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TWO NEW
BIFURCATION PHENOMENA

Herbert B. KELLER 23p.

Rapport de Recherche N° 369

Novembre 1979

Institut de Recherche
d'Informatique
et d'Automatique

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Rocquencourt
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Tél.: 954 90 20
TWO NEW BIFURCATION PHENOMENA

Herbert B. KELLER

California Institute of Technology

-0-

RESUME

On étudie la formation d'ISOLAS et de bifurcations multiples aux points limites. Dans le premier cas les centres des isolas sont caractérisés d'une manière analogue aux points elliptiques dans l'analyse utilisant le plan des phases. Les points de bifurcation ainsi que la bifurcation perturbée correspondante sont causés par les points hyperboliques.

Les points limites sur les branches de solution peuvent coïncider avec les points de bifurcation. Si la dimension du noyau associé au point de bifurcation est supérieure à un, toute solution passant par le point de bifurcation y présente un point limite. En dimension un, cependant, les bifurcations ne peuvent pas apparaître aux points limites.

ABSTRACT

The formation of ISOLA's and multiple bifurcation at limit points are studied. In the former case "centers" of isolas are characterized in exact analogy with elliptic points in phase plane analysis. Bifurcation points and the related perturbed bifurcation are due to hyperbolic points.

Limit points on solution branches can coïncide with bifurcation points. If the null space associated with the bifurcation point is more than one dimensional every solution path through the bifurcation point must have a limit point there. However in the one dimensional case bifurcation cannot occur at a limit point.
TWO NEW BIFURCATION PHENOMENA

Herbert B. KELLER

California Institute of Technology

# 1. Introduction

We examine here two new bifurcation phenomena namely : ISOLA formation and multiple limit point bifurcation. The former phenomenon has been observed many times in specific problems and indeed in even discussal in the literature. But we show here its intimate connection with perturbed bifurcation theory. The obvious analogy with hyperbolic -and elliptic- points is clearly apparent. The work on ISOLA formation is the result of discussions with B. Matkowski and E. Reiss who have a somewhat different approach using singular perturbations.

Limit points are usually defined (rather loosely) as solutions say $[u_0, \lambda_0]$ of $G(u, \lambda) = 0$ where $G_u(u_0, \lambda_0)$ is singular and $G_\lambda(u_0, \lambda_0)$ is not in the range of $G(u_0, \lambda_0)$. We show that such points may also be bifurcation points. But this only occurs if the null space of $G_u(u_0, \lambda_0)$ is higher dimensional. This may explain why they have not been observed before and we do not yet have a practical example of their occurrence. This work is joint with D. Decker [3, 4].

In Part I : Sections 2-5, we discuss the ISOLA problem and in Part II : Sections 6-9, we sketch the theory for multiple limit point bifurcations.
# Part I : Isolas and Perturbed Bifurcation

## 2. ISOLA Formation

An ISOLA is a closed isolated "loop" of solutions of some nonlinear operator equation. More properly it is a family of solutions that is homo- topic to a circle. Such families of solutions have been observed in many applications, particularly in nonlinear elasticity [9,10] and in chemical reactors [2, 5, 12].

Typically the equilibrium state, $u$, of some process is the solution of

$$ (2.1) \qquad G(u, \lambda, \tau) = 0. $$

Here $\lambda$ and $\tau$ are real parameters and, say, $G = X \times \mathbb{R}^2 \to Y$ for appropriate Banach spaces $X$ and $Y$. The solution

$$ (2.2) \qquad u = u(\lambda, \tau) \in X $$

depends upon two parameters, $(\lambda, \tau) \in \mathbb{R}^2$. In terms of a third real parameter $s$ we define a curve in $\mathbb{R}^2$ by specifying $\lambda = \lambda(s)$ and $\tau = \tau(s)$. Then we can represent a "curve" in $X$ by :

$$ (2.3) \qquad u(s) = u(\lambda(s), \tau(s)), $$

and this is an ISOLA if the curve in $\mathbb{R}^2$ is a simple closed curve. Other parametrizations are possible and in fact our theory in Sections 3-4 uses an implicit relationship of the form $f(\varepsilon, \lambda, \tau) = 0$ to determine solution families $u = u(\varepsilon, \lambda, \tau)$.

The determination of isolas in most practical problems has been by the process of "necking-off". That is as $\tau$ is varied the global solution curve or family of solutions in $X$ deforms and comes closer to a self-intersection. A value, say $\tau = \tau^0$, at which such a self-intersection occurs leads to a bifurcation phenomenon as well.
As $\tau$ varies beyond $\tau^0$ the local behavior (near the self-intersection) is that of perturbed bifurcation theory [6], and part of the perturbed branches may close to form an isola. It may happen that as $\tau$ continues to vary the corresponding isolas shrink to a point ! This suggests a new way to find isolas -by first locating such a point or "center" of isolas . In fact their characterization is a local one and so the global problem of necking-off can be eliminated.

We find an even deeper connection between perturbed bifurcation points and centers of isolas ; that is simply the distinction between hyperbolic and elliptic points in a family of plane quadratic curves.

