Period-doubling renormalization theory

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

We discuss some of the properties of maps and the techniques for analyzing them in Sections 10.1-10.5. The emphasis is on period-doubling and chaos in the logistic map. Section 10.6 introduces the amazing idea of universality, and summarizes experimental tests of the theory. Section 10.7 is an attempt to convey the basic ideas of Feigenbaum s renormalization technique. 

In this section we give an intuitive introduction to Feigenbaum s (1979) renormalization theory for period-doubling. For nice expositions at a higher mathematical level than that presented here, see Feigenbaum (1980), Collet and Eckmann (1980), Schuster (1989), Drazin (1992), and Cvitanovic (1989b). 

The renormalization theory is based on the self-similarity of the figtree—the twigs look like the earlier branches, except they are scaled down in both the x and r directions. This structure reflects the endless repetition of the same dynamical processes a 2"-cycle is born, then becomes superstable, and then loses stability in a period-doubling bifurcation. 

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See also in sourсe #XX -- [ , ]

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