Period-doubling bifurcation in nonlinear systems
The dependence of the critical exponents, such as the Feigenbaum ratios, of the period-doubling bifurcation on the order of critical point z is studied. The dependence is quantitatively given for one-dimensional dissipative maps. The scaling factor in the period-doubling power spectrum of a class of two-dimensional area-preserving maps is found to approach a universal limit by both fast Fourier transform and autocorrelationfunction analysis. The dependence of the fractal dimension on the critical-point order is studied. The variation of the three most commonly used definitions of dimension, viz., the capacity, the information dimension, and the correlation exponent, is computed as a function of z. The numerical values agree very well with analytical estimates. The dependence of the scaling of the period-doubling bifurcation on the dimensionality of the reduced phase space is considered. Especially, the investigation of period-doubling bifurcations in four-dimensional symplectic maps indicates the existence of an universally self-similar period-doubling sequence. The fixed-point map has two unstable directions under the period-doubling operator with two relevant eigenvalues. The four orbital scaling factors have been found. As an extension of the Feigebaum scaling law for parameter and orbital element, a many-term scaling law is suggested. For one-dimensional dissipative maps, two-dimensional area-preserving maps, and four-dimensional symplectic maps, the many-term scaling law is very well obeyed. New scaling factors have been found.