My dissertation research

The subject of my dissertation is the bifurcation problem for reversible dynamical systems and a specific application to the elliptical instability for fluid flows.

A system of differential equations

$$\dot{{x}} \in$$ (1)

is said to be reversible with respect to a linear involution R : V $\rightarrow$ V if and only if

RF(x, t) = - F(Rx, - t).

Similarly, the system $\dot{{x}}$ = F(x, t) is symmetric with respect to a linear operator S : V $\rightarrow$ V if SF(x, t) = F(Sx, t).

We investigate normal forms of system (1) at critical points. The normal forms are the ``simplest'' representation of Eq(1), consisting of those terms of the Taylor polynomial that cannot be eliminated via a series of near identity non-linear change of coordinates.

Linear Problem

In Eq(1), when V = $\mathbb {R}$ⁿ, F is autonomous and F(0) = 0, we obtain

$$\dot{{x}}$$ = Lx + N(x),

where L = DF(0). Reversibility implies LR = - RL. This means if $\lambda$ is an eigenvalue of L, so are - $\lambda$ and $\bar{{\lambda}}$. We consider cases where all eigenvalues have zero real parts.

We first show that L and the corresponding reversing matrix R can be simultaneously brought to their convenient ``standard'' forms, by a similarity transformation. Also see [Sev92]. We shall refer to the ``standard'' form of L at criticality as L₀.

Sevyruk's calculation on the versal deformations of reversible matrices [Sev92] allows us to compute the codimension of any reversible matrix L₀. We identify 3 codimension one cases and 6 codimension two cases.

The Nonlinear Problem

The nonlinear part N of the normal form is completely characterized by the homological equation:

D_xN(x) ^. L₀^*x - L₀^*N(x) = 0. (2)

This first order system of PDEs can be solved by the method of characteristics. However, we are not looking for the unique solution of an initial value problem. Instead, we are interested in finding polynomial solutions (in x) for N(x) [ETB⁺88].

Associated with Eq. (2) is the scalar differential equation:

D_xG(x) ^. L₀^*x = 0. (3)

This differential equation has the same base characteristic system as Eq(2). We seek a general solution G(x) $\in$ {u} where {u} are all first integrals polynomial in x. Let {u_B} be a minimal subset of {u} such that every u can be written as a polynomial in {u_B}. The task is to decide whether a finite set {u_B} exists and if it does, to compute it. Also, can we obtain a unique representation for {u} and if so, how? We will consider these questions in section 3.

We construct from this G the description for the normal form N(x). We then apply the reversing symmetries to the normal forms for further simplification. These normal forms contain unspecified coefficients that can be determined for specific cases, in particular, for the elliptical instability problem.