### 3. Theory of ISOLAS and Perturbed Bifurcation

We assume that $G$ is $C_u^k$ in all arguments and

$$ (3.1) \quad (a) \quad G_u^0 = G(u^0, \lambda^0, \tau^0) = 0. $$

The Fréchet derivative

$$ G_u^0 = \frac{\partial G}{\partial u}(u^0, \lambda^0, \tau^0) $$

and its formal adjoint or dual operator $G_u^{0*}$ have :

$$ \eta(G_u^0) = \operatorname{span} \{\Phi\} , \quad \Phi \in X^* ; $$

$$ (3.1) \quad (b) \quad \eta(G_u^{0*}) = \operatorname{span} \{\Psi^*\} , \quad \Psi^* \in X^* . $$

Further we require

$$ (3.1) \quad (c) \quad R(G_u^0) \text{ is closed, codim } R(G_u^0) = 1. $$

Thus $G_u^0$ is Fredholm of index zero. We can choose $\Phi^* \in X^*$ and $\Psi \in Y$ such that

$$ (3.2) \quad (a) \quad \Phi^* \Phi = 1 , \quad \Psi^* \Psi = 1. $$
Then X and Y have the decompositions :

$$
(3.2) \text{(b)} \quad X \equiv \operatorname{span} \{\Phi\} \oplus X_1, \quad Y = \operatorname{span} \{\Psi\} \oplus Y_1
$$

where $\Phi^*\Phi = \Psi^*\Psi = 1$ and

$$
(3.2) \text{(c)} \quad X_1 \equiv \{x \in X : \Phi^* x = 0\}, \quad Y_1 \equiv \{y \in Y : \Psi^* y = 0\}
$$

We note in particular that

$$
(3.3) \quad Y_1 = R(G_u^0)
$$

The crucial distinction between the bifurcation parameter, $\lambda$,
and the perturbation parameter, $\tau$, is contained in the requirements that

$$
(3.4) \quad \begin{array}{l} \Psi^* G_{\lambda}^{0} = 0 \quad , \text{i.e. } G_{\lambda}^{0} \in R(G_{u}^{0}) \ ; \\ \Psi^* G_{\tau}^{0} = d \neq 0 \quad , \text{i.e. } G_{\tau}^{0} \notin R(G_{u}^{0}) \ . \end{array}
$$

$$
b) \quad \Psi^* G_{\tau}^0 = d \neq 0, \text{ i.e. } G_{\tau}^0 \neq R(G_u^0).
$$

This basic difference was first clarified in [6]. Now with $G_{\lambda}^{0} \in R(G_{u}^{0})$ there
must exist a unique $\Phi_{0} \in X$ satisfying :

$$
(3.5) \quad G_u^0 \Phi_0 + G_\lambda^0 = 0 , \quad \Phi^* \Phi_0 = 0 .
$$

Thus $\Phi_0 \in X_1$ and $\Phi^*$ in (3.2) is chosen simply to make this solution unique.

Finally we recall the important discriminant

$$
(3.6) \quad \text{a)} \quad \Delta \equiv ac - b^2
$$

where

$$
(3.6) \text{ b)} \quad \left\{
\begin{array}{l}
a \equiv \Psi^* G_{uu}^0 \Phi \Phi \\
b \equiv \Psi^* [G_{u\lambda}^0 \Phi + G_{uu}^0 \Phi \Phi_0] \\
c \equiv \Psi^* [G_{\lambda\lambda}^0 + 2 G_{u\lambda}^0 \Phi_0 + G_{uu}^0 \Phi_0 \Phi_0]
\end{array}
\right.
$$
The quadratic form with discriminant $\Delta$ and its role in bifurcation theory (i.e. no $\tau$-dependence) was first pointed out in [7]. In the present context we have for :

$$ (3.7) \qquad \left. \begin{array}{l} < 0, \text{ perturbed bifurcation} \\ \Delta \\ > 0, \text{ isola formation.} \end{array} \right\} $$

More precisely the solution field of (2.1) in a neighborhood of $(u^0, \lambda^0, \tau^0)$ has the form :

$$ a) \quad u = u(\varepsilon, \lambda, \tau) \equiv u^0 + \varepsilon \Phi + v(\varepsilon, \lambda, \tau) ; $$

$$ (3.8) \quad b) \quad v(0, \lambda^0, \tau^0) = 0, \quad v(\varepsilon, \lambda, \tau) \in \mathcal{X}_1 $$

$$ c) \quad a \varepsilon^2 + 2b \varepsilon (\lambda - \lambda^0) + c(\lambda - \lambda^0)^2 = 2d(\tau^0 - \tau) + \text{h.o.t.'s} $$

Here $v(\varepsilon, \lambda, \tau)$ is $\mathcal{O}^k$. For $\Delta < 0$ the relation (3.8c) is satisfied for each fixed $\tau \neq \tau^0$, but near $\tau^0$, on two distinct "hyperbolic" curves in the $(\varepsilon, \lambda)$-plane. For $\tau = \tau^0$ these curve intersect at $(\varepsilon, \lambda) = (0, \lambda^0)$. This is precisely the phenomena of *perturbed bifurcation*. The orientation of the axis of the hyperbola is easily determined and the complete local structure is revealed by elementary considerations which we avoid here.

