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point in R+ x C+(M), i.e. a Jordan curve parameterized over R+ according to the
classification theorem of oriented I-dimensional manifolds. Let us summarize:
Theorem 1. Let (HO) - (H5) be satisfied. Then
a) ' 13 is the trace of a Jordan curve in R+ x C+(M): there exists a homeomorphism
1 : R+ -; ' 13 (onto), which is C 1 from (0,00) onto ' 13 \ {(O, O)} with l' (p) # (0,0) for
all P E (0,00).
b) ' 13 is "S-shaped": prl 01(P) -; 00, in! pr2 01(P) -; 00 as P -; 00, andprl 01 has
an even number of strict local extrema.
c) Given ({1, w) E ' 13, w is an asymptotically stable (unstable) stationary solution of
(1) if1'(?-1({1,w)) > 0 « 0).
Assertion c) states that equilibria located on forward bending segments of ' 13 are
asymptotically stable, whereas those lying on backward bending segments are unstable.
This is due to the fact that we have a principle of linearized stability at hand, which
relates the stability question to the sign of the principal eigenvalue ()..o(V, w) positive
implies asymptotic stability).
Details of the proof of the theorem can be found in [Hetzer (1990)].
B.E. Schmidt [Schmidt (1994)] investigated in her dissertation the case, where (H4)
is violated, i.e. a higher eigenvalue passes through zero. That can of course only occur
on a backward bending part of the principal component. She considered a semi linear
Legendre type equation
d
2 ,
-
--[k(x)(l - x )w (x)] = p, Q[l - a(w(x))]- g(w(x))
dx
-- 1 < :r; < 1,
which corresponds, so to speak, to a one-dimensiollal energy balance model with spatially homogeneous radiation fluxes and albedo. In that case the prillcipal branch consists of pairs (fl, p) with P being a constant, and only then it is possible to calculate the
eigenvalues of the linearized problem
-l
type, i.e. the solution set near such a hifurcation point consists of two simple curves,
point in R+ x C+(M), i.e. a Jordan curve parameterized over R+ according to the
classification theorem of oriented I-dimensional manifolds. Let us summarize:
Theorem 1. Let (HO) - (H5) be satisfied. Then
a) ' 13 is the trace of a Jordan curve in R+ x C+(M): there exists a homeomorphism
1 : R+ -; ' 13 (onto), which is C 1 from (0,00) onto ' 13 \ {(O, O)} with l' (p) # (0,0) for
all P E (0,00).
b) ' 13 is "S-shaped": prl 01(P) -; 00, in! pr2 01(P) -; 00 as P -; 00, andprl 01 has
an even number of strict local extrema.
c) Given ({1, w) E ' 13, w is an asymptotically stable (unstable) stationary solution of
(1) if1'(?-1({1,w)) > 0 « 0).
Assertion c) states that equilibria located on forward bending segments of ' 13 are
asymptotically stable, whereas those lying on backward bending segments are unstable.
This is due to the fact that we have a principle of linearized stability at hand, which
relates the stability question to the sign of the principal eigenvalue ()..o(V, w) positive
implies asymptotic stability).
Details of the proof of the theorem can be found in [Hetzer (1990)].
B.E. Schmidt [Schmidt (1994)] investigated in her dissertation the case, where (H4)
is violated, i.e. a higher eigenvalue passes through zero. That can of course only occur
on a backward bending part of the principal component. She considered a semi linear
Legendre type equation
d
2 ,
-
--[k(x)(l - x )w (x)] = p, Q[l - a(w(x))]- g(w(x))
dx
-- 1 < :r; < 1,
which corresponds, so to speak, to a one-dimensiollal energy balance model with spatially homogeneous radiation fluxes and albedo. In that case the prillcipal branch consists of pairs (fl, p) with P being a constant, and only then it is possible to calculate the
eigenvalues of the linearized problem
-l
type, i.e. the solution set near such a hifurcation point consists of two simple curves,