Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems

Abstract

ABSTRACT. A two-dimensional thermal convection problem in a circular annulus subject to a constant inward radial gravity and heated from the inside is considered. A branch of spatio-temporal symmetric periodic orbits that are known only numerically shows a multi-critical codimension-two point with a triple +l-Floquet multiplier. The weakly nonlinear analysis of the dynamics near such point is performed by deriving a system of amplitude equations using a perturbation technique, which is an extensión of the Lindstedt-Poincaré method, and solvability conditions. The results obtained using the amplitude equation are compared with those from the original system of partial differential equations showing a very good agreement.

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

“…The main objective of a weakly nonlinear analysis is to obtain an evolution equation for this unknown amplitude function. This method has been applied to a number of specific examples in physics, including shear flows, shallow water waves, thermoacoustics and magnetohydrodynamics; see, for example [32][33][34][35][36][37][38]. In these examples, the nature of the equations analysed means that the algebra often becomes very complicated.…”

“…In these examples, the nature of the equations analysed means that the algebra often becomes very complicated. Our approach (inspired by [35] who derived a weakly nonlinear model for a different type of bifurcation) is to perform the analysis for a general set of equations that can be applied to a wide range of systems so that an analytic approximation for the periodic orbits and steady states can be obtained. The analysis is based on a continuous set of equations that are independent of the nature of a discretization procedure.…”

“…Near this point of marginal stability the oscillations will occur on a fast time scale and will be modulated by a time-dependent amplitude function that operates on a slower time scale. Following [34,35,39] we define a fast time scale, t 0 = t, and a slow time scale, t 1 = ε 2 t, where ε 1 is a small unfolding parameter, and employ the method of multiple scales. The solution, u, is now a function of two different time scales as well as the spatial variables denoted by x.…”

Accelerating CFD via local POD plus Galerkin projections

“…For instance, particular bifurcations can be analyzed via projection onto the center manifold to derive the normal form equations (Kuznetsov, 1998). This can be performed for realistic, infinite dimensional systems by various techniques, either analytic (Martel et al, 2000) or purely numerical (Sánchez et al, 2006). However, the procedure must be customized for each type of bifurcation, which penalizes flexibility and makes the method impractical in complex bifurcation problems exhibiting various bifurcations.…”

“…Particular bifurcations can be analyzed/approximated calculating the associated center manifold to derive the normal form equations. This can be done in a quite computationally efficient way, even for complex bifurcations in complex systems involving the Navier-Stokes equations (Sánchez et al, 2006). Nevertheless, this procedure must be customized for each specific bifurcation, which penalizes flexibility.…”