Dynamics of two-cell systems with discrete delays
- PDF / 684,694 Bytes
- 24 Pages / 439.642 x 666.49 pts Page_size
- 49 Downloads / 226 Views
Dynamics of two-cell systems with discrete delays Z. Dadi1
Received: 6 July 2016 / Accepted: 17 November 2016 © Springer Science+Business Media New York 2017
Abstract We consider the system of delay differential equations (DDE) representing the models containing two cells with time-delayed connections. We investigate global, local stability and the bifurcations of the trivial solution under some generic conditions on the Taylor coefficients of the DDE. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including pitchfork, transcritical and Hopf bifurcation) and Takens-Bogdanov bifurcation as a codimension two bifurcation. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. Finally, we show that the analytical results agree with numerical simulations. Keywords Two-cell system · Delay differential equations · Stability · Liapunov function · Center manifold theory · Transcritical bifurcation · Pitchfork bifurcation · Takens-Bogdanov bifurcation · Hopf bifurcation Mathematics Subject Classification (2010) 34k99 · 34k20 · 34k18
1 Introduction Dynamical systems are used as models in a wide range of applications, see for example [1, 10–16, 28, 31, 34, 35] and so on. In this study, we follow the theory of coupled Communicated by: Karsten Urban Z. Dadi
[email protected]; [email protected] 1
Department of Mathematics, University of Bojnord, Bojnord, Iran
Z. Dadi
cell networks formalized in [19–22, 41]. In this theory, coupled cell network is a directed graph with vertices as individual systems (cells) and directed edges as specific output-input connections between cells. Indeed, this graph abstracts dynamical properties of systems are coupled together. It is important to distinguish between a coupled cell network which is an arrangement of cells and connections, and a coupled cell system which is a particular realization of a coupled cell network as a system of coupled differential equations (in our case typically a set of coupled delay differential equations). It is interesting that neural networks can be considered as an application of coupled cell networks which its theories and applications have been greatly developed after the work of [10] and [31]. It is important to note that every neuron can be considered as a cell in cell systems theory but a cell is not only a neuron, see [10, 19–22, 31, 41]. Introducing a single delay into Hopfield network by [36] showed that delay could destroy stability and cause sustained oscillations. Then many authors ([23, 44] and so on) have been interested in studying delay neural network. Studies in recently decades [4, 5, 14, 15, 27, 31, 39, 40, 42, 46–49] show that dynamics properties (including stable, unstable, oscillatory and chaotic behavior) of dynamical systems with delay have attracted great attention of many researchers.
Data Loading...
