A symplecticity-preserving RG analysis is carried out to study a resonance structure near an elliptic fixed point of a proto-type symplectic map in two dimensions. Through analyzing fixed points of a reduced RG map, a topology of the resonance structure such as a chain of resonant islands can be determined analytically. An application of this analysis to the Hénon map is also presented.
The Kuramoto-Sivashinsky (KS) equation is derived as a phase equation using the perturbative renormalization group method. In our derivation, it is not necessary to introduce an unusual small scale for the phase, which is crucial in deriving the KS equation by means of standard methods, such as the reductive perturbation method. The higher order KS equation is also derived consistently. §1. IntroductionIn the standard derivation of the Kuramoto-Sivashinsky (KS) equation as a phase equation by means of the reductive perturbation (RP) method, the phase must be scaled by a small quantity, that is, a small perturbation parameter. 1) It is unnatural to introduce a small scale for the leading order phase in the perturbational expansion, since absolute values of the phase have no physical meaning in unperturbed systems. Such a small scale is not necessary for other phase equations such as the diffusion equation and the Burgers equation. Therefore, the derivation of the KS equation without a small scale for the phase is desirable.Recently, a method based on perturbative renormalization group (RG) theory has been applied to derivations of amplitude equations 2), 3) and a phase equations of the diffusion type. 4) In comparison with the RP method and a multiple-scales analysis, the RG method does not require precise scales of various variables, but rather only a straightforward perturbation solution is necessary, assuming that a perturbative framework is set up.In this paper, taking advantage of the RG method, we focus on the derivation of the KS equation without using a small scale for the phase. In order to avoid calculational complexity, we consider the behaviour of a sinusoidal solution of the complex Ginzburg-Landau (cGL) equation. In §2, the phase of a sinusoidal solution of the cGL equation is generally described by the Burgers equation in the asymptotic sense using the RG method. In the case that a coefficient of the diffusion term in the Burgers equation is as small as the perturbation parameter, the phase is described by the KS equation, as shown in §3. In the final section, we compare the present derivation using the RG method with that using the RP method. by guest on March 21, 2015 http://ptp.oxfordjournals.org/ Downloaded from
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