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Explore the fourth lecture in the Dynamical Systems series delivered by Stefano Luzzatto. Delve into advanced concepts and theories within the field of dynamical systems, building upon the foundations established in previous lectures. Gain insights into complex mathematical principles and their applications in modeling and analyzing dynamic phenomena. Engage with rigorous mathematical proofs and demonstrations that elucidate key aspects of dynamical systems theory. Enhance your understanding of this fundamental area of mathematics through clear explanations and illustrative examples provided by the expert lecturer.