrt{z_{p} (t) \bar{z_{p} (t)}} = \sqrt{{(z_{p}^{R} (t))}^{2} + {(z_{p}^{I} (t))}^{2} + {(z_{p}^{J} (t))}^{2} + {(z_{p}^{K} (t))}^{2}}$ .

Definition 1

(Podlubny, 1999) Caputo fractional derivative of function x(t) is defined as

{}_{t_{0}}^{C}{D_{t}^{α}} x (t) = \frac{1}{Γ (1 - α)} \int_{t_{0}}^{t} \frac{x^{'} (ξ)}{{(t - ξ)}^{α}} d ξ .

(2)

Definition 2

(Podlubny, 1999) The Mittag-Leffler function with one parameter is defined as

E_{α} (x) = \sum_{p = 0}^{+ \infty} \frac{x^{p}}{Γ (p α + 1)} .

(3)

Definition 3

(Yang et al., 2018) An equilibrium point x* of system ${}_{t_{0}}^{C}{D_{t}^{α}} x (t) = f (x, t)$ is said to be globally Mittag-Leffler stable if there exist positive constants H, λ, and β such that for any solution x(t) of the system with initial value x₀, one has

‖ x (t) - x^{*} ‖ \leq H ‖ x_{0} - x^{*} ‖ {[E_{α} (- λ t^{α})]}^{β}, t \neq 0 .

Lemma 1

(Aguila-Camacho et al., 2014) Let $x (t) \in R^{n}$ be a continuous and derivable vector value function on [t₀, + ∞). Then, for any time instant t > t₀, it satisfies

{}_{t_{0}}^{C}{D_{t}^{α}} x^{T} (t) x (t) \leq 2 x^{T} {(t)}_{t_{0}}^{C} D_{t}^{α} x (t) .

(4)

The proof of Lemma 1 is based on the definition of fractional-order derivatives and the Lyapunov method.

Lemma 2

(Li et al., 2015b) Let $x (t) \in R^{n}$ be a continuous and derivable vector value function on [t₀, + ∞) and satisfying

{}_{t_{0}}^{C}{D_{t}^{α}} x (t) \leq γ x (t),

where

0 < α < 1, γ \in R

. Then

x (t) \leq x (t_{0}) E_{α} [γ {(t - t_{0})}^{α}] .

(5)

Lemma 3

(Li et al., 2019) Let $x (t) \in Q^{n}$ be a continuous and derivable vector value function on [t₀, + ∞). Then, for any time instant t > t₀ and 0 < α < 1, it satisfies

{}_{t_{0}}^{C}{D_{t}^{α}} [x^{T} (t) \bar{x (t)}] \leq x^{T} (t) \bar{{}_{t_{0}}^{C}{D_{t}^{α}} x (t)} + {\bar{x (t)}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} x (t) .

(6)

The proof of Lemma 3 is based on Lemma 1. If $x (t) \in C^{n}$ , Lemma 3 is also valid in Zhang et al., (2019b).

Lemma 4

(Aguila-Camacho et al., 2014; Zhang et al., 2019b) The fractional-order extension of the Lyapunov direct method.

Let x = 0 be an equilibrium point for the non-autonomous fractional-order system ${}_{t_{0}}^{C}{D_{t}^{α}} x (t) = f (x, t)$ . Assume that there exists a Lyapunov function V(x, t) and class-K functions γ_i, i = 1, 2, 3 satisfying

γ_{1} ‖ x (t) ‖ \leq V (x, t) \leq γ_{2} ‖ x (t) ‖,

(7)

and

{}_{t_{0}}^{C}{D_{t}^{α}} x (t) \leq - γ_{3} x (t) .

(8)

Then fractional-order system ${}_{t_{0}}^{C}{D_{t}^{α}} x (t) = f (x, t)$ is asymptotically stable.

Next, we consider the following fractional-order quaternion-valued neural networks

{}_{t_{0}}^{C}{D_{t}^{α}} x (t) = - C x (t) + A f (x (t)) + B f (x (t - τ)),

(9)

where t ≥ t₀,

x (t) = (x_{1} (t), x_{2} (t), \dots, x_{n} (t)) \in Q^{n}

is the state vector,

C = diag (c_{1}, c_{2}, \dots, c_{n}) \in R^{n \times n}

and c_i > 0 represent the self-feedback coefficients,

A, B \in Q^{n \times n}

is the connection weight matrix,

f (x (t)) \in Q^{n}

is the neuron activation function, and τ denotes the discrete time delay.

Impulsive controlled response fractional order quaternion-valued neural networks with different structures are defined as

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} y (t) = - D y (t) + W g (y (t)) + V g (y (t - τ)) \\ + u (t), t \neq t_{m} Δ y (t_{m}) = H_{m} y (t_{m}), m = 1,2, \dots, \end{array}

(10)

where

y (t) = (y_{1} (t), y_{2} (t), \dots, y_{n} (t)) \in Q^{n}

is the state vector,

D = diag (d_{1}, d_{2}, \dots, d_{n}) \in R^{n \times n}

and d_i > 0 represent the self-feedback coefficients,

W, V \in Q^{n \times n}

is the connection weight matrix, and

g (y (t)) \in Q^{n}

is the neuron activation function, which has a different structure with f(x(t)) in the drive system. And

u (t) \in Q^{n \times n}

is the designed controller.

{t_{m}}_{m = 0}^{\infty}

is the time point,

\lim_{m \to \infty} t_{m} = + \infty

. And

H_{m} = y (t_{m}^{+}) - y (t_{m}^{-})

, H_m ∈ C^n×n is the impulsive matrix at t = t_m, and

y (t_{m}^{+}) = \lim_{t \to t_{m}^{+}} y (t_{m})

Let e(t) = y(t) − x(t) be the error between two fractional-order quaternion-valued neural networks (9) and (10), then

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} e (t) = - D e (t) + (C - D) x (t) + W g (y (t)) \\ - A f (x (t)) + V g (y (t - τ)) \\ - B f (x (t - τ)) + u (t), t \neq t_{m} Δ e (t_{m}) \\ = H_{m} e (t_{m}), m = 1,2, \dots \end{array}

(11)

where

e (t) = e_{p} {(t)}_{p = 1}^{n}

and each

e_{p} (t) = e_{p}^{R} (t) + i e_{p}^{I} (t) + j e_{p}^{J} (t) + k e_{p}^{K} (t)

If $\lim_{t \to \infty} e (t) = 0$ , then we can realize the synchronization between two fractional-order quaternion-valued neural networks. Before analyzing the stability of the error system (11), we propose some essential assumptions about the nonlinear function vector, similar to the Lipschitz condition.

Assumption 1

(Zhang et al., 2019b) There exists two positive constants L^f and L^g such that the nonlinear neuron activation function satisfies

\begin{array}{l} ‖ f (y (t)) - f (x (t)) ‖ & \leq L^{f} ‖ y (t) - x (t) ‖, \\ ‖ g (y (t)) - g (x (t)) ‖ & \leq L^{g} ‖ y (t) - x (t) ‖ . \end{array}

(12)

Assumption 1 can also be written as the following.

Assumption 2

(Zhang et al., 2019b) There exists two positive constants L^h and L^f such that

e^{T} (t) \bar{f (y (t)) - f (x (t))} + {(f (y (t)) - f (x (t)))}^{T} \bar{e (t)} \leq 2 L^{f} e^{T} (t) \bar{e (t)},

e^{T} (t) \bar{g (y (t)) - g (x (t))} + {(g (y (t)) - g (x (t)))}^{T} \bar{e (t)} \leq 2 L^{g} e^{T} (t) \bar{e (t)} .

(13)

Lemma 5

(Aguila-Camacho et al., 2014) For any two vectors $ω_{1}, ω_{2} \in Q$ , there exists a positive constant ξ, such that

\bar{ω_{1}} ω_{2} + \bar{ω_{2}} ω_{1} \leq ξ \bar{ω_{1}} ω_{1} + ξ^{- 1} \bar{ω_{2}} ω_{2} .

(14)

3. Main result

3.1. Synchronization with known parameters

To analyze the stability of the error system, we need two more assumptions.

