Transient Dynamics on the Edge of Stability
Here we propose, on the basis of the winnerless competition (WLC) principle, which induces robust transient dynamics in open complex networks, and whose geometrical image in phase space is a heteroclinic sequence, to study the behavior of complex multiagent systems such as brain or ecological food networks that present transient dynamics in a network with active elements whose equilibria are in multidimensional unstable manifolds. In particular, we introduce and study numerically a characteristic of sequential transient dynamics, an uncertainty function that measures the level of nonreproducibility of generalized heteroclinic channels. For a Lotka–Volterra-type model, we describe the behavior of uncertainty functions and its dependence on parameters of the system. We analyze the probability to get a heteroclinic chain with fixed uncertainty and its dependence on the number of saddles of the chain and the number of elements in the network.
We thank Valentin Afraimovich for many constructive discussions and suggestions that helped us to make this piece of research stronger. I.T. acknowledges support by CONACYT. M.R. was supported by US Naval Research by grant ONR N00014-1-0741.
- 8.Hertz, J., Krogh, A., Palmer, R.: Introduction to the Theory of Neural Computation. Perseus Books Group, Cambridge (1991)Google Scholar
- 9.Hirsh, M., Pugh, C., Shub, M.: Invariant Manifolds (Lecture Notes in Mathematics, vol. 583) (1977)Google Scholar
- 11.Lotka, A.: Elements of Physical Biology. Williams & Wilkins company, Baltimore (1925)Google Scholar
- 14.Øksendal, B.: Stochastic Differential Equations: An Introduction With Applications. Springer, Berlin-Heidelberg (2003)Google Scholar
- 18.Rabinovich, M., Huerta, R., Varona, P., Afraimovich, V.: Transient cognitive dynamics, metastability, and decision making. PLoS Comput. Biol. 4(5) (2008)Google Scholar
- 19.Toro, M., Aracil, J.: Qualitative analysis of system dynamics ecological models. Syst. Dyn. Rev. 4(1–2), 56–80 (2006)Google Scholar