The lectures presented by the author are not reproduced here since that material is available in J. Marsden, Qualitative Methods in Bifurcation Theory, Bull. Am. Math. Soc. 84 (1978), 1125–1148, R. Abraham and J. Marsden, Foundations of Mechanics, Second Edition, Addison Wesley (1978), and in J. Marsden and M. McCracken, The Hopf Bifurcation and its Application

Marsden, Jerrold E.

English

Caltech Authors - Main

t On the Geometry of the Liapunov-Schmidt Procedure 
Jerrold E. Marsden* 
1. Introduction 
One of the most commonly used procedures in bifurcation theory 
is the Liapunov-Schmidt procedure (which we review in §2). However, 
in differential topology this procedure is also regularly used, but 
of course, under a different name, namely "transversality." The aim 
of this short note is to make this link explicit. 
There are two reasons for geometrizing the Liapunov-Schmidt 
procedure. First of all, it is useful in some applications the author, 
A. Fischer and V. Moncrief have made to relativity (work in progress). 
Secondly, the dynamic analogue of the procedure, namely center mani-
fold theory, already has a geometric flavor (i.e. it makes intrinsic 
sense on manifolds), so it is natural to bring the classical Liapunov-
Schmidt procedure in line with it. 
2. Review of the Liapunov-Schmidt Procedure 
Let x and Y be Banach spaces and f: X x RP + Y a map, 
k ~ 1. Let Dxf(x,A) be the (Frechet) derivative of f with respect 
to x, a continuous linear map of X to Y. Let f(xO,A O) = 0 and 
let 
t The lectures presented by the author are not reproduced here since 
that material is available in J. Marsden, Qualitative Methods in 
Bifurcation Theory, Bull. Am. Math. Soc. 84 (1978), 1125-1148, 
R. Abraham and J. Marsden, Foundations oTMechanics , Second Edition, 
Addison Wesley (1978), and in J. Marsden and M. McCracken, The Hopf 
Bifurcation and its Applications, Springer Applied Math Sciences 
#19 (1976). 
*Partially supported by the National Science Foundation. 
78 
Assume Xl is finire dimensional with a complement X2 so that 
X = Xl ~ X2 · Also, assume 
is closed and has a finite-dimensional complement Y 2 • In other words, 
Dxf(xO'YO) is a Fredholm operator. Write Y = YI ~ Y2 and let 
P: Y 7 YI be the projection. By the implicit function theorem, the 
equation 
has a unique solution x = 2 near where 
x = xl + x 2 E X = Xl ~ X2 . Thus, the equation f(x,A) = 0 is equiva-
lent to the bifurcation equation 
a system of dim Y2 equations in dim Xl unknowns. This reduction of 
f(X,A) = 0 to the bifurcation equation is the Liapunov-Schmidt 
procedure. 
For purposes of this procedure alone, the assumption that Xl 
and Y 2 are finite dimensional is, of course, irrelevant. This is 
made with the theory of Fredholm operators waiting in the wings. 
Similarly, the parameter space RP may be replaced by a Banach space 
Z. In fact, the parameter is just "along for the ride." 
3. A Topological Procedure 
Let X and Z be Banach manifolds, Y a Banach space and 
f: X x Z 7 Y be a map. (More generally, one can replace X x Z 
by a fiber bundle over Zl. We are interested in solving the equation 
79 
f(x,A) o for (X,A) E X x Z. Let (XO,A O) be a known solution and 
let 
and assume Xl splits; i.e. 
X2 C T X. Let Xo 
Xl ® X2 for a closed subspace 
and assume YI is closed and splits; i.e. Y = YI @ Y2 for a closed 
subspace Y 2 • (This, of course, involves a choice of Y 2 , as it 
did above. ) Let P: Y -+ YI be the projection. 
The map f is, for fixed A 0' transversal to the subspace Y2 
at (XO,A O). Therefore, in a neighborhood of (XO,A O)' 
{(X,A) I Pf(x,A) o} 
is a smooth submanifold of X x Z tangent to Xl x TA Z at 
0 
(XO,A O). [In the notation of § 2, Sp = { (xl + u (xl' A ) ,A ) } ] . 
Let fp denote the restriction of f to Sp. Clearly f(x,A) = 
iff (I - P)fp(x,A) = 0 iff fp(x,A) = O. The later condition is the 
geometric version of the bifurcation equation. It has proven to be 
useful, at least to the author. 
4. More General 
We can allow Y to be an arbitrary Banach manifold to clarify 
the choices involved. (I don't know any examples where this is 
necessary or useful.) Now we fix Yo E Y and attempt to solve 
f(x,A) = yO near a known solution (xo,yO)· Let Xl' X2 be as 
