Renormalization Group Study of Critical Transitions in Dynamical Systems.

Renormalization group concepts are used to investigate the properties of critical transitions in dynamical systems. Two systems are studied: the transition from quasiperiodic motion to chaos in the map of the circle, and the noisy period-doubling cascade to chaos in maps of the interval. Analogies between these transitions and second-order phase transitions in thermodynamic systems are exploited. A new renormalization group approach, akin to the 'momentum-space' approach to thermodynamic phase transitions, is developed and utilized in analyzing the transition from quasiperiodicity to chaos. Within this framework, a perturbative expansion in a universality-class parameter is possible. This expansion, which resmbles the (epsilon)-expansion for thermodynamic phase transitions, yields analytic approximations for the critical exponents. The results are in excellent agreement with values previously obtained by numerical methods. Functional techniques, related to those used in non-equilibrium statistical mechanics, are employed to study the noisy period-doubling transition to chaos. By means of a 'real-space' renormalization group approach, in which the path integral that describes the action is 'decimated,' the scaling behavior of the transition is derived and several universal features are determined. The noise amplitude and the nonlinearity parameter are seen to play roles near chaotic transition points that are analogous, respectively, to the roles played by the magnetic field and the temperature near magnetic critical points.