Ju n 20 11 Symmetries , Stability , and Control in Nonlinear Systems and Networks

This paper discusses the interplay of symmetries and stability in the analysis and control of nonlinear dynamical systems and networks. Specifically, it combines standard results on symmetries and equivariance with recent convergence analysis tools based on nonlinear contraction theory and virtual dynamical systems. This synergy between structural properties (symmetries) and convergence properties (contraction) is illustrated in the contexts of network motifs arising e.g. in genetic networks, of invariance to environmental symmetries, and of imposing different patterns of synchrony in a network.

[1]  American Journal of Mathematics , 1886, Nature.

[2]  E. B. Wilson.  PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES. , 1916, Science.

[3]  M. Kelsey.  “Annual Review of Microbiology” , 1946, Nature.

[4]  Journal of Molecular Biology , 1959, Nature.

[5]  G. Dahlquist.  Stability and error bounds in the numerical integration of ordinary differential equations , 1961 .

[6]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[7]  H. Kalmus.  Biological Cybernetics , 1972, Nature.

[8]  D. Signorini,et al.  Neural networks , 1995, The Lancet.

[9]  N. Brown,et al.  Molecular Microbiology , 1998, NATO ASI Series.

[10]  M. Hassoun,et al.  Neural processing letters , 2000 .

[11]  M. Golubitsky,et al.  The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space , 2002 .

[12]  N. Turner.  PLOS Biology , 2004, BMJ : British Medical Journal.

[13]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[14]  Fernando Paganini,et al.  IEEE Transactions on Automatic Control , 2006 .

[15]  Jean-Jacques Slotine,et al.  Neuronal networks and controlled symmetries, a generic framework , 2006, q-bio/0612049.

[16]  Yuri Orlov,et al.  International Journal of Adaptive Control and Signal Processing: Editorial , 2009 .

[17]  M. Arcak.  On spatially uniform behavior in reaction-diffusion PDE and coupled ODE systems , 2009, 0908.2614.

[18]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[19]  L. Christophorou.  Science , 2018, Emerging Dynamics: Science, Energy, Society and Values.

[20]  K. Miller,et al.  45 , 2014, Tao te Ching.