Jungho Park

Dynamic Bifurcation of the Ginzburg--Landau Equation

We study in this article the bifurcation and stability of the solutions of the Ginzburg--Landau equation, using a notion of bifurcation called attractor bifurcation. We obtain in particular a full classification of the bifurcated attractor and the global attractor as

Two-Dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions

\lambda

Dynamical Bifurcation of the Generalized Swift–Hohenberg Equation

crosses the first critical value of the linear problem. Bifurcations from the rest of the eigenvalues of the linear problem are obtained as well.

Thermosolutal convection at infinite prandtl number with or without rotation: Bifurcation and stability in physical space

This paper examines the bifurcation and structure of the bifurcated solutions of the two-dimensional infinite Prandtl number convection problem. The existence of a bifurcation from the trivial solution to an attractor Σ R was proved by Park (Disc. Cont. Dynam. Syst. B [2005]). We prove in this paper that the bifurcated attractor Σ R consists of only one cycle of steady-state solutions and that it is homeomorphic to S1. By thoroughly investigating the structure and transitions of the solutions of the infinite randtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In turn, this will corroborate and justify the suggested results with the physical findings about the presence of the roll structure. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using a new geometric theory of incompressible flows. Both theories were developed by Ma and Wang; see Bifurcation Theory and Applications (World Scientific, 2005) and Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics (American Mathematical Society, 2005).

Bifurcation to traveling waves in the cubic–quintic complex Ginzburg–Landau equation

In this paper, we prove that the generalized Swift–Hohenberg equation bifurcates from the trivial states to an attractor as the control parameter α passes through critical points. The bifurcation is divided into two groups according to the dimension of the center manifolds. We show that the bifurcated attractor is homeomorphic to S1 or S3 and it contains invariant circles of static solutions. We provide a criterion on the quadratic instability parameter μ which determines the bifurcation to be supercritical or subcritical.

Structure of bifurcated solutions of two-dimensional infinite Prandtl number convection with no-slip boundary conditions

We examine the nature of the thermosolutal convection with or without rotation in the infinite Prandtl number regime, which is applicable to magma chambers. The onset of bifurcation and the structure of the bifurcated solutions in this double diffusion problem are analyzed. The stress-free boundary condition is imposed at the top and bottom plates confining the fluid. For the rotation free case, two-dimensional Boussinesq equations are considered and we prove that there are bifurcating solutions from the basic solution and that the bifurcated solutions consist of only one cycle of steady state solutions that are homeomorphic to S1. By thoroughly investigating the structure and transitions of the solutions of the thermosolutal convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In the presence of rotation, we consider three-dimensional Boussinesq equations and we can get similar results as of the rotation free case. We also see how intensively the ...

Thermosolutal convection at infinite Prandtl number: initial layer and infinite Prandtl number limit

We consider the one-dimensional complex Ginzburg–Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg–Landau equation, Pure Appl. Anal. 7(5) (2008) 1237–1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling waves is examined and it is proved by means of the center manifold reduction method and some perturbation arguments, that the variable amplitude traveling waves are quasi-periodic and they connect two constant amplitude traveling waves.

Bifurcation and stability of the generalized complex Ginzburg--Landau equation

Abstract We consider the two-dimensional infinite Prandtl number convection problem with no-slip boundary conditions. The existence of a bifurcation from the trivial solution to an attractor Σ R was proved by Park [13] . The no-stress case has been examined in [14] . We prove in this paper that the bifurcated attractor Σ R consists of only one cycle of steady state solutions and it is homeomorphic to S 1 . By thoroughly investigating the structure and transitions the solutions of the infinite Prandtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. We show what the asymptotic structure of the bifurcated solutions looks like. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using geometric theories of incompressible flows.

Bifurcation of infinite Prandtl number rotating convection

We examine the initial layer problem and the infinite Prandtl number limit of the thermosolutal convection, which is applicable to magma chambers. We derive the effective approximating system of the Boussinesq system at large Prandtl number using two time scale approach [M. Holmes, Introduction to Perturbation Methods, Springer, New York, 1995, A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York, American Mathematical Society, Providence, RI, 2003]. We show that the effective approximating system is nothing but the infinite Prandtl number system with initial layer terms. We also show that the solutions of the Boussinesq system converge to solutions of the effective approximating system with the convergence rate of O(ϵ).