For $\Delta > 0$ real solutions of (3.8c) for $(\varepsilon, \lambda)$ near $(0, \lambda^0)$ exist only when $d(\tau^0 - \tau)/a \ge 0$. The curves of solutions in the $(\varepsilon, \lambda)$-plane are closed loops which are in fact perturbed ellipses. Of course the solutions of (2.1) given by (3.8a) are homotopic to the solutions of the algebraic system (3.8c). Thus the indicated geometric properties of the algebraic solution curves are inherited by the solution families of (2.1) near $(u^0, \lambda^0, \tau^0)$.

## 4. Proof Outline.

The above results are easily proven using the Liapunov-Schmidt procedure and the Morse Lemma as indicated in [11]. To get the isolas a slight but obvious extension of this technique is required. More direct proofs using implicit function techniques proceed as follows :
Using (3.8a) we seek $v(\epsilon, \lambda, \tau) \in X_1$, such that

$$ (4.1) \quad F(\epsilon, \lambda, \tau; v) \equiv G(u(\epsilon, \lambda, \tau), \lambda, \tau) - \Psi^* G(u(\epsilon, \lambda, \tau), \lambda, \tau) \Psi = 0. $$

For $(\epsilon, \lambda, \tau) = (0, \lambda^0, \tau^0)$ a solution is $v = 0$. We also note that

$$ F_v^0 \equiv F_v(0, \lambda^0, \tau^0, 0) = G_u^0. $$

However we have that $F : X_1 \to Y_1 \equiv R(G_u^0)$ and so $F_v^0$ is an isomorphism of $X_1$ onto $Y_1$. Thus the implicit function theorem insures the existence of a unique $C^k$ smooth $v(\epsilon, \lambda, \tau) \in X_1$ satisfying (4.1) for all $(\epsilon, \lambda, \tau)$ near $(0, \lambda^0, \tau^0)$.

To solve (2.1) with this $v(\epsilon, \lambda, \tau)$ we need only find $(\epsilon, \lambda, \tau)$ such that

$$ (4.2) \quad \text{a)} \quad f(\epsilon, \lambda, \tau) \equiv \Psi^* G(u^0 + \epsilon\Phi + v(\epsilon, \lambda, \tau), \lambda, \tau) = 0. $$

This is just the bifurcation equation and expanding about $(\epsilon, \lambda, \tau) = (0, \lambda^0, \tau^0)$ yields :

$$ (4.2) \quad \text{b)} \quad f(\epsilon, \lambda, \tau) = d(\tau - \tau^0) + \frac{1}{2}[a\epsilon^2 + 2b\epsilon(\lambda - \lambda^0) + c(\lambda - \lambda^0)^2] + \text{h.o.t.'s} $$

Our results clearly follow from the definiteness or indefiniteness of the quadratic form in $\epsilon$ and $(\lambda - \lambda^0)$.

## 5. Further Observations

If $G_\lambda^0 = 0$, as frequently occurs in many applications, then the only solution of (3.5) is $\Phi_0 = 0$. In this case b and c of (3.6) simplify considerably. If in addition $G_{\lambda\lambda}^0 = 0$ then $c = 0$ and it is impossible to have $\Delta > 0$. So in this case isolas cannot form as in the present theory. Of course the most common occurence of $G_\lambda^0 = G_{\lambda\lambda}^0 = 0$ is when $u = 0$ is assumed to be a solution of (2.1) for all $\lambda$ and some $\tau = \tau^0$. This is the case of the trivial solution in which
$$G(0, \lambda, \tau^0) = 0 \quad \forall \lambda \in \mathbb{R}.$$

We also note in this case that

$$\Delta = b^2 > 0$$

provided only that $b \neq 0$. Thus the bifurcation condition is frequently given as

$$\Psi^* G_{u\lambda}^0 \Phi \neq 0.$$

We see that the deeper implication of the true condition are not apparent in this statement.

An algebraic example is given in [5] which seems to have the "center" of isolas located on a trivial branch of solutions. These are not strictly elliptic points as in the present theory and they show that other types of isolas can exist.
# Part II : Multiple Limit Point Bifurcation

## 6. The Bifurcation Equations

We shall examine bifurcation or branching of solution of

$$ (6.1) \qquad G(u, \lambda) = 0 $$

for $G: X \times R \to Y$ at a solution $(u^0, \lambda^0) \in X \times R$ where the Fréchet derivative

$$ G_u^0 \equiv \frac{\partial G}{\partial u} (u^0, \lambda^0) $$

has a higher dimensional null space. Specifically we assume $G_u^0$ and the dual operator $G_u^{0*}$ have :

$$ (6.2) \text{ a)} \qquad \begin{aligned} \eta(G_u^0) &= \operatorname{span} \{\Phi_1, \Phi_2, \dots, \Phi_m\}, & \Phi_j &\in X \\ \eta(G_u^{0*}) &= \operatorname{span} \{\Psi_1^*, \Psi_2^*, \dots, \Psi_m^*\}, & \Psi_j^* &\in X^* \end{aligned} $$

and that

$$ (6.2) \text{ b)} \quad \mathrm{R}(G_u^0) \text{ is closed, codim } \mathrm{R}(G_u^0) = m. $$