Assumption 3

For the nonlinear part of the neural networks, we have the following inequality.

\begin{array}{l} e^{T} (t) \bar{W [g (y (t)) - h (x (t)]} + {[g (y (t)) - g (x (t))]}^{T} W^{T} \bar{e (t)} \\ \leq n M_{W} {(e (t))}^{T} \bar{e (t)}, \\ e^{T} (t) \bar{A [f (y (t)) - f (x (t))]} + {[f (y (t)) - f (x (t))]}^{T} A^{T} \bar{e (t)} \\ \leq n M_{A} {(e (t))}^{T} \bar{e (t)} \end{array},

(15)

where

M_{w} = \max_{p q} {1 / ρ_{p q}^{2} L_{p q}^{g} + ρ_{p q}^{2} \bar{w_{p q}} w_{p q}}

and

M_{A} = \max_{p q} {1 / ρ_{p q}^{1} L_{p q}^{f} + ρ_{p q}^{1} \bar{a_{p q}} a_{p q}}

ρ_{p q}^{1}

and

ρ_{p q}^{2}

are two positive constants.

Proof

The fractional-order quaternion-valued neural networks (9) can be rewritten as follows.

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} x_{p} (t) = - c_{p} (t) + \sum_{q = 1}^{n} a_{p q} f_{q} (x_{q} (t)) \\ + \sum_{q = 1}^{n} b_{p q} f_{q} (x_{q} (t - τ)) . \end{array}

(16)

Then the second inequality of equation (15) satisfies

\begin{array}{l} e^{T} (t) \bar{A [f (y (t)) - f (x (t))]} + {[(f (y (t)) - f (x (t)))]}^{T} A^{T} \bar{e^{T} (t)} \\ = \sum_{p = 1}^{n} \sum_{q = 1}^{n} [e_{p} (t) \bar{a_{p q} (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t)))} \\ + (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))) a_{p q} \bar{e_{p} (t)}] . \end{array} .

Let $ω_{1} = e_{p} (t) \bar{a_{p q}}$ and ω₂ = f_q(y_q(t)) − f_q(x_q(t)). Applying Lemma 5, there exists a positive constant $ρ_{p q}^{1}$ such that the above equation satisfies equation (14).

\begin{array}{l} e_{p} (t) \bar{a_{p q}} \bar{f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))} \\ + \bar{e_{p} (t)} a_{p q} (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))) \\ \leq \frac{1}{ρ_{p q}^{1}} (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))) \bar{(f_{q} (y_{q} (t)) - f_{q} (x_{q} (t)))} \\ + ρ_{p q}^{1} e_{p} (t) \bar{a_{p q}} a_{p q} \bar{e_{p} (t)} \\ \leq (\frac{1}{ρ_{p q}^{1}} L_{p q}^{f} + ρ_{p q}^{1} \bar{a_{p q}} a_{p q}) e_{p} (t) \bar{e_{p} (t)} . \end{array}

(17)

Hence, we have the following inequality.

\begin{array}{l} \sum_{p = 1}^{n} \sum_{q = 1}^{n} [e_{p} (t) \bar{a_{p q} (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t)))} \\ + (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))) a_{p q} \bar{e_{p} (t)}] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [\frac{1}{ρ_{p q}^{1}} (f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))) \bar{f_{q} (y_{q} (t)) - f_{q} (x_{q} (t))} \\ + ρ_{p q}^{1} \bar{e_{p} (t)} a_{p q} \bar{a_{p q}} e_{p} (t)] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [(\frac{1}{ρ_{p q}^{1}} L_{p q}^{f} + ρ_{p q}^{1} \bar{a_{p q}} a_{p q}) e_{p} (t) \bar{e_{p} (t)}] \\ \leq n M_{A} \sum_{p = 1}^{n} (e_{p} (t) \bar{e_{p} (t)}) \\ = n M_{A} {(e (t))}^{T} \bar{y (t) - x (t)}, \end{array}

(18)

where

M_{A} = \max_{p q} {1 / ρ_{p q}^{1} L_{p q}^{f} + ρ_{p q}^{1} \bar{a_{p q}} a_{p q}}

In the same way, we can get a similar equation for the nonlinear part of the response system. There also exists a positive $ρ_{p q}^{2}$ such that

\begin{array}{l} {(e (t))}^{T} \bar{W [g (y (t)) - g (x (t))]} + {[(g (y (t)) - g (x (t)))]}^{T} W^{T} \bar{e (t)} \\ = \sum_{p = 1}^{n} \sum_{q = 1}^{n} [e_{p} (t) \bar{w_{p q} (g_{q} (y_{q} (t)) - g_{q} (x_{q} (t)))} \\ + (g_{q} (y_{q} (t)) - g_{q} (x_{q} (t))) w_{p q} \bar{e_{p} (t)}] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [\frac{1}{ρ_{p q}^{2}} (g_{q} (y_{q} (t)) - g_{q} (x_{q} (t))) \bar{g_{q} (y_{q} (t)) - g_{q} (x_{q} (t))} \\ + ρ_{p q}^{2} \bar{e_{p} (t)} w_{p q} \bar{w_{p q}} e_{p} (t)] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [(\frac{1}{ρ_{p q}^{2}} L_{p q}^{g} + ρ_{p q}^{2} \bar{w_{p q}} w_{p q}) e_{p} (t) \bar{e_{p} (t)}] \\ \leq n M_{W} \sum_{p = 1}^{n} ((y_{p} (t) - x_{p} (t)) \bar{e_{p} (t)}) \\ = n M_{W} {(e (t))}^{T} \bar{e (t)}, \end{array}

(19)

where

M_{w} = \max_{p q} {1 / ρ_{p q}^{2} L_{p q}^{g} + ρ_{p q}^{2} \bar{w_{p q}} w_{p q}}

In neural networks, the time delay is a common phenomenon, which usually uses the same activation function. Then, it also generally meets the above assumptions. To simplify expressions, let f*(e(t − τ)) = f(y(t − τ)) − f(x(t − τ)) and g*(e(t − τ)) = g(y(t − τ)) − g(x(t − τ)). Then, we get Assumptions 4.

Assumption 4

For the time-delayed nonlinear part of the neural networks, we have the following inequality.

\begin{array}{l} e {(t)}^{T} \bar{B f^{*} (e (t - τ))} + {[f^{*} (e (t - τ))]}^{T} B^{T} \bar{e (t)} \\ \leq n M_{F} e {(t - τ)}^{T} \bar{e (t - τ)} + n M_{B} e {(t)}^{T} \bar{e (t)}, \\ e {(t)}^{T} \bar{V g^{*} (e (t - τ))} + {[g^{*} (e (t - τ))]}^{T} V^{T} \bar{e (t)} \\ \leq n M_{H} e {(t - τ)}^{T} \bar{e (t - τ)} + n M_{V} e {(t)}^{T} \bar{e (t)}, \end{array}

(20)

where

M_{F} = \max_{p q} {1 / ρ_{p q}^{3} L_{p q}^{f}}

M_{B} = \max_{p q} {ρ_{p q}^{3} \bar{b_{p q}} b_{p q}}

M_{F} = \max_{p q} {1 / ρ_{p q}^{4} L_{p q}^{g}}

, and

M_{V} = \max_{p q} {ρ_{p q}^{4} \bar{v_{p q}} v_{p q}}

ρ_{p q}^{3}

and

ρ_{p q}^{4}

are two positive constants.

Proof

The fractional-order quaternion-valued neural networks (9) can be rewritten as equation (16).

