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Nonlinear phenomena and instabilities arise in all fields of physics, chemistry, biology, and engineering. The book consists of six chapters. Chapter 1 contains preliminary topics from analysis: elements of set theory, measure theory, integration theory, asymptotic expansions, continued fractions and Pade approximations. Chapters 2 and 3 are devoted to basic and advanced methods of ODEs and their visualization: attractors, fractals, bifurcations, chaos, and elements of stability theory. Chapter 4 studies analytical and geometrical aspects of perturbation theory and contains many examples....