263
it is therefore of considerable interest to see that this S-shapedness will be preserved in
various settings in a hierarchy of models. E.g., assuming for a moment that such a model
would reflect the system's behavior on a long-term time-scale, we could conclude that
the climate system exhibits no low-frequency autooscillations, an issue of considerable
interest.
In coming back to (3) we can no longer expect that 6 consists of only one component,
but there is a distinguished one, the principal branch I,}}, which is the only unbounded
component of 6. Computer simulations of many one- and two-dimensional climate
models show that I,}} is S-shaped in R+ x C+(M). Note that as long as I,}} is trace of
a Jordan curve, turning points are a well-defined concept; namely they are the strict
extrema of pri 0, for any paramerization, of I,}} , if pri : R+ xC+(M) --> R+ denotes the
natural projection onto the parameter set. We are going to explain these computational
findings by means of two "structural hypotheses" , which correspond to the assumptions
about the zeroes of K, in our toy model.
The a priori estimates stated above under "absorbing property" apply in particular
to stationary solutions of (1), i.e. solutions of (3). Therefore we obtain immediately
from Rabinowitz's version of the Leray-Schauder principle that prl (~) = R+. Just
rewrite (3) as an operator equation
(5)
A w + p w = F(/1, w) + p w
and observe that (A + pld)-I is a compact linear operator for each p > O. Here,
Aw:= -div(kgradw) for allw E W2,2(M) with l1w E G(M), where W2,2(M) denotes
the well-known Sobolev space, and F(/1,w)(x) = /1Q(x)[l - a(x,w(x))]- g(w(x)) for
wE G(M) and x E M. Clearly, (O,w) E (5 implies w = 0, and it is an easy exercise
to show that the fixed point index of (A + pld)-l 0 F(O,') at 0 is nonzero. Moreover,
employing the last assumption under (H3) and the a priori estimates one derives for /1
in an interval [0, ~J (f!:.. > 0) that there exists an 1) > 0 with
0= 1M {k(x) I grad (WI - w2)(x)1 2 - [F(/1, wt)(x)-F(/1, W2)(x)][wl(:r) - w2(x)]}dx
~ 1) IWl(X) - w2(x)1 2 ,
whenever (/1, wt), (/1, W2) E 6, which shows unique solvability of (3) for small /1 > O.
Likewise, the asymptotic conditions at infinity stated under (H2) and (H3) guarantee
it is therefore of considerable interest to see that this S-shapedness will be preserved in
various settings in a hierarchy of models. E.g., assuming for a moment that such a model
would reflect the system's behavior on a long-term time-scale, we could conclude that
the climate system exhibits no low-frequency autooscillations, an issue of considerable
interest.
In coming back to (3) we can no longer expect that 6 consists of only one component,
but there is a distinguished one, the principal branch I,}}, which is the only unbounded
component of 6. Computer simulations of many one- and two-dimensional climate
models show that I,}} is S-shaped in R+ x C+(M). Note that as long as I,}} is trace of
a Jordan curve, turning points are a well-defined concept; namely they are the strict
extrema of pri 0, for any paramerization, of I,}} , if pri : R+ xC+(M) --> R+ denotes the
natural projection onto the parameter set. We are going to explain these computational
findings by means of two "structural hypotheses" , which correspond to the assumptions
about the zeroes of K, in our toy model.
The a priori estimates stated above under "absorbing property" apply in particular
to stationary solutions of (1), i.e. solutions of (3). Therefore we obtain immediately
from Rabinowitz's version of the Leray-Schauder principle that prl (~) = R+. Just
rewrite (3) as an operator equation
(5)
A w + p w = F(/1, w) + p w
and observe that (A + pld)-I is a compact linear operator for each p > O. Here,
Aw:= -div(kgradw) for allw E W2,2(M) with l1w E G(M), where W2,2(M) denotes
the well-known Sobolev space, and F(/1,w)(x) = /1Q(x)[l - a(x,w(x))]- g(w(x)) for
wE G(M) and x E M. Clearly, (O,w) E (5 implies w = 0, and it is an easy exercise
to show that the fixed point index of (A + pld)-l 0 F(O,') at 0 is nonzero. Moreover,
employing the last assumption under (H3) and the a priori estimates one derives for /1
in an interval [0, ~J (f!:.. > 0) that there exists an 1) > 0 with
0= 1M {k(x) I grad (WI - w2)(x)1 2 - [F(/1, wt)(x)-F(/1, W2)(x)][wl(:r) - w2(x)]}dx
~ 1) IWl(X) - w2(x)1 2 ,
whenever (/1, wt), (/1, W2) E 6, which shows unique solvability of (3) for small /1 > O.
Likewise, the asymptotic conditions at infinity stated under (H2) and (H3) guarantee