Stability analysis of fixed point of fractional-order coupled map lattices

Abstract

We study the stability of synchronized fixed-point state for linear fractional-order coupled map lattice (CML). We observe that the eigenvalues of the connectivity matrix determine the stability of this system as in integer-order CML. These eigenvalues can be determined exactly in certain cases. We find exact bounds in a one-dimensional lattice with translationally invariant coupling using the theory of circulant matrices. This can be extended to any finite dimension. Similar analysis can be carried out for the synchronized fixed point of nonlinear coupled fractional maps where eigenvalues of the Jacobian matrix play the same role. The analysis is generic and demonstrates that the eigenvalues of connectivity matrix play a pivotal role in the stability analysis of synchronized fixed point even in coupled fractional maps.