above and let T Y = YI @ Y2 where YI = Range Dxf(xO,A O). yo 
0 
80 
Now choose a submanifold M C Y tangent to Y2 at The proce-
dure only depends on this choice; it is analogous to the choice in §2 
of a linear complement to Yl . 
Again, f is transversal to M (with A. a parameter), so 
SM {(x,A.) I f(x,A.) E M} 
is a submanifold of X x Z. Let fM be the restriction of f to SM' 
regarded as a map of SM to M. Then the obvious assertion that 
f(x,A.) Yo E M 
is the abstract Liapunov-Schmidt procedure. 
5. A Sample Calculation 
In the usual Liapunov-Schmidt theory of §2, to analyze zeros 
of the map 
we need to compute its derivatives at (x lO ,1,.0) (x lO is the first 
component of x O). It is usually assumed that DAf(xO,k O) = O. Thus 
(xlO,A. O) is a critical point for g. By implicit differentiation 
one finds that 
(The right hand side appropr£ately restricted.) 
In the context of f3, it is clear that if DA.f(xO'A.O) = 0 then 
(xO,A. O) is a critical point (in the sense of zero derivative) of f p . 
Moreover, it is now obvious from the fact that Hessians are well-
81 
defined at critical points that 
restricted to 
i.e. we recover the same conclusion as above. The procedure can, 
of course, be repeated to obtain a formula for if the k-l 
jet of fp vanishes, as is well-known in bifurcation theory. Note 
that once the structure of the zeros of g is found, this still has 
to be lifted via the graph of u to obtain the zeros of f. In the 
geometric setting this is not necessary; the zeros of fp are the 
zeros of f. 
While these results are essentially nothing more than new 
language for well-known material, the geometric setting seems to 
clarify and even simplify what is going on. 
6. Pbtential Operators and Vector Fields 
If X = Yare Hilbert spaces and the original map f in 
Section 2 is the gradient of a potential function ~ in the variable 
x, then the reduced function g(xl,A) = (l-P)f(xl + u(xl,A) ,A)) is 
also a gradient, with a modified potential ~ (depending on u); 
see, for instance. P. Rabinowitz, J. Funct. An. ~ (1977) 412-424. 
We wish to explain the geometric meaning of this. In such cases, the 
function f may be thought of as a vector field on a manifold X, 
depending on the parameters A. 
For vector fields the procedure in §3 can be refined somewhat. 
At a zero (XO,A O) of a (parametrized) vector field f: X x Z + TX, 
the derivative Dxf(xO,A O) makes intrinsic sense as a linear map of 
T X to itself. One can now seek an invariant manifold SA for Xo 0 
f tangent to the subspace Xl = ker Dxf(xO,A O)' To do this for 
nearby A as well~ it is convenient to suspend f to the vector 
field I: X x Z +T(X x Z) by setting I(x,A) = (f(x,A),O) and 
find an invariant manifold S associated to ker DI(xO,A O); one can 
82 
then take SA = S n X x {A}, an invariant manifold for each A. One 
may regard SA as implicitly defined in' the same way as the function 
U(Xl,A) in the Liapunov-Schmidt procedure is implicitly defined. 
Zeros of f near (XO,A O) necessarily lie on S, so the 
problem reduces to finding zeros of fls, the analogue of the bifur-
cation equation. For finding fixed points , this is a geometric 
formulation of the Liapunov-Schmidt procedure. The fact that we are 
dealing with vector fields entails that the choice of Y2 (or M in 
Section 4) is now automatically made; both Sp and M are now re-
placed by S. 
In order to capture dynamic bifurcations as well as static ones, 
it is necessary to enlarge S to the full center manifold, as is 
explained in, for example, J. Marsden and M. McCracken, The Hopf 
Bifurcation and Its Applications 3 Springer Appl. Math. Sciences #19 
(1976). (For operators with real eigenvalues, such as potential 
operators, S equals the center manifold.) 
The fact that the reduction of a potential operator by the 
Liapunov-Schmidt procedure results in a potential operator is now 
clear. In fact, if one uses the space S, a modification of ~ is 
not necessary; one needs only to restrict it to S. This is because 
of the following obvious fact: the restriction of a gradient vector 
field to an invariant submanifold is a gradient vector field whose 
potential is the restriction of the original one; i.e. 
if V~ is tangent to S. 
(V ~) I S = V (~ I S) 


