Dynamics and Control of a Delayed Oscillator Network by Pinning Strategy

Large scale networks may exhibit complicated dynamical behaviors, such as instability, bifurcation, and chaos. It is not easy to get which nodes or links need to be controlled for ensuring the desired dynamical behavior. This paper presents a detailed analysis on the dynamics and control of a delayed network by a pinning strategy. The network system is first mapped to a simple system by means of an orthogonal transformation. The control signals are then exerted only on a fraction of nodes in the transformed system. We analyze how to get the desired dynamics of the original controlled system (such as zeros solutions, periodic solutions, quasi-periodic solutions and chaos) via controlling the transformed system. It shows that the solutions of the original system can be obtained by the local dynamical behavior exhibiting in the transformed system. The results show that the proposed pinning control strategy is an effective approach for dynamics control.

[1]  Min Shi,et al.  Stability and Hopf bifurcation control of a fractional-order small world network model , 2013 .

[2]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[3]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[4]  Jie Wu,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2003 .

[5]  Huizhong Yang,et al.  Pinning control of a generalized complex dynamical network model , 2009 .

[6]  A. Goldbeter,et al.  Biochemical Oscillations And Cellular Rhythms: Contents , 1996 .

[7]  Xiaofeng Liao,et al.  Local stability, Hopf and resonant codimension-Two bifurcation in a harmonic oscillator with Two Time delays , 2001, Int. J. Bifurc. Chaos.

[8]  Guanrong Chen,et al.  Pinning control of scale-free dynamical networks , 2002 .

[9]  Yuanyuan Wang,et al.  Numerical time-delayed feedback control in a neural network with delay , 2016 .

[10]  XU XU,et al.  Dynamics of a Two-Dimensional Delayed Small-World Network under Delayed Feedback Control , 2006, Int. J. Bifurc. Chaos.

[11]  Roy,et al.  Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system. , 1992, Physical review letters.

[12]  Niloy Ganguly,et al.  Dynamics On and Of Complex Networks , 2009 .

[13]  Jing Zhou,et al.  Stability, Instability and Bifurcation Modes of a Delayed Small World Network with Excitatory or Inhibitory Short-Cuts , 2016, Int. J. Bifurc. Chaos.

[14]  Duncan J. Watts,et al.  Six Degrees: The Science of a Connected Age , 2003 .

[15]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[16]  Jinde Cao,et al.  Bifurcation control of complex networks model via PD controller , 2016, Neurocomputing.

[17]  Baowen Li,et al.  Thermodynamic stability of small-world oscillator networks: a case study of proteins. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Wenwu Yu,et al.  Synchronization via Pinning Control on General Complex Networks , 2013, SIAM J. Control. Optim..

[19]  P. Cai,et al.  Hopf bifurcation and chaos control in a new chaotic system via hybrid control strategy , 2017 .

[20]  Zhen Chen,et al.  Hopf bifurcation Control for an Internet Congestion Model , 2005, Int. J. Bifurc. Chaos.

[21]  Xu Xu,et al.  Dynamical model and control of a small-world network with memory , 2013 .

[22]  Linying Xiang,et al.  Pinning control of complex dynamical networks with general topology , 2007 .

[23]  Xu Xu,et al.  Collective Dynamics and Control of a 3-d Small-World Network with Time delays , 2012, Int. J. Bifurc. Chaos.

[24]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[25]  Jin Zhou,et al.  Synchronization of coupled harmonic oscillators with local instantaneous interaction , 2012, Autom..

[26]  Albert-László Barabási,et al.  Controllability of complex networks , 2011, Nature.