Stability and Hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays

Rakkiyappan, R.; Udhayakumar, K.; Velmurugan, G.; Cao, Jinde

doi:10.1186/s13662-017-1266-3

Cited by 37 publications

(16 citation statements)

References 41 publications

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“…The equilibrium point X * of system (13) is the solution of the equation F(X, X) = 0. The associated linearized system of system (13) at an equilibrium point X * can be written as…”

Section: Consider a General Delayed Fractional-order Systemmentioning

confidence: 99%

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Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

2019

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This paper proposes a delayed fractional-order model of glucose-insulin interaction in the sense of the Caputo fractional derivative with incommensurate orders. This fractional-order model is developed from the first-order model of glucose-insulin interaction. Firstly, we investigate the non-negativity and the boundedness of solutions of the fractional-order model. Secondly, the stability and the bifurcation of the model are studied by separating the associated characteristic equation of the model into its real and imaginary parts and taking a time delay as the bifurcation parameter. The asymptotic stability and the Hopf bifurcation are discussed via the condition of creation of the bifurcation. Furthermore, it is shown that the onset of the bifurcation is related to the fractional orders of the model. Finally, some numerical simulations of the model using the Adam-Bashforth-Moulton predictor corrector scheme are demonstrated to support our obtained theoretical results.

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“…The equilibrium point X * of system (13) is the solution of the equation F(X, X) = 0. The associated linearized system of system (13) at an equilibrium point X * can be written as…”

Section: Consider a General Delayed Fractional-order Systemmentioning

confidence: 99%

“…Recently, fractional-order differential equations, which are used to model complex phenomena, have been extensively applied in many fields [13][14][15][16]. This is because the behaviors of many biological systems have memory or hereditary properties which can be better described using fractional-order derivatives [17].…”

Section: Introductionmentioning

confidence: 99%

Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

2019

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“…Remark 3.2. In this paper, we generalized the assumptions in [22][23][24]35] to a bivariate activation function in A(1) and proposed the generalized linear growth condition for a discontinuous function with two variables in A (2). In addition, we adopted the set-valued map for bivariate activation function defined in [26,27], which generalized the corresponding ones in [22][23][24]35].…”

Section: System Transformation and Some Preliminariesmentioning

confidence: 99%

Generalized stability for discontinuous complex-valued Hopfield neural networks via differential inclusions

et al. 2018

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Some dynamical behaviours of discontinuous complex-valued Hopfield neural networks are discussed in this paper. First, we introduce a method to construct the complex-valued set-valued mapping and define some basic definitions for discontinuous complex-valued differential equations. In addition, Leray-Schauder alternative theorem is used to analyse the equilibrium existence of the networks. Lastly, we present the dynamical behaviours, including global stability and convergence in measure for discontinuous complex-valued neural networks (CVNNs) via differential inclusions. The main contribution of this paper is that we extend previous studies on continuous CVNNs to discontinuous ones. Several simulations are given to substantiate the correction of the proposed results.

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“…On account of this, the research of fractional calculus has fascinated the interest of many scholars in science and engineering [11,12]. In recent years, fractional derivatives have been brought into neural networks, in which fractional-order equations can describe their behaviors [13][14][15]. Thereafter, the dynamics of fractional-order neural networks (FNNs) has been a topic of attention in control and system engineering.…”

Section: Introductionmentioning

confidence: 99%

LMI conditions for some dynamical behaviors of fractional-order quaternion-valued neural networks

Lin

Chen

et al. 2019

Adv Differ Equ

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This paper addresses the issue of three dynamical behaviors including global Mittag-Leffler stability, robust stability and projection synchronization for fractional-order quaternion-valued neural networks (FQVNNs). Some linear matrix inequality conditions for these dynamical behaviors of FQVNNs are given by Lyapunov stability theory, quaternion matrix theory, Homeomorphic mapping theory and fractional differential equation theory. Furthermore, these obtained sufficient conditions for stability and synchronization are superior to those in existing literature. Finally, three examples are given to illustrate the effectiveness of the theoretical results.

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Stability and Hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays

Cited by 37 publications

References 41 publications

Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

Generalized stability for discontinuous complex-valued Hopfield neural networks via differential inclusions

LMI conditions for some dynamical behaviors of fractional-order quaternion-valued neural networks

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