Remarks about bifurcation and stability of quasi-periodic solutions which bifurcate from periodic solutions of the Navier Stokes equations

Abstract

L. D. Landau (1944) and E. Hopf (1948) have conjectured that the transition to turbulence may be described as repeated branching of quasi-periodic solutions into quasi-periodic solutions with more frequencies. The simplest case is the bifurcation of periodic solutions from steady solutions. The next hardest problem is the bifurcation of quasi-periodic solutions from basic time periodic solutions of fixed frequency. This problem is treated in the lecture by a generalization of the Poincaré-Lindstedt perturbation which is successful in the simplest case. It is assumed that the Floquet exponents are simple eigenvalues of the spectral problem for the basic flow.If the Floquet exponent is zero at criticality, the formal construction gives two bifurcating solutions of the same frequency as the basic flow. The small amplitude solutions which bifurcate supercritically are stable; subcritical solutions with small amplitudes are unstable.When the Floquet exponents at criticality are complex and rationally independent of the fixed frequency, the solution of the spectral problem is quasi-periodic. The formal construction then gives the natural frequency of the bifurcating solution as a power series in the amplitude. The stability of the two-frequency power series is studied using a Floquet representation, generalizing a suggestion of Landau and using the Poincaré-Lindstedt method. Again the bifurcating solution is stable when supercritical and is unstable when subcritical.When the Floquet exponents and the fixed frequency are rationally dependent at criticality, the perturbation problems whose solutions give the coefficients of the Poincare-Lindstedt series cannot be solved unless certain additional orthogonality conditions are satisfied. Though these "extra" conditions do not arise exactly when the frequencies are rationally independent, they may be viewed as limiting forms of conditions associated with small divisors. In general, a bounded inverse cannot be expected even in the quasi-periodic case.KeywordsPeriodic SolutionBasic FlowPeriodic MotionFixed FrequencyPoincare MappingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.