On the Geometry of the Liapunov-Schmidt Procedure

t On the Geometry of the Liapunov-Schmidt Procedure 
Jerrold E. Marsden* 
1. Introduction 
One of the most commonly used procedures in bifurcation theory 
is the Liapunov-Schmidt procedure (which we review in §2). However, 
in differential topology this procedure is also regularly used, but 
of course, under a different name, namely "transversality." The aim 
of this short note is to make this link explicit. 
There are two reasons for geometrizing the Liapunov-Schmidt 
procedure. First of all, it is useful in some applications the author, 
A. Fischer and V. Moncrief have made to relativity (work in progress). 
Secondly, the dynamic analogue of the procedure, namely center mani-
fold theory, already has a geometric flavor (i.e. it makes intrinsic 
sense on manifolds), so it is natural to bring the classical Liapunov-
Schmidt procedure in line with it. 
2. Review of the Liapunov-Schmidt Procedure 
Let x and Y be Banach spaces and f: X x RP + Y a map, 
k ~ 1. Let Dxf(x,A) be the (Frechet) derivative of f with respect 
to x, a continuous linear map of X to Y. Let f(xO,A O) = 0 and 
let 
t The lectures presented by the author are not reproduced here since 
that material is available in J. Marsden, Qualitative Methods in 
Bifurcation Theory, Bull. Am. Math. Soc. 84 (1978), 1125-1148, 
R. Abraham and J. Marsden, Foundations oTMechanics , Second Edition, 
Addison Wesley (1978), and in J. Marsden and M. McCracken, The Hopf 
Bifurcation and its Applications, Springer Applied Math Sciences 
#19 (1976). 
*Partially supported by the National Science Foundation. 
78 
Assume Xl is finire dimensional with a complement X2 so that 
X = Xl ~ X2 · Also, assume 
is closed and has a finite-dimensional complement Y
2
• In other words, 
Dxf(xO'YO) is a Fredholm operator. Write Y = YI ~ Y2 and let 
P: Y 7 YI be the projection. By the implicit function theorem, the 
equation 
has a unique solution x = 2 near where 
x = xl + x
2 
E X = Xl ~ X
2
. Thus, the equation f(x,A) = 0 is equiva-
lent to the bifurcation equation 
a system of dim Y2 equations in dim Xl unknowns. This reduction of 
f(X,A) = 0 to the bifurcation equation is the Liapunov-Schmidt 
procedure. 
For purposes of this procedure alone, the assumption that Xl 
and Y
2 
are finite dimensional is, of course, irrelevant. This is 
made with the theory of Fredholm operators waiting in the wings. 
Similarly, the parameter space RP may be replaced by a Banach space 
Z. In fact, the parameter is just "along for the ride." 
3. A Topological Procedure 
Let X and Z be Banach manifolds, Y a Banach space and 
f: X x Z 7 Y be a map. (More generally, one can replace X x Z 
by a fiber bundle over Zl. We are interested in solving the equation 
79 
f(x,A) o for (X,A) E X x Z. Let (XO,A O) be a known solution and 
let 
and assume Xl splits; i.e. 
X2 C T X. Let Xo 
Xl ® X2 for a closed subspace 
and assume YI is closed and splits; i.e. Y = YI @ Y2 for a closed 
subspace Y
2
• (This, of course, involves a choice of Y
2
, as it 
did above. ) Let P: Y -+ YI be the projection. 
The map f is, for fixed A 0' transversal to the subspace Y2 
at (XO,A O). Therefore, in a neighborhood of (XO,A O)' 
{(X,A) I Pf(x,A) o} 
is a smooth submanifold of X x Z tangent to Xl x TA Z at 
0 
(XO,A O). 
[In the notation of § 2, Sp = { (xl + u (xl' A ) ,A ) } ] . 
Let fp denote the restriction of f to Sp. Clearly f(x,A) = 
iff (I - P)fp(x,A) = 0 iff fp(x,A) = O. The later condition is the 
geometric version of the bifurcation equation. It has proven to be 
useful, at least to the author. 
4. More General 
We can allow Y to be an arbitrary Banach manifold to clarify 
the choices involved. (I don't know any examples where this is 
necessary or useful.) Now we fix Yo E Y and attempt to solve 
f(x,A) = yO near a known solution (xo,yO)· Let Xl' X2 be as 
above and let T Y = YI @ Y2 where YI = Range Dxf(xO,A O). yo 
0 
80 
Now choose a submanifold M C Y tangent to Y2 at The proce-
dure only depends on this choice; it is analogous to the choice in §2 
of a linear complement to Yl . 
Again, f is transversal to M (with A. a parameter), so 
SM {(x,A.) I f(x,A.) E M} 
