HOPF BIFURCATION OF A FRACTIONAL-ORDER OCTONION-VALUED NEURAL NETWORKS WITH TIME DELAYS

. In this paper, the hopf bifurcation of a fractional-order octonion-valued neural networks with time delays is investigated. With this constructed model all the parameters would belong to the normed division algebra of octo-nians. Because of the non-commutativity of the octonians, the fractional-order octonian-valued neural networks can be decomposed into four-dimensional real-valued neural networks. Furthermore, the conditions for the occurrence of Hopf bifurcation for the considered model are ﬁrstly given by taking time delay as a bifurcation parameter. Also we investigate their bifurcation when the sys- tem loses its stability. Finally, we give one numerical simulation to verify the eﬀectiveness of the our proposed method.


1.
Introduction. Fractional-order neural networks has gained interest over the last few years due to their wide range of applications such as electro magnetic waves, visco elastic systems, dielectric polarization and biological systems [4,14,23,27,30]. Fractional calculus is the generalization of ordinary differentiation and integration to arbitrary fractional(non-integer) order. In recent years, fractional calculus has been extensively used to model the system behavior and physical system exhibiting memory and hereditory properties [28] and hence it gains more dominance than the classical integer-order systems [11,19]. Taking all these facts into account, the incorporation of the memory term in the network model is an important improvement. Recently the dynamic behaviors of fractional-order neural networks have gained research interest and there were several literatures discussing the stability of the fractional-order neural networks which can be found in [3,12,16,23].
In recent decades, the multidimensional neural network model becomes more popular, because of their increasing applications in the field of radar imaging, antena design, quantum waves, filtering, communications signal processing, speech synthesis and so on [17,21,22]. The complex-valued neural networks(CVNNs) which is an extension of real-valued neural networks(RVNNs) [25] has received a rapid growth in these few years and has produced significant results in real life problems. Meanwhile, the quaternion-valued neural networks(QVNNs) are another multidimensional networks, which were discussed in [26] and both of theses type of neural networks are special cases of Clifford-valued neural networks. Octonionvalued neural networks(OVNNs) are generalization of both CVNNs and QVNNs. Moreover, the OVNNs have an important property of being a normed division algebra, which means that a norm and a multiplicative inverse can be defined on it [5]. Based on these facts, it is imperative that the octonions are introduced in the neural networks and it should be noted that results on feed forward octonions were reported in [15] and they have comprehensive applications in signal and high-dimensional data processing.
The dynamic properties of OVNNs were intensively discussed in recent days in which the global exponential stability of neutral-type OVNNs with time varying delays were studied in [21] and in [26] using Cayley-Dickson construction, the octonion numbers were decomposed into their complex components to establish the global exponential stability of OVNNs with delay. which gave sufficient criteria in terms of linear matrix inequalities. In [22], notable delay-dependent criteria in terms of complex-valued linear matrix inequality for global exponential stability of OVNNs with leakage and mixed time delays were investigated. In this case, the activation functions are separated into real and imaginary parts for discussing the problem of Hopf bifurcation, in which the discrete time delay is taken as the bifurcation parameter in [20] and in [7,10] sum of the time delay is taken as the bifurcation parameter. Stability and bifurcation direction are discussed by using the central manifold theorem addressed in [6]. To the best of our knowledge so far, the problem of Hopf bifurcation is mostly concentrated on RVNNs with time delays [1,2,18,29,31] and CVNNs with time delays [8,12,20,24]. Taking all the considerations, in this paper our main aim is to deal with the problem of bifurcation for a two-dimensional OVNNs with time delays. Inspired by the above mentioned reasons, this paper is formulated as follows: 1. Firstly, the fractional-order OVNNs model is transformed into an four ndimensional fractional-order CVNNs for the activation functions considered here, the origin is taken as the equilibrium point and thus the decomposed fractional-order CVNNs model has been linearized with zero as the equilibrium point. The linearized system undergoes Laplase transformation through which we can be able to formulate the characteristic matrix of the fractional-order OVNNs. 2. Using, certain results and conditions available, the bifurcation analysis of the eigenvalues of the obtained characteristic equation. That is, the global asymptotically stability of the proposed model could be guaranteed when all the roots of the characteristic equation have negative real parts. 3. When the system has unstable position with respect to zero equilibrium, the Hopf bifurcation will occurs, and in this case the considered time-delay is taken to be a bifurcation parameter along with the corresponding critical frequency of the time-delay. The remaining part of this paper is arranged as follows. In Section 2, some definitions and properties of Caputo-derivative are provided. In Section 3, the model of the octonion-valued network was given with some assumptions, and using these assumptions, the characteristic matrix was found. In Section 4, the conditions of Hopf bifurcation are established. In Section 5, numerical simulations are given to show the effectiveness of our theoretical results. Finally, conclusion is given in Section 6.

