Complex systems research is a young branch of Science. It is widely recognized as a markedly interdisciplinary venture covering an unusually wide spectrum of problems.

The research carried out in the Center on this topic follows a fundamental perspective, stemming largely from the close interaction between statistical mechanics, thermodynamics and nonlinear dynamics that has taken place in recent years and has imposed itself since then as one of the most dynamic and innovative orientations of these disciplines. This work has led to, among others, the characterization of complex systems undergoing bifurcations and chaotic dynamics in terms of the spectral properties of the underlying Liouville and Frobenius-Perron operators. A classification of chaotic versus stochastic processes has also been performed on the basis of dynamical entropies, and the connections with algorithmic complexity in the sense of Kolmogorov have been analyzed.

Current topics of interest in this area include :

• Connections between entropy, complexity and automaticity of a sequence, with applications in the analysis of biosequences.
• Generic properties of discrete networks of interacting elements, including the stability versus complexity issue.
• Closure schemes and constitutive relations for nonlinear dynamical systems.
• Dynamical basis and characterization of coarse-graining and symbolic dynamics in non-hyperbolic systems.
• Computation of correlation times from the spectral properties of Liouville and Fokker-Planck operators or of their coarse-grained forms.

Another aspect of basic complex systems research is to elucidate the mechanisms of appearance of complex behaviors. Research in the Center has contributed to the understanding of the onset of homoclinic chaos, of its connection with multi-modal oscillations and of its role in modeling phenomena arising in the context of autocatalytic chemical reactions, electrochemistry, and thermal combustion. The generic properties of coupled normal forms arising from the interference of instabilities have also been addressed.

Currently, the local aspects of dynamical instability in non-hyperbolic and in non-autonomous systems are being considered. This should provide access to the local stretching rates and fractal dimensions, and hence to the "most dangerous" phase space directions for instability.