So $G_u^0$ is Fredholm of index zero. In addition we can choose $\Phi_j^* \in X^*$ and $\Psi_j \in Y$ such that

$$ (6.3) \text{ a)} \quad \Phi_i^* \Phi_j = \delta_{ij}, \quad \Psi_i^* \Psi_j = \delta_{ij}, \quad i,j = 1,2,\dots,m. $$

Then $X$ and $Y$ have the decompositions :

$$ (6.3) \text{ b)} \quad X = \operatorname{span} \{\Phi_j\} \oplus X_1, \quad Y = \operatorname{span} \{\Psi_j\} \oplus Y_1; $$

where

$$ (6.3) \text{ c)} \quad X_1 = \{x \in X : \Phi_j^* x = 0, \quad \le j \le m\}, \quad Y_1 = \{y \in Y : \Psi_j^* y = 0, \quad \le j \le m\}. $$
Again we note that

$$ (6.4) \qquad Y_1 = R(G_u^0). $$

We shall first assume the existence of an arc or branch of solutions $[u(t), \lambda(t)]$, with $[u(0), \lambda(0)] = [u^0, \lambda^0]$, depending smoothly on $t$ in $|t| < t_0$. Then we derive various necessary conditions which must be satisfied at $[u^0, \lambda^0]$ by such an arc. These distinguish two basic cases as we shall see. On the arc in question we have that

$$ (6.5) \qquad G(u(t), \lambda(t)) = 0, \quad |t| < t_0. $$

Then all existing derivatives with respect to $t$ vanish identically on the given interval. Assuming at least second Fréchet derivatives to exist we then must have, using the obvious notation $\dot{u}(t) = du(t)/dt$, etc, differentiated twice and evaluating at $t=0$:

$$ (6.6) \qquad G_u^0 \dot{u}(0) + G_\lambda^0 \dot{\lambda}(0) = 0 $$

$$ (6.7) \qquad G_u^0 \ddot{u}(0) + G_\lambda^0 \ddot{\lambda}(0) = -[G_{uu}^0 \dot{u}(0)\dot{u}(0) + 2G_{u\lambda}^0 \dot{u}(0)\dot{\lambda}(0) + G_{\lambda\lambda}^0 \dot{\lambda}(0)\dot{\lambda}(0)] $$

If we have $C^\infty$ smoothness then the corresponding infinite set of perturbation equations can be defined.

But as we shall see only the first two can suffice for a rigorous existence theory.

Since we have assumed that the arc is smooth so that the "tangent" $[\dot{u}(0), \dot{\lambda}(0)]$ exists it follows from (6.6) that we must have

$$ \dot{\lambda}(0) G_\lambda(u^0, \lambda^0) \in R(G_u^0). $$

There are thus two distinct possibilities:
$$
\begin{align}
(i) \quad & G_{\lambda}^{0}(u^{0}, \lambda^{0}) \in \mathbb{R}(G_{u}^{0}) , \notag \\
(6.8) \quad & G_{\lambda}^{0}(u^{0}, \lambda^{0}) \notin \mathbb{R}(G_{u}^{0}) \text{ but } \dot{\lambda}(0) = 0. \notag
\end{align}
$$

Case (i) is the one usually considered and so we call it a regular bifurcation point. Case (ii) will be called a limit point.

6.1. Regular Bifurcation Equations

We proceed with the first case briefly and consider

$$
(6.9) \quad G_u^0 \Phi_0 + G_\lambda^0 = 0 ; \Phi_j^* \Phi_0 = 0, j = 1, 2, \dots, m.
$$

This problem has a unique solution, $\Phi_0 \in X_1$, and it follows from (6.3)
and (6.6) that $\dot{u}(0)$ must have the form :

$$
(6.10) \quad \dot{u}(0) = \sum_{j=0}^{m} \xi_j \Phi_j , \quad \xi_0 = \dot{\lambda}(0).
$$

The other scalars $\xi_j$, $1 \le j \le m$ are at present arbitrary. Now assuming
that the arc has second derivatives $[\ddot{u}(0), \ddot{\lambda}(0)]$ exists, so $G_u^0 \ddot{u}(0) \in R(G_u^0)$
and we get from (6.7) using (6.10), (6.3c) and (6.4) :

$$
(6.11) \quad \sum_{j=1}^{m} \sum_{k=1}^{m} a_{ijk} \xi_j \xi_k + 2 \sum_{j=1}^{m} b_{ij} \xi_j \xi_0 + c_i \xi_0^2 = 0
$$