Then the second inequality of equation (20) satisfies

\begin{array}{l} e {(t)}^{T} \bar{B f^{*} (e (t - τ))} + {[f^{*} (e (t - τ))]}^{T} B^{T} \bar{e (t)} \\ = \sum_{p = 1}^{n} \sum_{q = 1}^{n} [e_{p} (t) \bar{b_{p q} f_{q}^{*} (e_{q} (t - τ))} + f_{q}^{*} (e_{q} (t - τ)) b_{p q} \bar{e_{p} (t)}] . \end{array}

(21)

Let $ω_{1} = e_{p} (t) \bar{b_{p q}}$ and $ω_{2} = f_{q}^{*} (e (t - τ))$ . Applying Lemma 5, there exists a positive constant $ρ_{p q}^{3}$ such that the above equation satisfies equation (14). Combining Assumptions 1 and 2, we have the following.

\begin{array}{l} e_{p} (t) \bar{b_{p q} f_{q}^{*} (e_{q} (t - τ))} + f_{q}^{*} (e_{q} (t - τ)) b_{p q} \bar{e_{p} (t)} \\ \leq \frac{1}{ρ_{p q}^{3}} f_{q}^{*} (e_{q} (t - τ)) \bar{f_{q}^{*} (e_{q} (t - τ))} + ρ_{p q}^{3} e_{p} (t) \bar{b_{p q}} b_{p q} \bar{e_{p} (t)} \\ \leq \frac{1}{ρ_{p q}^{3}} L_{p q}^{f} e_{q} (t - τ) \bar{e_{q} (t - τ)} + ρ_{p q}^{3} \bar{b_{p q}} b_{p q} e_{p} (t) \bar{e_{p} (t)} . \end{array}

(22)

Hence, we have the following inequality.

\begin{array}{l} \sum_{p = 1}^{n} \sum_{q = 1}^{n} [e_{p} (t) \bar{b_{p q} f_{q}^{*} (e_{q} (t - τ))} + f_{q}^{*} (e_{q} (t - τ)) b_{p q} \bar{e_{p} (t)}] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [\frac{1}{ρ_{p q}^{3}} f_{q}^{*} (e_{q} (t - τ)) \bar{f_{q}^{*} (e_{q} (t - τ))}] \\ + \sum_{p = 1}^{n} \sum_{q = 1}^{n} [ρ_{p q}^{3} e_{p} (t) \bar{b_{p q}} b_{p q} \bar{e_{p} (t)}] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [\frac{1}{ρ_{p q}^{3}} L_{p q}^{f} e_{q} (t - τ) \bar{e_{q} (t - τ)} + ρ_{p q}^{3} \bar{b_{p q}} b_{p q} e_{p} (t) \bar{e_{p} (t)}] \\ \leq n M_{F} \sum_{q = 1}^{n} e_{q} (t - τ) \bar{e_{q} (t - τ)} + n M_{B} \sum_{p = 1}^{n} e_{p} (t) \bar{e_{p} (t)} \\ = n M_{F} e {(t - τ)}^{T} \bar{e (t - τ)} + n M_{B} e {(t)}^{T} \bar{e (t)}, \end{array}

(23)

where

M_{F} = \max_{p q} {1 / ρ_{p q}^{3} L_{p q}^{f}}

and

M_{B} = \max_{p q} {ρ_{p q}^{3} \bar{b_{p q}} b_{p q}}

In the same way, we can get a similar equation for the nonlinear part of the response system. There also exists a positive $ρ_{p q}^{4}$ such that

\begin{array}{l} e {(t)}^{T} \bar{V g^{*} (e (t - τ))} + {[g^{*} (e (t - τ))]}^{T} V^{T} \bar{e (t)} \\ = \sum_{p = 1}^{n} \sum_{q = 1}^{n} [e_{p} (t) \bar{v_{p q} g_{q}^{*} (e_{q} (t - τ))} + g_{q}^{*} (e_{q} (t - τ)) v_{p q} \bar{e_{p} (t)}] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [\frac{1}{ρ_{p q}^{4}} g_{q}^{*} (e_{q} (t - τ)) \bar{g_{q}^{*} (e_{q} (t - τ))}] \\ + \sum_{p = 1}^{n} \sum_{q = 1}^{n} [ρ_{p q}^{4} \bar{e_{p} (t)} v_{p q} \bar{v_{p q}} e_{p} (t)] \\ \leq \sum_{p = 1}^{n} \sum_{q = 1}^{n} [\frac{1}{ρ_{p q}^{4}} L_{p q}^{h} e_{q} (t - τ) \bar{e_{q} (t - τ)} + ρ_{p q}^{4} \bar{v_{p q}} v_{p q} e_{p} (t) \bar{e_{p} (t)}] \\ \leq n M_{H} \sum_{q = 1}^{n} e_{q} (t - τ) \bar{e_{q} (t - τ)} + n M_{V} \sum_{p = 1}^{n} e_{p} (t) \bar{e_{p} (t)} \\ = n M_{H} e {(t - τ)}^{T} \bar{e (t - τ)} + n M_{V} e {(t)}^{T} \bar{e (t)}, \end{array}

(24)

where

M_{F} = \max_{p q} {1 / ρ_{p q}^{4} L_{p q}^{H}}

and

M_{V} = \max_{p q} {ρ_{p q}^{4} \bar{v_{p q}} v_{p q}}

To realize stability, we design controller u(t).

\begin{array}{l} u (t) & = - (C - D) x (t) - K e (t) - W g (x (t)) + A f (y (t)) \\ - V g (x (t - τ)) + B f (y (t - τ)), \end{array}

(25)

when t ≠ t_m,

K \in R^{n \times n}

is the designed control gain matrix.

Under the designed controller (25), the error system of fractional-order quaternion-valued neural networks (11) is defined as

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} e (t) = - (D + K) e (t) + W [g (y (t)) - g (x (t))] \\ + A f (y (t)) - A f (x (t)) + V g (y (t - τ)) \\ - V g (x (t - τ)) + B f (y (t - τ)) \\ - B f (x (t - τ)), t \neq t_{m} Δ e (t_{m}) \\ = H_{m} e (t_{m}), m = 1,2, \dots . \end{array}

(26)

Then we get the first theorem. Let β_m be the largest eigenvalue of matrix ${(I + H_{m})}^{T} \bar{I + H_{m}}$ and σ = −2λ_D + nM_W + nM_V + nM_A + nM_B + η(nM_F + nM_H).

Theorem 1

The controller is designed as equation (25). The following two conditions hold:

(A1): σ < 0 and β_m < 1 for m = 0, 1, 2, ⋯ and

(A2): there exists a positive constant δ > 1 for m = 0, 1, 2, ⋯ such that

β_{m} E_{α} {(σ (t_{m + 1} - t_{m})}^{α}) < \frac{1}{δ} .

(27)

Then the error system (26) is asymptotically stable, which means the synchronization of two fractional-order quaternion-valued neural networks (equations (9) and (10)).

Proof

Consider the following Lyapunov function

V (t) = e^{T} (t) \bar{e (t)} .

(28)

When t ∈ [t₀, t₁), calculating the Caputo derivative of Lyapunov function equation (28) along the trajectory, we have the following inequality.

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} V (t) & \leq e^{T} {(t)}_{t_{0}}^{C} D_{t}^{α} \bar{e (t)} + {\bar{e (t)}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} e (t) \\ \leq e^{T} (t) \bar{- (D + K) e (t)} - {\bar{e (t)}}^{T} (D + K) e (t) \\ + e^{T} (t) \bar{W (g (y (t)) - g (x (t)))} \\ + {\bar{e (t)}}^{T} [W (g (y (t)) - g (x (t)))] \\ + e^{T} (t) \bar{V (g (y (t - τ)) - g (x (t - τ)))} \\ + {\bar{e (t)}}^{T} [V (g (y (t - τ)) - g (x (t - τ)))] \\ + e^{T} (t) \bar{A (f (y (t)) - f (x (t)))} \\ + {\bar{e (t)}}^{T} [A (f (y (t)) - f (x (t)))] \\ + e^{T} (t) \bar{B (f (y (t - τ)) - f (x (t - τ)))} \\ + {\bar{e (t)}}^{T} [B (f (y (t - τ)) - f (x (t - τ)))] . \end{array}

Applying Assumptions 3 and 4, we simplify the above inequality.

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} V (t) \leq - 2 λ_{D} e^{T} (t) \bar{e (t)} + n M_{W} e^{T} (t) \bar{e (t)} \\ + n M_{H} e {(t - τ)}^{T} \bar{e (t - τ)} + n M_{V} e {(t)}^{T} \bar{e (t)} \\ + n M_{A} e^{T} (t) \bar{e (t)} + n M_{F} e {(t - τ)}^{T} \bar{e (t - τ)} \\ + n M_{B} e {(t)}^{T} \bar{e (t)} \\ \leq (- 2 λ_{D} + n M_{W} + n M_{V} + n M_{A} + n M_{B}) e^{T} (t) \bar{e (t)} \\ + (n M_{F} + n M_{H}) e^{T} (t - τ) \bar{e (t - τ)} \\ \leq (- 2 λ_{D} + n M_{W} + n M_{V} + n M_{A} + n M_{B}) V (t) \\ + (n M_{F} + n M_{H}) V (t - τ) . \end{array}

(29)

Combining inequality (29) and fractional-order Razumikhin theorem (Liu et al., 2019), there exists a positive constant η > 1 such that

(n M_{F} + n M_{H}) V (t - τ) \leq η (n M_{F} + n M_{H}) V (t) .

