Non-uniform small-gain theorems for systems with unstable invariant sets

We consider the problem of small-gain analysis of asymptotic behavior in interconnected nonlinear dynamic systems. Mathematical models of these systems are allowed to be uncertain and time-varying. In contrast to standard small-gain theorems that require global asymptotic stability of each interacting component in the absence of inputs, we consider interconnections of systems that can be critically stable and have infinite input-output Linfin gains. For this class of systems we derive small-gain conditions specifying state boundedness of the interconnection. The estimates of the domain in which the systempsilas state remains are also provided. Conditions that follow from the main results of our paper are non-uniform in space. That is they hold generally only for a set of initial conditions in the systempsilas state space. We show that under some mild continuity restrictions this set has a non-zero volume, hence such bounded yet potentially globally unstable motions are realizable with a non-zero probability. Proposed results can be used for the design and analysis of intermittent, itinerant and meta-stable dynamics which is the case in the domains of control of chemical kinetics, biological and complex physical systems, and non-linear optimization

Results for a network of Hindmarsh-Rose neurons.

<p>(a) Expected value of the local mean field of the node against the node degree . The error bar indicates the variance () of . (b) Black points indicate the value of and for <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048118#pone.0048118.e248" target="_blank">Eq. (13)</a> to present a stable periodic orbit (no positive Lyapunov exponents). The maximal values of the periodic orbits obtained from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048118#pone.0048118.e248" target="_blank">Eq. (13)</a> is shown in the bifurcation diagram in (c) considering and . (d) The CAS pattern for a neuron with degree  = 25 (with and ). In the inset, the same CAS pattern of the neuron and some sampled points of the trajectory for the neuron and another neuron with degree . (e) The difference between the first coordinates of the trajectories of neurons and , with a time-lag of . (f) Phase difference between the phases of the trajectories for neurons and .</p

Results for a network of coupled maps.

<p>(a) Expected value of the local mean field of the node against the node degree . The error bar indicates the variance () of . (b) A bifurcation diagram of the CAS pattern [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048118#pone.0048118.e060" target="_blank">Eq. (6)</a>] considering . (c) Probability density function of the trajectory of a node with degree  = 80 (therefore, , ). (d) A return plot considering two nodes ( and ) with the same degree 80.</p