Here we have introduced

$$
a_{ijk} = a_{ikj} \equiv \Psi_i^* G_{uu}^0 \Phi_j \Phi_k
$$

$$
(6.12) \quad b_{ij} \equiv \Psi_i^*[G_{uu}^0 \Phi_0 + G_{u\lambda}^0] \Phi_j \\
\phantom{b_{ij} \equiv \Psi_i^*[} c_i \equiv \Psi_i^*[G_{uu}^0 \Phi_0 \Phi_0 + 2 G_{u\lambda}^0 \Phi_0 + G_{\lambda\lambda}^0]
$$

The system (6.11) is homogeneous and thus admits rays of solutions. To
avoid this and to fix a definition of the parameter $\epsilon$ we adjoin the
normilization

$$ \xi_0^2 + \xi_1^2 + \dots + \xi_m^2 = 1 . $$

Now we write (6.11) and the above in the compact form :

$$ (6.13) \qquad \begin{aligned} & \text{a)} & A(\xi)\xi + 2 B \xi \xi_0 + C \xi_0^2 &= 0 ; \\ & \text{b)} & \xi_{\sim}^T \xi + \xi_0^2 &= 1 . \end{aligned} $$

Here we have introduced the vectors and matrices :

$$ (6.14) \qquad \begin{aligned} \xi &= (\xi_1, \dots, \xi_m)^T, \quad A(\xi) = (A_{ij}(\xi)) = \left(\sum_{k=1}^m a_{ijk} \xi_k\right), \\ B &= (b_{ij}), \quad C = (c_1, \dots, c_m)^T. \end{aligned} $$

We call the system (6.13) the **Regular Bifurcation Equations (RBE)**. They are (m+1) quadratic equations for the (m+1) scalars $\xi_0, \dots, \xi_m$. It also follows that each $C^2$ arc of solutions of (6.1) through $[u^0, \lambda^0]$ yields a solution of the RBE. Further each pair of such solution arcs that are not tangent to each other at $[u^0, \lambda^0]$ yield distinct solutions of the RBE. A form of converse is also known : for each isolated root of the RBE there corresponds a $C^2$ solution arc of (6.1) passing through $[u^0, \lambda^0]$ with tangent (6.10). The statement and proof of this result is indicated in Section 8.

## 6.2. Limit Point Bifurcation Equations

We return now to case (ii) of (6.8). It is not possible in this case to solve (6.9) : Thus the tangent to the solution arc, or the solution of (6.6) must have the form

$$ (6.15) \qquad \dot{u}(0) = \sum_{j=1}^{m} \xi_j \Phi_j, \quad \dot{\lambda}(0) = 0. $$

Also we must have, with
$$
(6.16) \quad \text{a)} \quad d_i \equiv \psi_i^* G_\lambda^0 , \quad i = 1, 2, \dots, m ,
$$

that

$$
(6.16) \quad \text{b)} \quad d \equiv (d_1, \dots, d_m)^T \neq 0 ,
$$

from (6.1) and (6.8ii). With η ≡ λ(0) we now get from (6.7) and the assumed
existence of [u(0), λ(0)], using (6.15) and (6.14) ;

$$
\begin{equation} \tag{6.17}
\begin{split}
a) \quad A(\xi)\xi + \eta \, d &= 0 , \\
\multicolumn{2}{@{}l@{}}{(\xi^T \xi = 1.)}
\end{split}
\end{equation}
$$

$$
b) \quad \xi \xrightarrow{\sim} \xi^T = 1.
$$

Again we have determined ε by adjoining the last, normalizing, equation
above.

We call the system (6.17) the _Limit Point Bifurcation Equations_
(LPBE). Again all non-tangential arcs of solutions through [u⁰,λ⁰] yield
distinct roots of the LPBE. Conversely we will show in Section 7 that
each _isolated_ root of the LPBE yields the existence of an arc of solutions
of (6.1) through [u⁰,λ⁰]. We recall that a root, say [ξ⁰,η⁰], of (6.17)
is isolated if the Jacobian matrix at the root is nonsingular.
This matrix is of order m+1 and has the form

$$
(6.18) \qquad J = 2 \begin{pmatrix} A(\xi^0) & \frac{1}{2} d \\ \xi^{\text{OT}} & 0 \end{pmatrix}
$$

Some necessary and sufficient conditions for isolation are given in the
next section. It is also easy to show that isolation of a root of the
LPBE insures that a *formal* perturbation expansion can be carried out to
all orders [4]. Our existence theory techniques could then be used to show
that the resulting perturbation expansion was indeed an asymptotic repre-
sentation of a solution. We do not carry out these details here as they are
similar to the treatment in [8].

7. Multiple Bifurcation and the LPBE

We demonstrate the existence of smooth solution arcs using implicit
function theoretic techniques. In the present study this is done in a manner
motivated by the following lemma on the non-singularity of linear operators
on product spaces.

