Instability and controllability of linearly coupled oscillators: Eigenvalue analysis

The stability and controllability of synchronous chaos of linearly coupled oscillators are investigated in detail, based on eigenvalue analysis. The complicated coupled problem is reduced to two independent problems: clarifying the unstable regions of a modified single-site system and specifying the eigenvalue distribution of the linear coupling and control matrix; both problems can be easily solved. Local injections (pinnings) are suggested to synchronize chaos. The dependence of the controllability on pinning density and the diffusive and gradient couplings is studied. It is shown that high control efficiency can be achieved in the strong gradient coupling case for certain classes of unstable region distributions.