Hence

{}_{t_{0}}^{C}{D_{t}^{α}} V (t) \leq [- 2 λ_{D} + n M_{W} + n M_{V} + n M_{A} + n M_{B} + η (n M_{F} + n M_{H})] V (t) .

(30)

And let σ = −2λ_D + nM_W + nM_V + nM_A + nM_B + η(nM_F + nM_H). Based on Lemma 2, there exists a Mittag-Leffler function such that equation (30) satisfies

V (t) \leq V (t_{0}) E_{α} [σ {(t - t_{0})}^{α}] .

(31)

Therefore, the following inequality holds on the interval t ∈ [t₀, t₁).

\begin{array}{l} V (t) & = e^{T} (t) \bar{e (t)} = \sum_{m = 1}^{n} e_{m} (t) \bar{e_{m} (t)} \\ \leq V (t_{0}) E_{α} [σ {(t - t_{0})}^{α}] \\ = \sum_{m = 1}^{n} e_{m} (t) \bar{e_{m} (t)} E_{α} [σ {(t - t_{0})}^{α}] \\ \leq \sup_{- τ \leq s \leq 0} \sum_{m = 1}^{n} {| e_{m} (t) |}^{2} E_{α} [σ {(t - t_{0})}^{α}] \end{array}

which implies that the error system is Mittag-Leffler stable from Definition 3 on interval t ∈ [t₀, t₁). The analyses can also be seen at You et al. (2020). Similarly, we can obtain that the error system is Mittag-Leffler stable within each impulsive interval. However, we need to consider the overall stability including impulsive points.

When t = t₁, the Lyapunov function (27) satisfies

\begin{array}{l} Δ V (t_{1}) = e^{T} (t_{1}^{+}) \bar{e (_{1}^{+})} - e^{T} (t_{1}) \bar{e (t_{1})} \\ = {((I + H_{1}) e (t_{1}))}^{T} \bar{((I + H_{1}) e (t_{1}))} - e^{T} (t_{1}) \bar{e (t_{1})} \\ = e^{T} (t_{1}) {((I + H_{1})}^{T} \bar{(I + H_{1})} - I) \bar{e (t)} \\ \leq (β_{1} - 1) e^{T} (t_{1}) \bar{e (t_{1})} \\ = (β_{1} - 1) V (t_{1}) \\ \leq 0 . \end{array}

where β₁ is the largest eigenvalue of matrix

{(I + H_{1})}^{T} \bar{I + H_{1}}

Besides, at impulsive point t = t₁, the Lyapunov function (28) also satisfies

\begin{array}{l} V (t_{1}^{+}) & = e^{T} (t_{1}^{+}) \bar{e (t_{1}^{+})} \\ = {((I + H_{1}) e (t_{1}))}^{T} \bar{(I + H_{1}) e (t_{1})} \\ = e^{T} (t_{1}) {((I + H_{1})}^{T} \bar{(I + H_{1})}) \bar{e (t_{1})} \\ \leq β_{1} e^{T} (t_{1}) \bar{e (t_{1})} \\ \leq β_{1} V (t_{1}) \\ \leq β_{1} V (t_{0}) E_{α} {(σ (t_{1} - t_{0})}^{α}) . \end{array}

(32)

Then applying the inequality (31) on interval t ∈ [t₁, t₂)

\begin{array}{l} V (t) & \leq V (t_{1}^{+}) E_{α} {(σ (t - t_{1})}^{α}) \\ \leq β_{1} V (t_{0}) E_{α} {(σ (t_{1} - t_{0})}^{α}) E_{α} {(σ (t - t_{1})}^{α}) . \end{array}

Applying the inequality (32) at impulsive point t = t₂

\begin{array}{l} V (t_{2}^{+}) & \leq β_{2} V (t_{2}) \\ \leq β_{2} β_{1} V (t_{0}) E_{α} {(σ (t_{1} - t_{0})}^{α}) E_{α} {(σ (t_{2} - t_{1})}^{α}) . \end{array}

In the same way, when t ∈ [t_m, t_m+1)

\begin{array}{l} V (t) = V (t_{m}^{+}) E_{α} {(σ (t - t_{m})}^{α}) \\ \leq β_{m - 1} \dots β_{2} β_{1} V (t_{0}) E_{α} {(σ (t_{1} - t_{0})}^{α}) E_{α} {(σ (t_{2} - t_{1})}^{α}) \\ \dots E_{α} {(σ (t_{m} - t_{m - 1})}^{α}) E_{α} {(σ (t - t_{m})}^{α}) \\ = V (t_{0}) β_{1} E_{α} {(σ (t_{1} - t_{0})}^{α}) β_{2} E_{α} {(σ (t_{2} - t_{1})}^{α}) \\ \dots β_{m - 1} E_{α} {(σ (t_{m} - t_{m - 1})}^{α}) E_{α} {(σ (t - t_{m})}^{α}) . \end{array}

(33)

Following the condition (A2) in the theorem, there exists a positive constant δ > 1 for m = 0, 1, 2, ⋯ such that

β_{m} E_{α} {(σ (t_{m + 1} - t_{m})}^{α}) < \frac{1}{δ} .

Then inequality (33) satisfies

\begin{array}{l} V (t) & \leq V (t_{0}) \frac{1}{δ^{m - 1}} E_{α} {(σ (t - t_{m})}^{α}) \\ \leq V (t_{0}) \frac{1}{δ^{m - 1}} E_{α} {(σ (t_{m + 1} - t_{m})}^{α}) . \end{array}

(34)

Hence, V(t) → 0 as m → ∞. The error system of fractional-order quaternion-valued neural network (26) is asymptotically stable, which means the synchronization of two fractional-order quaternion-valued neural networks with different structures (equations (9) and (10)) is realized.

In addition, we need to explain that condition (A2) is tenable. In condition (A2), the β_m and σ can be controlled by designing an appropriate control gain matrix and impulse matrix. As introduced in equation (3), the Mittag-Leffler function is a function series. And let $ℵ = σ {(t_{m + 1} - t_{m})}^{α}$ , then

\begin{array}{l} β_{m} E_{α} (σ {(t_{m + 1} - t_{m})}^{α}) \\ = β_{m} \sum_{p = 0}^{+ \infty} \frac{{(σ {(t_{m + 1} - t_{m})}^{α})}^{p}}{Γ (p α + 1)} \\ \begin{array}{l} = β_{m} \sum_{p = 0}^{+ \infty} \frac{ℵ^{p}}{Γ (p α + 1)} \\ = β_{m} (\frac{1}{Γ (1)} + \frac{ℵ}{Γ (α + 1)} + \frac{ℵ^{2}}{Γ (2 α + 1)} + \frac{ℵ^{3}}{Γ (3 α + 1)} \\ + \frac{ℵ^{4}}{Γ (4 α + 1)} + \frac{ℵ^{5}}{Γ (5 α + 1)} + \dots) . \end{array} \end{array}

In condition (A1), we designed the control gain matrix and impulse matrix such that σ < 0 and 0 < β_m < 1. And the impulsive interval Δt_m = t_m+1 − t_m is usually small; hence, we can design a control gain matrix such that the value of ℵ is easy to locate in the interval (−1, 0), then 0 < 1 + ℵ < 1 and 0 < ℵ² < 1. Besides, the gamma function Γ(α) is monotonically increasing; therefore

\begin{array}{l} E_{α} (ℵ) & < \frac{1 + ℵ}{Γ (1)} + \frac{ℵ^{2} (1 + ℵ)}{Γ (2 α + 1)} + \frac{ℵ^{4} (1 + z)}{Γ (4 α + 1)} + \dots \\ < (1 + ℵ) (1 + ℵ^{2} + ℵ^{4} + \dots) \\ = \lim_{n \to + \infty} \frac{(1 + ℵ) (1 - ℵ^{2 n + 2})}{1 - ℵ^{2}} \\ = \frac{1}{1 - ℵ} < 1 . \end{array}

Then there exists a positive constant δ > 1 such that condition (A2) holds.

3.2. Synchronization with unknown parameters

In Theorem 1, we mainly discuss the case that the self-feedback coefficients C, D and connection matrices A, B, W, V are known. However, in many neural networks, these connection matrices are unknown, increasing the difficulty in achieving synchronization and stability analysis. The fractional-order quaternion-valued neural networks (9) can be rewritten as follows.