Lemma 7.1 : Let X and Y be Banach spaces and consider a linear operator

A : X × ℝ$^r \to Y \times \mathbb{R}^r$, of the form

$$
a \equiv \begin{pmatrix} A & B \\ * & D \end{pmatrix},
$$

where

$$
A : X \to Y, \quad B : \mathbb{R}^r \to Y
$$

$$
C^* : X \to \mathbb{R}^r, \quad D : \mathbb{R}^r \to \mathbb{R}^r
$$

i) If A is nonsingular then $\mathcal{A}$ is nonsingular iff :

$$
(7.1) \quad \text{a) } (D - C_A^{-1} B) \text{ is nonsingular}
$$

ii) If *A* is singular with

$$
(7.1) \quad \text{b) } \dim \eta(A) = \dim R(A) = r \ge 1,
$$

then $\mathcal{A}$ is nonsingular iff :

$$
\begin{equation} \tag{7.1}
\begin{alignedat}{2}
& c_0) \dim R(B) &&= r \\
& c_1) R(B) \cap R(A) &&= 0 \\
& c_2) \dim R(C^*) &&= r \\
& c_3) \eta(A) \cap \eta(C^*) &&= 0
\end{alignedat}
\end{equation}
$$

<footnote>* We use here and in the following the convention that a linear operator is "nonsingular" if it is an isomorphism from its domain onto its range.</footnote>
iii) If A is singular with $dim \eta(A) > r$ then $A$ is singular.

**Proof** : This lemma with X = Y was first proven in the unpublished Bifurcation Theory Lecture Notes of H. B. Keller. It is also stated in [7] and a detailed proof is given in [3]. ■

This lemma suggests how a system with singular Fréchet derivative A
can be "embedded" into a larger system with derivative A which is non-singular.
Hence the implicit function theorem can apply to the larger system. This is
precisely how we shall employ it for bifurcation theory. First however we
use the lemma to study the isolation of roots of the LPBE. Specifically with
the Jacobian matrix as partitioned in (6.18) to operate on $\mathbb{R}^m \times \mathbb{R}$ into it-
self, it readily follows that _a root_ $(\xi^0, \eta^0)$ _of_ (6.17) _is isolated iff_ :

$$
(7.2) \quad \text{i) } A(\xi^0) \text{ is nonsingular,}
$$

or

$$
(7.2) \quad \text{ii) } \eta^0 = 0, \eta(A(\xi^0)) = \operatorname{span}\{\xi^0\}, d \notin R(A(\xi^0))
$$

We now state the result showing that a smooth solution arc
[u(t),λ(t)] through [u⁰,λ⁰] is determined by each isolated root of the LPBE.

**Theorem 7.3.** : Let $G(u,\lambda)$ be a $C^k$ **mapping** ($k \ge 2$) **from** $X \times R$ **to** $Y$. Let $[u^0, \lambda^0]$ be a **solution of** (6.1) for which $G_u^0$ satisfies (6.2) and $G_\lambda^0 \notin R(G_u^0)$. Let $(\xi^0, \eta^0)$ be an **isolated root of the LPBE**, (6.17). Let $v^0$ **be the unique solution of**

$$
(7.3) \text{ a) } G_u^0 v^0 = -[G_{uu}^0 \Phi^0 \Phi^0 + \eta^0 G_\lambda^0] ,
$$

$$
\text{b) } v^0 \in X_1 ,
$$

where

$$
(7.3) \text{ c) } \Phi^0 \equiv \sum_{j=1}^{m} \xi_j^0 \Phi_j .
$$
Then for some $\epsilon_0 > 0$ there exist unique $c^{k-2} \in \text{ smooth functions}$
$[\xi(\epsilon), \eta(\epsilon), v(\epsilon)]$ such that

for all $|\epsilon| < \epsilon_0$ :

$G(u(\epsilon), \lambda(\epsilon)) = 0$ ,

where

$$
\begin{equation} \tag{7.4}
\begin{aligned}
\text{a)} \quad & u(\epsilon) \equiv u^0 + \epsilon\Phi(\epsilon) + \frac{\epsilon^2}{2} v(\epsilon), \quad \lambda(\epsilon) \equiv \lambda^0 + \frac{\epsilon^2}{2} \eta(\epsilon) ; \\
& \qquad \sum_{j=1}^{m} \xi_j(\epsilon)\Phi_j \quad , \quad v(\epsilon) \in X_1 ;
\end{aligned}
\end{equation}
$$

$$
b) \quad \Phi(\epsilon) \equiv \sum_{j=1}^{m} \xi_j(\epsilon) \Phi_j \quad , \quad v(\epsilon) \in X_1 \ ;
$$

and

$$
(7.4) \quad \text{c)} \quad \xi(0) = \xi^0, \eta(0) = \eta^0, v(0) = v^0.
$$

**Proof** : We first note that the right hand side of (7.3a) is in $R(G_u^0)$ since $(\xi^0, \eta^0)$ is a root of the LPBE.
Then (7.3b) makes $v^0$ unique by (6.3b).