Preliminaries.
Definition 2.1. [13] The fractional-order integral of non-integer order ν for an integral function w(t) is defined as follows: where, t ≥ t 0 , ν > 0, Γ(.) is the Gamma function and is defined as Definition 2.2.
[13] The Caputo-fractional derivative of order ν for a function w(t) is defined by where, n − 1 ≤ ν < n, n ∈ Z + , t = t 0 is the initial time.
Especially, when 0 It follows from (3) that taking the Laplace transform: where, W (s) is the Laplace transform of w(t) i.e. W (s) = L {w(t)} if w (k) (0) = 0, k = 1, 2, · · · , n, then L C t0 D ν t w(t) : s = s ν W (s). Property 2.3. C t0 D ν t µ = 0 holds, where µ is any constant. Property 2.4. For any constant α and β, the linearity of Caputo fractional-order derivative gives, where, e 0 is the scalar or real element(it may be identified with the real number 1), if e 0 = 0 then the octonion is said to be pure. An octonion number x ∈ O (set of octonions) can be written in the form where, x l (l = 0, 1, · · · , 7) are real coefficients. The addition and subtraction of octonions is defined by x + y = 7 l=0 (x l + y l )e l , x − y = 7 l=0 (x l − y l )e l and the multiplication of the unit octonions is given in the following table, which describes the results of multiplying the elements in the ith row by the element in the jth column. It follows from table that the octonion multiplication is neither commutative (e i e j = −e j e i = e j e i for if i, j are distinct and non zero)nor associative ((e i e j )e k = −e i (e j e k ) = e i (e j e k ) for if i, j, k are distinct, non zero or e i e j = ±e k ).
The octonion conjugate is denoted as x * and is defined by x l e l .
The norm of the octonion number x is defined by ||x|| = √ xx * = 7 l=0 x 2 l and the inverse of the octonion is defined by where, w p (t) ∈ O is the state of the pth neuron, d p > 0, a ∈ O and b p ∈ O are connection weights for neuron p to p neuron p to p−1 respectively; f p : O → O is the nonlinear octonion-valued activation of pth neuron without time delay; g p : O → O is the nonlinear octonion-valued activation of pth neuron with time delay; τ is the time delay; ∀p = 2, 3, · · · , n. Using the Cayley-Dickson construction, the octonion units are written in the following form e 0 = 1, e 1 = i, e 2 = j, e 3 = ij, e 4 = k, e 5 = ik, e 6 = jk, e 7 = ijk.
Since C = z = z 1 + iz 2 |z 1 , z 2 ∈ R, i 2 = −1 is the set of complex number. Now we can write any octonion number x as In this case the product of two octonions is defined as: The neuron activations f and g can be written in the form (M 4 ) : There exists constants γ p , p = 1, 2, . . . n, such that the following conditions hold: By the Cayley-Dickson construction, system (5) can be decomposed into the 4ndimensional complex valued systems: Under the hypothesis (M 2 ) the origin is an equilibrium point of the above equations. By linearizing the systems (6) and (7) at the equilibrium point, we get the following: Then by taking Laplace transform on both sides of the above equations, the characteristic matrix can be obtained as where, Therefore the bifurcation analysis of the system (5) can be absolutely determined by the distribution of the eigenvalues of det(∆(s)).
Lemma 3.1. [9] If all the roots of the characteristic equation det(∆(s)) have negative real parts, then the zero solution of system (5) is Lyapunov globally asymptotically stable. Remark 1. The dynamical behavior of the fractional-order neural networks with time delays were studied in [3]. The occurrence of Hopf bifurcation results with respect to the octonion parameters have not been discussed yet for fractional-order systems.
5. Numerical simulation. In this section, one numerical example is provided to affirm the effectiveness of our proposed results. The simulation results all are based on Adams-Bash forth-Moulton predictor-corrector scheme.
The initial value is selected as w(0) = (0.1 + 0.6i) + j(0.2 + 0.4i) + k(0.5 + 0.5i) + jk(0.1 + 0.3i), v(0) = (0.1 + 0.6i) + j(0.2 + 0.4i) + k(0.5 + 0.5i) + jk(0.1 + 0.3i). By some calculation, it is obtained that the critical frequency ω 0 = 1.48, and the bifurcation parameter τ 0 = 0.15. Hence, from Theorem. 4.1., the zero equilibrium of the system (34) is locally asymptotically stable when τ = 0.14 < τ 0 = 0.15. The system (34) is unstable with the zero equilibrium and the bifurcation occurs when τ 0 = 0.15 < τ = 0.39. 6. Conclusions. In this paper, we investigate the problem of Hopf bifurcation of a fractional-order octonion-valued neural networks with time delay. It was shown that some sufficient criteria for the equilibrium solution of the model has loses its stability and a bifurcation occurs. In this case, the considered time delay is taken to be a bifurcating parameter along with the corresponding critical frequency of the time delay, thus it guarantees the behaviors of the Hopf bifurcation. Finally, one numerical example is given to demonstrate the occurrence of the bifurcation for the considered model.