{}_{t_{0}}^{C}{D_{t}^{α}} x (t) = - Φ (x (t)) ξ_{D} + F_{A} (x, t) ξ_{A} + F_{B} (x, t - τ) ξ_{B},

(35)

where

ξ_{D} \in R^{n \times 1}

ξ_{A}, ξ_{B} \in R^{n^{2} \times 1}

denote the unknown parameters,

Φ (x (t)) \in R^{n \times n}

is a diagonal matrix corresponding to x(t), and

F_{A} (x, t), F_{B} (x, t - τ) \in R^{n \times n^{2}}

are two matrices corresponding to nonlinear part f(x(t)) and time-delay part f(x(t − τ)).

Compared with equations (9) and (35), we can easily get the following rules.

\begin{array}{l} Φ (x (t)) ξ_{D} = D x (t), \\ F_{A} (x, t) ξ_{A} = A f (x (t)), \\ F_{B} (x, t - τ) ξ_{B} = B f (x (t - τ)) . \end{array}

Then, the impulsive controlled response fractional-order quaternion-valued neural networks (10) with different structures are

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} y (t) = - Ψ (y (t)) ζ_{D} + G_{W} (y, t) ζ_{W} \\ + G_{V} (y, t - τ) ζ_{V} + u (t), t \neq t_{m} Δ y (t_{m}) \\ = H_{m} y (t_{m}), m = 1,2, \dots, \end{array}

(36)

where

ζ_{D} \in R^{n \times 1}

ζ_{W}, ζ_{V} \in R^{n^{2} \times 1}

denote the unknown parameters,

Φ (y (t)) \in R^{n \times n}

is a diagonal matrix corresponding to y(t), and

G_{w} (y, t), G_{V} (y, t - τ) \in R^{n \times n^{2}}

are two matrices corresponding to nonlinear part g(y(t)) and time-delay part g(y(t − τ)).

Similarly, we can also get the equation by combining equations (10) and (36).

\begin{array}{l} Ψ (y (t)) ζ_{C} & = C y (t), \\ G_{W} (y, t) ζ_{W} & = W g (y (t)), \\ G_{V} (y, t - τ) ζ_{B} & = V g (y (t - τ)) \end{array}

Then, the controller u(t) is designed as

\begin{array}{l} u (t) & = - K e (t) + Ψ (y (t)) \hat{ζ_{D}} - Φ (x (t)) \hat{ξ_{C}} - G_{W} (y, t) \hat{ζ_{W}} \\ - G_{V} (y, t - τ) \hat{ζ_{V}} + F_{A} (x, t) \hat{ξ_{A}} + F_{B} (x, t - τ) \hat{ξ_{B}}, \end{array}

(37)

where

K \in R^{n \times n}

is the designed control gain matrix and

\hat{ζ_{D}}, \hat{ζ_{W}}, \hat{ζ_{V}}, \hat{ξ_{C}}, \hat{ξ_{A}}, \hat{ξ_{B}}

are the approximation vectors corresponding to ζ_D, ζ_W, ζ_V, ξ_C, ξ_A, ξ_B. Generally, we can evaluate them via the following adaptive laws.

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{D}} = {\begin{cases} - {\bar{Ψ (y (t))}}^{T} e (t), & t \neq t_{m} \\ 0, & t = t_{m} \end{cases}, \\ {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{W}} = {\begin{cases} {\bar{G_{W} (y, t)}}^{T} e (t), & t \neq t_{m} \\ 0, & t = t_{m} \end{cases}, \\ {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{V}} = {\begin{cases} {\bar{G_{V} (y, t - τ)}}^{T} e (t), & t \neq t_{m} \\ 0, & t = t_{m} \end{cases}, \\ {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{C}} = {\begin{cases} {\bar{Φ (x (t))}}^{T} e (t), & t \neq t_{m} \\ 0, & t = t_{m} \end{cases}, \\ {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{A}} = {\begin{cases} - {\bar{F_{A} (x, t)}}^{T} e (t), & t \neq t_{m} \\ 0, & t = t_{m} \end{cases}, \\ {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{B}} = {\begin{cases} - {\bar{F_{B} (x, t - τ)}}^{T} e (t), & t \neq t_{m} \\ 0, & t = t_{m} \end{cases} . \end{array}

(38)

Then we can get the theorem of fractional-order quaternion-valued neural networks with random parameters and different structures.

Theorem 2

The controller and adaptive law are designed as equations (37) and (38). Let λ be the largest eigenvalue of matrix $\bar{K} + K$ .

There exists a positive constant δ > 1 for m = 0, 1, 2, ⋯, such that

β_{m} E_{α} {(λ (t_{m + 1} - t_{m})}^{α}) < \frac{1}{δ} .

(39)

Then synchronization of fractional-order quaternion-valued neural networks with random parameters and different structures (equations (36) and (37)) is realized.

Proof

Constructing the following Lyapunov function

\begin{array}{l} V (t) & = e^{T} (t) \bar{e (t)} + {(\hat{ζ_{D}} - ζ_{D})}^{T} \bar{\hat{ζ_{D}} - ζ_{D}} \\ + {(\hat{ζ_{W}} - ζ_{W})}^{T} \bar{\hat{ζ_{W}} - ζ_{W}} + {(\hat{ζ_{V}} - ζ_{V})}^{T} \bar{\hat{ζ_{V}} - ζ_{V}} \\ + {(\hat{ξ_{C}} - ξ_{C})}^{T} \bar{\hat{ξ_{C}} - ξ_{C}} + {(\hat{ξ_{A}} - ξ_{A})}^{T} \bar{\hat{ξ_{A}} - ξ_{A}} \\ + {(\hat{ξ_{B}} - ξ_{B})}^{T} \bar{\hat{ξ_{B}} - ξ_{B}} . \end{array}

(40)

Applying Lemma 3, the Caputo derivative of Lyapunov function (40) satisfies

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} e^{T} (t) \bar{e (t)} \leq e^{T} (t) \bar{{}_{t_{0}}^{C}{D_{t}^{α}} e (t)} + {\bar{e (t)}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} e (t) \leq e^{T} (t) \bar{- K e (t)} + {\bar{e (t)}}^{T} (- K e (t)) \\ + e^{T} (t) \bar{Ψ (y (t)) (\hat{ζ_{D}} - ζ_{D}) - G_{W} (y, t) (\hat{ζ_{W}} - ζ_{W})} \\ + e^{T} (t) \bar{G_{V} (y, t - τ) (- \hat{ζ_{V}} - ζ_{V}) - Φ (y (t)) (\hat{ξ_{C}} - ξ_{C})} \\ + e^{T} (t) \bar{F_{A} (x, t) (\hat{ξ_{A}} - ξ_{A}) - F_{B} (x, t - τ) (\hat{ξ_{B}} - ξ_{B})} \\ + {\bar{e (t)}}^{T} [Ψ (y (t)) (\hat{ζ_{D}} - ζ_{D}) - G_{W} (y, t) (\hat{ζ_{W}} - ζ_{W}) - G_{V} (y, t - τ) (\hat{ζ_{V} - Φ (y (t)) (\hat{ξ_{C}} - ξ_{C})} - ζ_{V})] \\ + {\bar{e (t)}}^{T} [F_{A} (x, t) (\hat{ξ_{A}} - ξ_{A}) - F_{B} (x, t - τ) (\hat{ξ_{B}} - ξ_{B})] = e^{T} (t) \bar{- K e (t)} + {\bar{e (t)}}^{T} (- K e (t)) \\ + e^{T} (t) \bar{Ψ (y (t)) (\hat{ζ_{D}} - ζ_{D})} + {\bar{e (t)}}^{T} Ψ (y (t)) (\hat{ζ_{D}} - ζ_{D}) \\ - e^{T} (t) \bar{G_{W} (y, t) (\hat{ζ_{W}} - ζ_{W})} - {\bar{e (t)}}^{T} G_{W} (y, t) (\hat{ζ_{W}} - ζ_{W}) \\ - e^{T} (t) \bar{G_{V} (y, t - τ) (\hat{ζ_{V}} - ζ_{V})} - {\bar{e (t)}}^{T} G_{V} (y, t - τ) (\hat{ζ_{V}} - ζ_{V}) \\ - e^{T} (t) \bar{Φ (x (t)) (\hat{ξ_{C}} - ξ_{C})} + {\bar{e (t)}}^{T} Φ (x (t) (\hat{ξ_{C}} - ξ_{C}) \\ + e^{T} (t) \bar{F_{A} (x, t) (\hat{ξ_{A}} - ξ_{A})} - {\bar{e (t)}}^{T} F_{A} (x, t) (\hat{ξ_{A}} - ξ_{A}) + e^{T} (t) \bar{F_{B} (x, t - τ) (\hat{ξ_{B}} - ξ_{B})} - {\bar{e (t)}}^{T} F_{B} (x, t - τ) (\hat{ξ_{B}} - ξ_{B}) . \end{array}

(41)

Combining Lemma 2 and adaptive law, we can get the following rule for the self-feedback matrix of the response neural network based on the nature of Caputo calculus.