In order to generate the appropriate solution arc we consider
the system of equations from X × $\mathbb{R}^{m+1}$ into Y × $\mathbb{R}^{m+1}$ :

$$
\begin{equation} \tag{7.5}
\begin{aligned}
\text{a)} \quad & 0 = g(\xi, \eta, v; \epsilon) \equiv \begin{cases} \frac{1}{\epsilon^2} G(u^0 + \epsilon\Phi(\epsilon) + \frac{\epsilon^2}{2}v\lambda^0 + \frac{\epsilon^2}{2}\eta) & , \epsilon \neq 0 , \\ \frac{1}{2}[G_u^0 v + G_{uu}^0 \Phi(0)\Phi(0) + \eta G_\lambda^0] & , \epsilon = 0 ; \end{cases} \\
& h_j(\xi, \eta, v; \epsilon) \equiv \Phi_j^* v \quad , \quad j = 1, 2, \dots, m ,
\end{aligned}
\end{equation}
$$

$$
b) \ 0 = h_j(\xi, \eta, v; \epsilon) \equiv \Phi_j^* v \ , \quad j = 1, 2, \dots, m \ ,
$$

$$
c) 0 = h_{m+1}(\xi, \eta, v; \epsilon) \equiv \xi^T \xi - 1 .
$$

It is clear that, for $\epsilon = 0$, using (7.4c) :

$$
g(\xi^0, \eta^0, v^0; 0) = 0 ,
$$

$$
h_j(\xi^0, \eta^0, v^0; 0) = 0 , \quad j = 1, 2, \dots, m+1 .
$$
Now our result will easily follow if we can apply the implicit function theorem to the system $g = 0, h = 0$ at the root $(\xi^0, \eta^0, v^0)$. So we need to demonstrate the "nonsingularity" of the Fréchet derivative :

$$ F = \frac{\partial(g,h_1, \dots, h_{m+1})}{\partial(v, \xi, \eta)} \begin{vmatrix} \varepsilon = 0 \\ v = v^0 \\ \xi = \xi^0 \\ \approx \eta = \eta^0 \end{vmatrix}. $$

That is we must show that $F$ is an isomorphism from $X \times \mathbb{R}^{m+1}$ into $Y \times \mathbb{R}^{m+1}$.

From (7.5) and (7.4b) it follows that $F$ has the structure :

$$ (7.6) \quad \text{a)} \quad F = \frac{1}{2} \begin{bmatrix} G_u^0 & G_{uu}^0 & \Phi_1^0 & \cdots & G_{uu}^0 & \Phi_m^0 & G_\lambda^0 \\ \Phi_1^* & \cdots & \Phi_m^* & \cdots & \Phi_1^0 & \cdots & \Phi_m^0 \\ \hline 0 & 4\xi_1^0 & \cdots & 4\xi_m^0 & 0 & 0 & 0 \end{bmatrix} $$

By relabelling we can write this in the more compact form :

$$ 2F = \begin{bmatrix} \hat{A} & \hat{B} \\ \hat{C}^* & 0 \end{bmatrix}, \quad \hat{A} = \begin{bmatrix} A^0 & B^0 \\ C^{0*} & 0 \end{bmatrix}, \quad \hat{B} = \begin{bmatrix} G_\lambda^0 \\ 0 \end{bmatrix}, $$

$$ (7.6) \quad \text{b)} \quad A^0 = G_u^0, \quad B^0 = [G_{uu}^0 \Phi_1^0 \dots G_{uu}^0 \Phi_m^0]^T, $$

$$ C^{0*} = \begin{bmatrix} \Phi_1^* \\ \vdots \\ \Phi_m^* \end{bmatrix}, \quad \hat{C}^* = [0, 4\xi_1^0, \dots, 4\xi_m^0]^T. $$

We show that $F$ is nonsingular by two applications of Lemma 7.1 ; first to $\hat{A}$ and then to $F$. The fact that $(\xi^0, \eta^0)$ is an *isolated*
solution of the LPBE, and hence (7.2) holds is crucial in both parts of
this proof. The details can be found in [3] or [4]. ■

Note that each isolated root of the LPBE generates a distinct
branch of solutions of (6.1) through $(u^0, \lambda^0)$. They each have the form
in (7.4a) with higher order terms in $\epsilon$ given by distinct forms in (7.4b,c).
In particular it should be noted that $\lambda(\epsilon)$ for every such solution has
$\dot{\lambda}(0) = 0$. Thus in the usual "bifurcation diagrams" drawn in $\mathbb{R}^2$ all the
branches seem to be tangent; but of course they are not tangent in the
full space $X \times \mathbb{R}$.

## 8. Multiple Bifurcation and the RBE.

It has been established previously, see [8, 13] that multiple
bifurcations occur when the RBE, (6.13) have several isolated roots and
case i) of (6.8) holds. However we state the appropriate result in our
present formulation and indicate the proof using Lemma 7.1. This is a
new and somewhat simpler proof of the standard multiple bifurcation theory.
We first determine the conditions for the isolation of a root of the RBE.

