Classification of a singularity using the Liapounov-Schmidt reduction

The Lyapunov–Schmidt reduction is a procedure used to study solutions to nonlinear equations at singular points, where the implicit function theorem does not work. In several cases of interest, this procedure permits the reduction of infinite-dimensional equations in a Banach space to a system of equations in finite dimension. Once this system of equations has been obtained, singularity theory can be used to classify singular point.

A bifurcation problem

We shall work with a Hilbert space $H$, and

• we shall denote by $u$ the typical element of $H$.

We consider a Lagrangian $\mathcal L(u,\lambda)$, where $\lambda$ is the loading parameter. We shall suppose that $\mathcal L:H\times\mathbb R\to\mathbb R$ is Frechét differentiable as many times as is needed, and we shall use the notation

• $D_u\mathcal L$ to denote the Fréchet derivative with respect to $u$.

Moreover, we shall use

• we shall denote by $\mathcal L_\lambda$ the derivative of $\mathcal L$ with respect to the loading parameter $\lambda$.

We introduce the operator $\Phi:H\times \mathbb R\to H$ defined by

We suppose that $\{(0,\lambda):\lambda\in\mathbb R\}$ is a set of solutions of the equation $\Phi(u,\lambda)=0$. This implies, in particular, that

The Lyapounov-Schmidt reduction

For each $\lambda$ we consider the linear operator define the operator $L(\lambda)$ defined by:

We assume that there exists a critical load $\lambda_c$ such that the operator

has one-dimensional kernel, and that the range of $L$ is the orthogonal complement $(ker L)^\perp$. Hence

is one to one.

The correction $W(\alpha,\lambda)$

We denote by $E: H\rightarrow(ker L)^{\perp}$ the orthogonal projection. We select one element $v$ of $ker L$ and we define

The map

is bounded and surjective. Hence by the implicit function theorem, we deduce that

• there is a neighborhood of the point $(\alpha,\lambda)=(0,\lambda_c)$ such that the equation $F((0,\lambda_c),w)=0$ with respect to the unknown $w$ has a unique solution

Moreover, the function $W(\alpha,\lambda)$ is smooth in this neighborhood and vanishes at the critical point

Thus, for $(\alpha,\lambda)$ in a small neighborhood of $(0,\lambda_c)$ we have the identity

The effective function $f(\alpha,\lambda)$

We define the equivalent function

In a small neighborhood of the critical point $(u,\lambda)=(0,\lambda_c)$ , the solutions of the equation $\Phi(u,\lambda)=0$ are in one-to-one correspondence with the solutions of the equation $f(\alpha,\lambda)=0$.

The following properties are important:

• in a sufficiently small neighborhood of $(\alpha,\lambda)=(0,\lambda_c)$, if $f(\alpha,\lambda)=0$, then $u=\alpha v+W(\alpha,\lambda)$ is a solution of the equation $\Phi(u,\lambda)=0$;
• in a sufficiently small neighborhood of the critical point $(u,\lambda)=(0,\lambda_c)$ every solution of the equation $\Phi(u,\lambda)=0$ has the form $u=\alpha v+W(\alpha,\lambda)$ with $(\alpha,\lambda)$ a solution of $f(\alpha,\lambda)=0$;
• the solution $u=\alpha v+W(\alpha,\lambda)$ is stable if $f_\alpha<0$ and unstable if $f_\alpha>0$;
• a sufficient condition for $(0,\lambda_c)$ to be a bifurcation point is that $f_{\alpha\lambda}(0,\lambda_c)\neq 0$ (this is the so-called transversality condition);
• the sign of $f_{\alpha\lambda}(0,\lambda_c)$ determines the switch between stability or instability in the fundamental branch; in particular, if $f_{\alpha\lambda}(0,\lambda_c)>0$ then the fundamental branch $(0,\lambda)$ is stable for $\lambda<\lambda_c$ and unstable for $\lambda>\lambda_c$; the transition is from unstable to stable if $f_{\alpha\lambda}$ has negative sign.

Derivatives of the effective function

Although the function $f(\alpha,\lambda)$ is not known explicitly, it is possible to compute its derivative at the bifurcation point $(0,\lambda_c)$.

Calculation of $W_\alpha$ and $f_\alpha$

Taking the derivative of $\eqref{eq:1}$ with respect to $\alpha$ we obtain:

Note that

Moreover,

Hence, $\eqref{eq:3}$ becomes:

The last equation allows us to compute $W_\alpha(\alpha,\lambda)$ from the knowledge of $W(\alpha,\lambda)$ by solving the linear equation:

Once this solution is known, we can substitute into

which is obtained from $\eqref{eq:2}$ by differentiation with respect to $\alpha$.

Calculation of $f_\lambda$

A calculation similar to the previous one allows us to compute $f_\lambda$. Indeed, we find, differentiating with respect to $\lambda$:

where $W_\lambda(\alpha,\lambda)$ is the solution of the following equation

which is obtained by differentiation of $\eqref{eq:1}$.

Calculation of $f_{\alpha\lambda}$

Differentiating $\eqref{eq:4}$ with respect to $\lambda$ we obtain

where $W_{\alpha\lambda}(\alpha,\lambda)$ is obtained by solving the equation

which follows form $\eqref{eq:7}$.

Calculation of $f_{\alpha\alpha}$

Differentiation of $\eqref{eq:4}$ with respect to $\alpha$ yields

Differentiation of $\eqref{eq:7}$ with respect to $\alpha$ yields an equation for $W_{\alpha\alpha}(\alpha,\lambda)$:

Calculation of $f_{\alpha\alpha\alpha}$

Differentiating

where the unknown $W_{\alpha\alpha\alpha}(\alpha,\lambda)$ is determined by solving

which follows from $\eqref{eq:12}$ upon differentiation with respect to $\alpha$.

Calculation of the derivatives of $f$ and $W$ at the critical point

We first show that:

In fact, we recall that $L=ED_u\Phi(0,\lambda_c)$ and that $Lv=0$, hence $\eqref{eq:5}$ implies in particular that $W_\alpha(0,\lambda_c)$ solves the equation $LW_\alpha(0,\lambda_c)=0$, and hence $W(\alpha,\lambda)\in (ker L)$. On the other hand, also $W_\alpha\in (ker L)$ cf. $\eqref{eq:11}$). Thus, the only possible solution of the above equation is

From $\eqref{eq:5}$ we obtain the first of $\eqref{eq:5}$. A similar argument, based on the observation that $\Phi_{\lambda}(0,\lambda)=0$ yields

This implies the second of $\eqref{eq:6}$.

Transversality condition

The fact that $D_u\Phi$ is singular at the critical point implies that the implicit function theorem cannot be applied at that point. However, it is not a sufficient condition to guarantee that the critical point is an actual bifurcation point. A sufficient condition can be formulated, however, which goes by the name of transversality condition.

Evaluating $\eqref{eq:15}$ at $(0,\lambda_c)$ and making use of $\eqref{eq:13}$ and $\eqref{eq:14}$ and on setting

we obtain

Recalling that for a system of one degree of freedom the transversality condition is $f_{\alpha\lambda}(0,\lambda_c)\neq 0$ we obtain the transversality condition $(v,L_\lambda v)\neq 0$.

Calculation of $f_{\alpha\alpha}(0,\lambda_c)$

We find from $\eqref{eq:18}$:

where $W_{\alpha\alpha}(0,\lambda_c)$ can be computed by solving the equation:

which follows from $\eqref{eq:12}$. From this we deduce, using the definition of $L$, and with some calculation, the formula:

Calculation of $f_{\alpha\alpha\alpha}(0,\lambda_c)$

Finally,