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} {(\hat{ζ_{D}} - ζ_{D})}^{T} \bar{\hat{ζ_{D}} - ζ_{D}} \\ \leq {(\hat{ζ_{D}} - ζ_{D})}^{T} \bar{{}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{D}}} + {\bar{\hat{ζ_{D}} - ζ_{D}}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{D}} \\ = {(\hat{ζ_{D}} - ζ_{D})}^{T} Ψ^{T} (y (t)) \bar{- e (t)} \\ + {\bar{\hat{ζ_{D}} - ζ_{D}}}^{T} (- \bar{Ψ^{T} (y (t))} e (t)) . \end{array}

(42)

Similarly, we can also get the analyses for other unknown parameters. For the connection matrix of the response neural network

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} {(\hat{ζ_{W}} - ζ_{W})}^{T} \bar{\hat{ζ_{W}} - ζ_{W}} \leq {(\hat{ζ_{W}} - ζ_{W})}^{T} \bar{{}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{W}}} + {\bar{\hat{ζ_{W}} - ζ_{W}}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{W}} \\ = {(\hat{ζ_{W}} - ζ_{W})}^{T} G_{W}^{T} (y, t) \bar{e (t)} + {\bar{\hat{ζ_{W}} - ζ_{W}}}^{T} \bar{G_{W}^{T} (y, t)} e (t) . \end{array}

(43)

For the connection matrix of the response neural network (time-delay part)

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} {(\hat{ζ_{V}} - ζ_{V})}^{T} \bar{\hat{ζ_{V}} - ζ_{V}} \leq {(\hat{ζ_{V}} - ζ_{V})}^{T} \bar{{}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{V}}} + {\bar{\hat{ζ_{V}} - ζ_{V}}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ζ_{V}} \\ = {(\hat{ζ_{V}} - ζ_{V})}^{T} G_{V}^{T} (y, t - τ) \bar{e (t)} + {\bar{\hat{ζ_{V}} - ζ_{V}}}^{T} \bar{G_{V}^{T} (y, t - τ)} e (t) . \end{array}

(44)

For the self-feedback matrix of the drive neural network

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} {(\hat{ξ_{C}} - ξ_{C})}^{T} \bar{\hat{ξ_{C}} - ξ_{C}} \leq {(\hat{ξ_{C}} - ξ_{C})}^{T} \bar{{}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{C}}} + {\bar{\hat{ξ_{C}} - ξ_{C}}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{C}} \\ = {(\hat{ξ_{C}} - ξ_{C})}^{T} Φ^{T} (x (t)) \bar{e (t)} + {\bar{\hat{ξ_{C}} - ξ_{C}}}^{T} \bar{Φ^{T} (x (t))} e (t) . \end{array}

(45)

For the connection matrix of the drive neural network

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} {(\hat{ξ_{A}} - ξ_{A})}^{T} \bar{\hat{ξ_{A}} - ξ_{A}} \leq {(\hat{ξ_{A}} - ξ_{A})}^{T} \bar{{}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{A}}} + {\bar{\hat{ξ_{A}} - ξ_{A}}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{A}} \\ = {(\hat{ξ_{A}} - ξ_{A})}^{T} F_{A}^{T} (x, t) \bar{- e (t)} + {\bar{\hat{ξ_{A}} - ξ_{A}}}^{T} \bar{F_{A}^{T} (x, t)} (- e (t)) . \end{array}

(46)

For the connection matrix of the drive neural network (time-delay part)

\begin{array}{l} {}_{t_{0}}^{C}{D_{t}^{α}} {(\hat{ξ_{B}} - ξ_{B})}^{T} \bar{\hat{ξ_{B}} - ξ_{B}} \leq {(\hat{ξ_{B}} - ξ_{B})}^{T} \bar{{}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{B}}} + {\bar{\hat{ξ_{B}} - ξ_{B}}}^{T} {}_{t_{0}}^{C}{D_{t}^{α}} \hat{ξ_{B}} \\ = {(\hat{ξ_{B}} - ξ_{B})}^{T} F_{B}^{T} (x, t - τ) \bar{- e (t)} + {\bar{\hat{ξ_{B}} - ξ_{B}}}^{T} \bar{F_{B}^{T} (x, t - τ)} (- e (t)) . \end{array}

(47)

Based on the characteristics of matrix multiplication, we can easily get

{(e^{T} (t) \bar{Ψ (y (t)) (\hat{ζ_{D}} - ζ_{D})})}^{T} = {\bar{\hat{ζ_{D}} - ζ_{D}}}^{T} \bar{Ψ^{T} (y (t))} e (t)

Similarly, other unknown parameters also satisfy the same conclusion.

Combining equations (41)–(47), we can get the Caputo derivative of Lyapunov function (40) when t ∈ [t₀, t₁).

{}_{t_{0}}^{C}{D_{t}^{α}} V (t) \leq - e^{T} (t) (\bar{K} + K) \bar{e (t)} \leq - λ e^{T} (t) \bar{e (t)} .

(48)

Based on Lemma 2, there exists a Mittag-Leffler function such that equation (48) satisfies

V (t) \leq V (t_{0}) E_{α} [λ {(t - t_{0})}^{α}] .

(49)

Therefore, the error system is Mittag-Leffler stable from Definition 3 on the interval t ∈ [t₀, t₁). Similarly, we can obtain that the error system is Mittag-Leffler stable within each impulsive interval. However, we need to consider the overall stability including impulsive points.

Besides, at impulsive point t = t₁, the Lyapunov function (41) also satisfies

\begin{array}{l} V (t_{1}^{+}) & = e^{T} (t_{1}^{+}) \bar{e (t_{1}^{+})} \\ = e^{T} (t_{1}) {((I + H_{1})}^{T} \bar{(I + H_{1})}) \bar{e (t_{1})} \\ \leq β_{1} e^{T} (t_{1}) \bar{e (t_{1})} \\ \leq β_{1} V (t_{1}) \\ \leq β_{1} V (t_{0}) E_{λ} {(σ (t_{1} - t_{0})}^{α}) . \end{array}

(50)

In the same way, when t ∈ [t_m, t_m+1)

\begin{array}{l} V (t) = V (t_{m}^{+}) E_{α} {(λ (t - t_{m})}^{α}) \\ \leq β_{m - 1} \dots β_{2} β_{1} V (t_{0}) E_{α} {(λ (t_{1} - t_{0})}^{α}) E_{α} {(λ (t_{2} - t_{1})}^{α}) \\ \dots E_{α} {(λ (t_{m} - t_{m - 1})}^{α}) E_{α} {(λ (t - t_{m})}^{α}) \\ = V (t_{0}) β_{1} E_{α} {(λ (t_{1} - t_{0})}^{α}) β_{2} E_{α} {(λ (t_{2} - t_{1})}^{α}) \\ \dots β_{m - 1} E_{α} {(λ (t_{m} - t_{m - 1})}^{α}) E_{α} {(λ (t - t_{m})}^{α}) . \end{array}

(51)

Following the condition (A2) in the theorem, there exists a positive constant δ > 1 for m = 0, 1, 2, ⋯ such that

β_{m} E_{α} {(λ (t_{m + 1} - t_{m})}^{α}) < \frac{1}{δ} .

Then inequality (51) satisfies

\begin{array}{l} V (t) & \leq V (t_{0}) \frac{1}{δ^{m - 1}} E_{α} {(λ (t - t_{m})}^{α}) \\ \leq V (t_{0}) \frac{1}{δ^{m - 1}} E_{α} {(λ (t_{m + 1} - t_{m})}^{α}) \end{array}

(52)

Hence, V(t) → 0 as m → ∞. The error system of fractional-order quaternion-valued neural network is asymptotically stable. The synchronization of two fractional-order quaternion-valued neural networks with unknown parameters and different structures (equations (36) and (37)) is realized.

