Invariance and computation of the extended fractal dimension for the attractor of CGL on RR

The main goal of this paper is to analyze the complexity of the asymptotic behavior of dissipative systems. More precisely, we want to explain how we can introduce the notion of extended fractal dimension in the case of infinite dimensional sets. In particular, we study the global attractor associated with the extended dynamical system induced by the complex Ginzburg–Landau equation on the line CGL. Furthermore, we compute and investigate the invariance of these quantities under an infinite type of metrics. As a direct consequence, we found that the attractor is similar in terms of complexity to an L∞(R)L∞(R)-ball in the space of band-limited functions.