Dynamic Transitions in Small World Networks: Approach to Equilibrium

We study the transition to phase synchronization in a model for the spread of infection defined on a small world network. It was shown (Phys. Rev. Lett. {\bf 86} (2001) 2909) that the transition occurs at a finite degree of disorder $p$, unlike equilibrium models where systems behave as random networks even at infinitesimal $p$ in the infinite size limit. We examine this system under variation of a parameter determining the driving rate, and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to $p=0$ in the very slow driving limit, just as in the equilibrium case.Comment: 8 pages, 2 figure

On fractional-order maps and their synchronization

We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map $f(x)=ax$ is stable for $a_c(\alpha)<a<1$ where $\alpha$ is a fractional order parameter and $a_c(\alpha)\approx -\alpha$. For coupled linear fractional maps, we can obtain `normal modes' and reduce the evolution to effectively one-dimensional system. If the eigenvalues are real the stability of the coupled system is dictated by the stability of effectively one-dimensional normal modes. For complex eigenvalues, we obtain a much richer picture. However, in the stable region, the evolution of modulus is dictated by Mittag-Leffler function and the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to synchronized fixed points of fractional nonlinear maps.Comment: 12 pages, 9 figure

Stability and Dynamics of Complex Order Fractional Difference Equations

We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C}$. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of equilibrium points for linear systems. For the one-dimensional case where $A =\lambda \in \mathbb {C}$, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for $\lambda \in \mathbb{R}$, the solutions can be complex and dynamics in one-dimension is richer than the case for $\alpha\in \mathbb{R}$. These results can be extended to $n$-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.Comment: 21 pages, 17 figure