4. Numerical simulation

4.1. Numerical result for FQVNNs with known parameters

Consider the following FQVNNs as the drive system:

{}_{t_{0}}^{C}{D_{t}^{α}} x (t) = - C x (t) + A f (x (t)) + B f (x (t - τ)),

(53)

where α = 0.9 is the fractional order,

x (t) = {(x_{1} (t), x_{2} (t))}^{T} \in Q^{2}

, and

x_{p} (t) = x_{p}^{R} (t) + i x_{p}^{I} (t) + j x_{p}^{J} (t) + k x_{p}^{K} (t)

with

x_{p}^{R} (t), x_{p}^{I} (t), x_{p}^{J} (t), x_{p}^{K} (t) \in R

for p = 1, 2,

f (x (t)) = (f_{1} (x_{1} (t)), f_{2} (x_{2} (t))) \in Q^{2}

with

f_{p} (x_{p} (t)) = \tanh (x_{p} (t)) = \tanh (x_{p}^{R} (t)) + \tanh (x_{p}^{I} (t)) i + \tanh (x_{p}^{J} (t)) j + \tanh (x_{p}^{K} (t)) k

. And matrix C = diag(1, 1) and the connection weight matrix A, B are

(\begin{array}{l} 2.1 + 2.1 i + 2.1 j + 2 k, & - 0.1 - 0.1 i - 0.1 j - 0.1 k \\ - 2.1 - 3 i - 2.1 j - 3 k, & 1 + i + j + k \end{array})

and

(\begin{array}{l} - 1.4 - 1.5 i - 1.4 j - 1.4 k, & - 0.1 - 0.2 i - 0.1 j - 0.2 k \\ - 0.3 - 0.2 i - 0.3 j - 0.2 k, & 2.4 + 2.5 i + 2.4 j + 2.5 k \end{array})

The FQVNNs equation (53) shows chaotic attractors with the initial value x(0) = [−0.2 + 0.2i − 0.2j + 0.2k, 1 + 0.2i + j + 0.2k] (Lin et al., 2019).

Then we choose an FQVNN with different structures as the response system with an impulsive controller, whose structure is

{}_{t_{0}}^{C}{D_{t}^{α}} D_{t}^{α} y (t) = - D x (t) + W g (x (t)) + V g (x (t - τ)) + u (t), t \neq t_{m} Δ y (t_{m}) = H_{m} y (t_{m}),

(54)

where q has the same fractional order as equation (53),

y (t) = {(y_{1} (t), y_{2} (t))}^{T} \in Q^{2}

, and for p = 1, 2.

y_{p} (t) = y_{p}^{R} (t) + i y_{p}^{I} (t) + j y_{p}^{J} (t) + k y_{p}^{K} (t)

with each parts satisfies

y_{p}^{R} (t), y_{p}^{I} (t), y_{p}^{J} (t), y_{p}^{K} (t) \in R

. The nonlinear activation function should also satisfy

g (y (t)) = (h_{1} (y_{1} (t)), h_{2} (y_{2} (t))) \in Q^{2}

with

g_{q} (y_{p} (t)) = sigmoid (y_{p} (t)) = sigmoid (y_{p}^{R} (t)) + sigmoid (y_{p}^{I} (t)) i + sigmoid (y_{p}^{J} (t)) j + sigmoid (y_{p}^{K} (t)) k

. And matrix C = diag(2, 2) and the connection weight matrix W, V are

(\begin{array}{l} 0.4 - 0.5 i + 0.4 j - 0.5 k & - 0.9 + 0.2 i - 0.8 j + 0.1 k \\ 0.6 + i - 0.6 j + 1 k & 0.05 - 0.5 i + 0.05 j - 0.5 k \end{array}),

(\begin{array}{l} 0.7 i + 0.7 k & 0.5 - 0.4 i + 0.5 j - 0.4 k \\ 0.4 + 0.6 i + 0.4 j + 0.6 k & - 0.5 - 1 i - 0.5 j + 0.1 k \end{array}) .

4.1.1. Conditions for achieving synchronization

For Theorem 1, we choose [1.2 + 2.2i − 3.2j + 1.7, − 1 − 3.2i + 1.9j + 2.2] as the initial value of equation (54). Then we realize the synchronization between equations (53) and (54) with the impulsive matrix H_m = diag(0.2, 0.2) and the control gain matrix as K = diag(15, 15), whose result is shown in Figures 1 and 2.

4.1.2. Synchronization of another example

If the nonlinear activation function in equation (53) is another function, so as the sin(·), we can also reach synchronization via Theorem 1, as illustrated in Figure 3.

4.2. Numerical result for FQVNNs with unknown parameters

In Theorem 2, we apply the adaptive impulsive methods to realize synchronization between two FQVNNs with unknown parameters. Then, we also choose the FQVNNs as equations (53) and (54). However, the parameter matrices A, B and W, V corresponding to the nonlinear activation function in these two equations are unknown, which are evaluated via the adaptive law (38).

We use the same initial value as the previous experiment. The impulsive matrix is set to H_m = diag(−0.2, − 0.2) and the impulsive interval is 60 steps. The error of synchronization can also realize stability, as shown in Figures 4 –6.

4.3. Detailed analysis of synchronization conditions

4.3.1. Impact of the impulsive matrix on synchronization

As shown in Theorem 1, we give two conditions corresponding to the control gain matrix and impulsive matrix. First, we fix the control gain matrix to analyze the impact of the impulsive matrix on synchronization. The largest eigenvalue β_m of matrix ${(I + H_{m})}^{T} \bar{I + H_{m}}$ satisfies β_m < 1. Then we analyze three situations: H_m = diag(−0.2, − 0.2), H_m = diag(0.2, 0.2), and H_m = diag(1.2, 1.2), as shown in Figures 7 and 8.

When H_m is greater than 1, the error system cannot reach stable, as shown by the green line in Figures 7 and 8. It can also be seen from the red and blue curves in Figures 7 and 8 that the smaller the matrix value, the shorter the time for the error system to become stable.

4.3.2. Impact of the control gain matrix on synchronization

Then we set the impulsive matrix H_m = diag(−0.2, − 0.2) to analyze the control gain matrix with K = diag(5, 5), K = diag(15, 15), and K = diag(25, 25), whose result is shown in Figure 9. In fact, the error curve with the control gain matrix of 5 or 15 does not tend to be stable. This shows the necessity of the control gain matrix.

4.3.3. Impact of the impulsive interval on synchronization

In the method of impulsive synchronization, the impulsive interval is also very important. In this experiment, we take the step size of numerical simulation as the pulse interval, namely, 60 steps, 120 steps, and 180 steps, as shown in Figures 10 and 11. The longer the impulsive interval is, the less stable the error is, as shown in the green and blue curves in Figures 10 and 11.

5. Conclusion

In this paper, we address the Mittag-Leffler stability between two fractional-order quaternion-valued neural networks (FQVNNs) with different structures via adaptive impulsive synchronization. Based on the Mittag-Leffler stability and fractional-order Lyapunov direct method, we designed an efficient impulsive control scheme and proposed a sufficient condition to ensure the global Mittag-Leffler synchronization of FQVNNs. We also analyzed the situation when the self-feedback matrix and connection matrix of quaternion-valued fractional-order neural networks with different structures are unknown and designed adaptive impulsive controllers to achieve synchronization. Finally, numerical examples are also presented to manifest the feasibility and validity of the obtained results. We analyzed the influence of the impulsive matrix, impulsive interval, and control gain matrix on synchronization in detail.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest for the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Natural Science Starting Project of SWPU (No. 2022QHZ023), the Sichuan Scientific Innovation Fund (No. 2022JDRC0009), the Sichuan Provincial Department of Science and Technology Project (No. 2022NSFSC0283), and Key Laboratory of Internet Natural Language Intelligent Processing in Sichuan Provincial Higher Education Institutions (No. INLP202202).

ORCID iDs

Xingpeng Zhang

Dong He

References

Aguila-Camacho

Duarte-Mermoud

Gallegos

(2014) Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation 19: 2951–2957.

