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

TL;DR

This paper investigates the stability of synchronized fixed points in fractional-order coupled map lattices, highlighting the role of eigenvalues of the connectivity matrix and providing exact bounds in specific cases.

Contribution

It extends stability analysis techniques from integer-order to fractional-order coupled map lattices, offering exact eigenvalue bounds and a generic framework for both linear and nonlinear cases.

Findings

Eigenvalues of the connectivity matrix determine stability.

Exact bounds are derived for 1D translationally invariant lattices.

The analysis applies to both linear and nonlinear fractional maps.

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 as for integer-order CML. These eigenvalues can be determined exactly in certain cases. We find exact bounds in 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 stability analysis of synchronized fixed point even in coupled fractional maps.