is a submanifold of X x Z. Let fM be the restriction of f to SM' 
regarded as a map of SM to M. Then the obvious assertion that 
f(x,A.) Yo E M 
is the abstract Liapunov-Schmidt procedure. 
5. A Sample Calculation 
In the usual Liapunov-Schmidt theory of §2, to analyze zeros 
of the map 
we need to compute its derivatives at (x lO ,1,.0) (x lO is the first 
component of x O). It is usually assumed that DAf(xO,k O) = O. Thus 
(xlO,A. O) is a critical point for g. By implicit differentiation 
one finds that 
(The right hand side appropr£ately restricted.) 
In the context of f3, it is clear that if DA.f(xO'A.O) = 0 then 
(xO,A. O) is a critical point (in the sense of zero derivative) of f p . 
Moreover, it is now obvious from the fact that Hessians are well-
81 
defined at critical points that 
restricted to 
i.e. we recover the same conclusion as above. The procedure can, 
of course, be repeated to obtain a formula for if the k-l 
jet of fp vanishes, as is well-known in bifurcation theory. Note 
that once the structure of the zeros of g is found, this still has 
to be lifted via the graph of u to obtain the zeros of f. In the 
geometric setting this is not necessary; the zeros of fp are the 
zeros of f. 
While these results are essentially nothing more than new 
language for well-known material, the geometric setting seems to 
clarify and even simplify what is going on. 
6. Pbtential Operators and Vector Fields 
If X = Yare Hilbert spaces and the original map f in 
Section 2 is the gradient of a potential function ~ in the variable 
x, then the reduced function g(xl,A) = (l-P)f(xl + u(xl,A) ,A)) is 
also a gradient, with a modified potential ~ (depending on u); 
see, for instance. P. Rabinowitz, J. Funct. An. ~ (1977) 412-424. 
We wish to explain the geometric meaning of this. In such cases, the 
function f may be thought of as a vector field on a manifold X, 
depending on the parameters A. 
For vector fields the procedure in §3 can be refined somewhat. 
At a zero (XO,A O) of a (parametrized) vector field f: X x Z + TX, 
the derivative Dxf(xO,A O) makes intrinsic sense as a linear map of 
T X to itself. One can now seek an invariant manifold SA for 
Xo 0 
f tangent to the subspace Xl = ker Dxf(xO,A O)' To do this for 
nearby A as well~ it is convenient to suspend f to the vector 
field I: X x Z +T(X x Z) by setting I(x,A) = (f(x,A),O) and 
find an invariant manifold S associated to ker DI(xO,A O); one can 
82 
then take SA = S n X x {A}, an invariant manifold for each A. One 
may regard SA as implicitly defined in' the same way as the function 
U(Xl,A) in the Liapunov-Schmidt procedure is implicitly defined. 
Zeros of f near (XO,A O) necessarily lie on S, so the 
problem reduces to finding zeros of fls, the analogue of the bifur-
cation equation. For finding fixed points , this is a geometric 
formulation of the Liapunov-Schmidt procedure. The fact that we are 
dealing with vector fields entails that the choice of Y2 (or M in 
Section 4) is now automatically made; both Sp and M are now re-
placed by S. 
In order to capture dynamic bifurcations as well as static ones, 
it is necessary to enlarge S to the full center manifold, as is 
explained in, for example, J. Marsden and M. McCracken, The Hopf 
Bifurcation and Its Applications 3 Springer Appl. Math. Sciences #19 
(1976). (For operators with real eigenvalues, such as potential 
operators, S equals the center manifold.) 
The fact that the reduction of a potential operator by the 
Liapunov-Schmidt procedure results in a potential operator is now 
clear. In fact, if one uses the space S, a modification of ~ is 
not necessary; one needs only to restrict it to S. This is because 
of the following obvious fact: the restriction of a gradient vector 
field to an invariant submanifold is a gradient vector field whose 
potential is the restriction of the original one; i.e. 
if V~ is tangent to S. 
(V ~) I S = V (~ I S) 


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On the geometry of the Liapunov-Schmidt procedure

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On the Geometry of the Liapunov-Schmidt Procedure

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