Al-Sharman

Zweiri

Jaradat

MAK

, et al. (2020) Deep-learning based neural network training for state estimation enhancement: application to attitude estimation. IEEE Transactions on Instrumentation and Measurement 69(1): 24–34.

Bao

Park

Cao

(2016) Synchronization of fractional-order complex-valued neural networks with time delay. Neural Networks: The Official Journal of the International Neural Network Society 81: 16–28.

Cao

(2022) Synchronization of multiple neural networks with reaction–diffusion terms under cyber–physical attacks. Knowledge-Based Systems 239: 107939.

Cao

Guo

, et al. (2020) Global exponential synchronization of delayed memristive neural networks with reaction-diffusion terms. Neural Networks: The Official Journal of the International Neural Network Society 123: 70–81.

Cao

Liu

Zhang

, et al. (2022) Synchronization of multiple reaction-diffusion memristive neural networks with known or unknown parameters and switching topologies. Knowledge-Based Systems 254: 109595.

Chen

Song

, et al. (2017) Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties. Neural Networks 91: 55–65.

Chen

Song

, et al. (2018) Stability analysis of continuous-time and discrete-time quaternion-valued neural networks with linear threshold neurons. IEEE Transactions on Neural Networks and Learning Systems 29: 2769–2781.

Chen

Zeng

, et al. (2020) An adaptive fractional-order BP neural network based on extremal optimization for handwritten digits recognition. Neurocomputing 391: 260–272.

10.

Fan

Huang

, et al. (2019) Aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: an interval matrix and matrix measure combined method. IEEE Transactions on Systems, Man, and Cybernetics: Systems 49(11): 2254–2265.

11.

Gokyildirim

Calgan

Demirtas

(2023) Fractional-order sliding mode control of a 4D memristive chaotic system. Journal of Vibration and Control 0(0): 1–17. doi: 10.1177/10775463231166187.

12.

Wang

(2020) Synchronization for commensurate Riemann-Liouville fractional-order memristor-based neural networks with unknown parameters. Journal of the Franklin Institute 357: 8870–8898.

13.

Zhang

(2016) Impulsive synchronization of fractional order chaotic systems with time-delay. Neurocomputing 216: 39–44.

14.

Zhang

, et al. (2015) Adaptive impulsive synchronization of fractional order chaotic system with uncertain and unknown parameters. Neurocomputing 167: 165–171.

15.

Jiang

Wang

, et al. (2015) Global Mittag–Leffler stability of coupled system of fractional-order differential equations on network. Applied Mathematics and Computation 270: 269–277.

16.

Zhang

, et al. (2020) Global Mittag-Leffler synchronization of fractional-order delayed quaternion-valued neural networks: direct quaternion approach. Applied Mathematics and Computation 373: 125020.

17.

Liu

Zhang

, et al. (2016) Global μ-tability criteria for quaternion-valued neural networks with unbounded time-varying delays. Information Sciences 360: 273–288.

18.

Lin

Chen

, et al. (2019) LMI conditions for some dynamical behaviors of fractional-order quaternion-valued neural networks. Journal of the Franklin Institute 26: 1–29.

19.

Liu

Wang

, et al. (2018) Adaptive fuzzy control for a class of unknown fractional-order neural networks subject to input nonlinearities and dead-zones. Information Sciences 454-455: 30–45.

20.

Liu

Zhang

Lou

, et al. (2018) Stability analysis of quaternion-valued neural networks: decomposition and direct approaches. IEEE Transactions on Neural Networks and Learning Systems 29: 4201–4211.

21.

Liu

Yang

Zhou

, et al. (2019) Stability analysis of fractional delayed equations and its applications on consensus of multi-agent systems. Communications in Nonlinear Science and Numerical Simulation 73: 351–362.

22.

Luo

Wang

(2013) Chaos in the fractional-order complex Lorenz system and its synchronization. Nonlinear Dynamics 71: 241–257.

23.

Mirrezapour

Zare

Hallaji

(2022) A new fractional sliding mode controller based on nonlinear fractional-order proportional integral derivative controller structure to synchronize fractional-order chaotic systems with uncertainty and disturbances. Journal of Vibration and Control 28(7-8): 773–785.

24.

Bao

(2022) Finite-time synchronization for fractional-order quaternion-valued coupled neural networks with saturated impulse. Chaos, Solitons & Fractals 164: 112714.

25.

Podlubny

(1999) Fractional Differential Equations. Cambridge, MA: Academic Press.

26.

Pratap

Raja

Alzabut

, et al. (2020) Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. Mathematical Methods in the Applied Sciences 43: 6223–6253.

27.

Pratap

Raja

Alzabut

, et al. (2020) Finite-time Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with impulses. Neural Processing Letters 51: 1485–1526.

28.

Bao

Cao

(2019) Exponential input-to-state stability of quaternion-valued neural networks with time delay. Applied Mathematics and Computation 358: 382–393.

29.

Singh

Kumar

Das

, et al. (2022) Synchronization of quaternion-valued neural networks with mixed time delays using Lyapunov function method. Neural Processing Letters 54: 785–801.

30.

Sun

Liu

(2020) Adaptive synchronization of fractional-order chaotic neural networks with unknown parameters and time-varying delays. International Journal of Innovative Computing, Information and Control 16: 649–661.

31.

Sun

Liu

(2021) Adaptive synchronization control and parameters identification for chaotic fractional neural networks with time-varying delays. Neural Processing Letters 53: 2729–2745.

32.

Cao

Alsaedi

, et al. (2018) Stability analysis for delayed quaternion-valued neural networks via nonlinear measure approach. Nonlinear Analysis Modelling and Control 23: 361–379.

33.

Udhayakumar

Rakkiyappan

Rihan

, et al. (2022) Projective multi-synchronization of fractional-order complex-valued coupled multi-stable neural networks with impulsive control. Neurocomputing 467: 392–405.

34.

Wang

Cao

Cai

, et al. (2019) Periodicity and finite-time periodic synchronization of discontinuous complex-valued neural networks. Neural Networks: The Official Journal of the International Neural Network Society 119: 249–260.

35.

Wang

Zhang

Yang

, et al. (2019) Adaptive synchronization of time delay chaotic systems with uncertain and unknown parameters via aperiodically intermittent control. International Journal of Control, Automation and Systems 18: 696–707.

36.

Wei

Cao

(2019) Fixed-time synchronization of quaternion-valued memristive neural networks with time delays. Neural Networks: The Official Journal of the International Neural Network Society 113: 1–10.

37.

Xia

Jahanchahi

Mandic

(2015) Quaternion-valued echo state networks. IEEE Transactions on Neural Networks and Learning Systems 26: 663–673.

38.

Liao

, et al. (2021) Impact of leakage delay on bifurcation in fractional-order complex-valued neural networks. Chaos, Solitons & Fractals 142: 110535.

39.

Yang

Song

, et al. (2018) Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons. Neural Networks: The Official Journal of the International Neural Network Society 105: 88–103.

40.

You

Song

Zhao

(2020) Global Mittag-Leffler stability and synchronization of discrete-time fractional-order complex-valued neural networks with time delay. Neural Networks: The Official Journal of the International Neural Network Society 122: 382–394.

41.

Zhang

Zeng

(2021) Synchronization of nonidentical neural networks with unknown parameters and diffusion effects via robust adaptive control techniques. IEEE Transactions on Cybernetics 51(2): 660–672.

42.

Zhang

, et al. (2019) Adaptive synchronization for a class of fractional order time-delay uncertain chaotic systems via fuzzy fractional-order neural network. International Journal of Control, Automation, and Systems 17: 1209–1220.

43.

Zhang

(2019) Adaptive impulsive synchronization for a class of fractional-order complex chaotic systems. Journal of Vibration and Control 25: 1614–1628.

44.

Zhang

Zhao

Sha

, et al. (2021) Finite time synchronization of delayed quaternion valued neural networks with fractional order. Neural Processing Letters 53: 3607–3618.

45.

Zhang

Cao

, et al. (2022) Global projective synchronization in fractional-order quaternion-valued neural networks. Asian Journal of Control 24: 227–236.

46.

Zhou

Zhang

Zeng

(2021) Synchronization of memristive neural networks with unknown parameters via event-triggered adaptive control. Neural Networks: The Official Journal of the International Neural Network Society 139: 255–264.

47.

Zou

Kou

Wang

(2016) Quaternion collaborative and sparse representation with application to color face recognition. IEEE Transactions on Image Processing 25: 3287–3302.