If $(\xi_0^0, \xi_\infty^0)$ is a root of the RBE, (6.13), then the Jacobian matrix
evaluated at this root is

$$ K \equiv 2 \begin{pmatrix} A(\xi_\infty^0) + \xi_0^0 B & B\xi_\infty^0 + \xi_0^0 C \\ \xi_\infty^{0T} & \xi_0^0 \end{pmatrix} . $$

Now an application of Lemma 7.1 easily yields that $(\xi_0^0, \xi_\infty^0)$ is an isolated root of (6.13) iff :

$$ (8.1) \quad \text{i)} \quad A(\xi_\infty^0) + \xi_0^0 B \quad \text{is non-singular,} $$

or

$$ (8.1) \quad \text{ii)} \quad \xi_0^0 = 0, \eta(A(\xi_\infty^0)) = \operatorname{span}\{\xi_\infty^0\}, B\xi_\infty^0 \notin \operatorname{R}(A(\xi_\infty^0)). $$
Compare these conditions with those in (7.2) for the isolation of roots of the LPBE.

We now state

**Theorem 8.2.** Let $G(u,\lambda)$ be a $C^k$ map ($k \ge 2$) from $X \times \mathbb{R}$ into $Y \times \mathbb{R}$. Let $[u^0, \lambda^0]$ be a solution of (6.1) for which $G_u^0$ satisfies (6.2) and $G_\lambda^0 \in R(G_u^0)$. Let $(\xi_0^0, \xi^0)$ be an isolated root of the RBE, (6.13). Let $v^0$ be the unique solution of :

$$ (8.2) \qquad \begin{aligned} \text{a)} \quad & G_u^0 v^0 = -[G_{uu}^0 \Phi^0 \Phi^0 + 2G_{u\lambda}^0 \Phi^0 \xi_0^0 + G_{\lambda\lambda}^0 \xi_0^{02}] , \\ \text{b)} \quad & v^0 \in X_1 \end{aligned} $$

where

$$ (8.2) \qquad \text{c)} \quad \Phi^0 = \sum_{j=0}^{m} \xi_j^0 \Phi_j $$

and $\Phi_0$ is defined in (6.9).

Then for some $\epsilon_0 > 0$ there exist unique $C^{k-2}$ smooth functions $[\xi(\epsilon), \xi_0(\epsilon), v(\epsilon)]$ such that for all $|\epsilon| < \epsilon_0$ :

$$ G(u(\epsilon), \lambda(\epsilon)) = 0, $$

where

$$ (8.3) \qquad \begin{aligned} \text{a)} \quad & u(\epsilon) = u^0 + \epsilon \Phi(\epsilon) + \frac{\epsilon^2}{2} v(\epsilon) , \quad \lambda(\epsilon) = \lambda^0 + \epsilon \xi_0(\epsilon) ; \\ \text{b)} \quad & \Phi(\epsilon) = \sum_{j=0}^{m} \xi_j(\epsilon) \Phi_j , \quad v(\epsilon) \in X_1 ; \end{aligned} $$

and

$$ (8.4) \quad \text{c)} \quad \xi(0) = \xi^0, \quad \xi_0(0) = \xi_0^0, \quad v(0) = v^0. $$
**Proof.** The proof follows that of Theorem 7.3 very closely and is thus omitted. ■

It is of interest to note that in the indicated proofs of Theorem 7.3 and 8.2 the Implicit Function Theorem is employed only once. However the basic Lemma 7.1 is used three times in each proof.

## 9. Bounds on the Number of Branches

We can get bounds on the maximum number of isolated roots that the RBE and LPBE may have. Since (6.13a) is m homogeneous equations of degree two in $m+1$ variables Bezout's theorem, see [1], implies that it has at most $2^m$ isolated roots in the m-dimensional complex projective space. The normalization condition (6.13b) selects two particular points on each such ray (differing only in sign). However each such pair of points yields the same solution branch of (6.1), one parametrized by $\epsilon$ and the other by $(-\epsilon)$. Thus for $m=1$, simple bifurcation, there are at most two branches through a regular bifurcation point $(G_{\lambda}^0 \in R(G_u^0))$.

To apply this analysis to the LPBE we need only introduce $\eta = \zeta^2$ to write (6.17a) as

$$ (9.1) \qquad A(\xi)\xi + \zeta^2 d = 0. $$

Thus again there are at most $2^m$ isolated roots in the m-dimensional complex projective plane. Now however if $(\xi, \zeta)$ is a root of (9.1) and (6.17b) then so is $(\xi, -\zeta)$. These roots are distinct in $\mathbb{CP}^m$, that is they do not lie on the same ray. But they yield the same branch of solutions of (6.1) ; see (7.4). Thus we can have at most $2^{m-1}$ isolated real roots of the LPBE. So for $m=1$ we can have at most one branch through $[u^0, \lambda^0]$ if $G_{\lambda}^0 \not\subset R(G_u^0)$.

We stress that we have only been concerned here with solution branches determined by isolated roots of the bifurcation equations. It is also known that "sheets" of solutions may eminate from bifurcation points when the bifurcation equations have non-isolated roots.
Acknowledgments

This work was supported, in part, by D.O.E. under contract
EY-76-S-03-0767 Project Agreement No 12 and by U.S.A.R.O. under contract
No. DAAG 29-78-C-0011.

It was written while on the staff at IRIA/LABORIA and was most
ably typed there by Mrs. Barny.

A Guggenheim Fellowship helped make possible the stimulating
visit in France.
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