262
As argued before the solution flow fJ I--t e I' (.; fJ) (fJ E R+) is gradient-like and absorbed
by a bounded interval for each P E R+, hence each solution trajectory converges to the
set of equilibria. Again, the stable stationary solutions (of (4) should be associated with
the climates of the earth. Now consider 6 = {(p, 0 E R+ x R+ : pO [1- a( OJ = g( O}.
Setting p,( 0 = Q[Ig~~(()l we can write 6 = {(p,( 0, 0 : ( E R+}. We assume that a
satisfies the hypotheses (H2) without "x-dependence" and that 9 fulfils (H3). It follows
from limy-->o !:f~~ = 0 (cf. (H3)) that II:(() := r(()[l - a(()J + g(()iY(O > 0 for
( E (0, p) and some p > 0, hence p,' > 0 near O. Clearly, p,( 0 -+ 00 as ( -+ 00 because
of g(O -+ 00. Moreover, the hypotheses lim yiY(y) = 0 and lim !\Y(ll < 00 imply for
y-->oo
y-->oo yg y
some TJ > 0 that
for ( sufficiently large. Thus p, is also strictly increasing near infinity.
Typically, II: will have a finite (even) number of simple zeroes, e.g. two, say (1 < (2,
and no other zeroes. Setting pj := p,( (j) for j=1,2, we conclude that PI is a local
maximum, whereas P2 is a local minimum of p,. Thus, 6 displayed in the p, ( - plane
is S-shaped in the sense that it is trace of a simple (Jordan) curve, which connects
(0,0) and (00,00) and has two (an even number) of turning points, the strict .local
extrema of p,. Moreover, we find asymptotically stable equilibria on the forward bending
segments of the curve and unstable ones on the backward bending segment ( s). The
latter can be seen by invoking the principle of linearized 8tability, which asserts that
( is asymptotically stable (unstable) under the solution flow fJ I--t ejt(()(-; fJ) provided
that -P,(OQiY(O _g'(() < 0 (> 0), which is equivalent to II:(() > 0 « 0) thanks to
P,(OQ[l- a(OJ = g(O· But, 11:(0 > 0 « 0) characterizes forward (backward) bending
curve segments.
What is the climatological significance of the S-shapedness of 6? Computer simulations of various models show (cf. [Ghil/Childress (1987), Hetzer et al. (1989), North
et al. (1981)J and the references therein) that the present climate state should be identified with a solution pair (p,O on the upper stable branch, i.e. (> (2 rather close
to the turning point (P,( (2), (2) and passing through such a turning point (caused by a
reduction in p ) would be a "climate catastrophe". On the other hand, S-shapedness
allows only for this one simple type of catastrophes, the turning point catastrophe, and
As argued before the solution flow fJ I--t e I' (.; fJ) (fJ E R+) is gradient-like and absorbed
by a bounded interval for each P E R+, hence each solution trajectory converges to the
set of equilibria. Again, the stable stationary solutions (of (4) should be associated with
the climates of the earth. Now consider 6 = {(p, 0 E R+ x R+ : pO [1- a( OJ = g( O}.
Setting p,( 0 = Q[Ig~~(()l we can write 6 = {(p,( 0, 0 : ( E R+}. We assume that a
satisfies the hypotheses (H2) without "x-dependence" and that 9 fulfils (H3). It follows
from limy-->o !:f~~ = 0 (cf. (H3)) that II:(() := r(()[l - a(()J + g(()iY(O > 0 for
( E (0, p) and some p > 0, hence p,' > 0 near O. Clearly, p,( 0 -+ 00 as ( -+ 00 because
of g(O -+ 00. Moreover, the hypotheses lim yiY(y) = 0 and lim !\Y(ll < 00 imply for
y-->oo
y-->oo yg y
some TJ > 0 that
for ( sufficiently large. Thus p, is also strictly increasing near infinity.
Typically, II: will have a finite (even) number of simple zeroes, e.g. two, say (1 < (2,
and no other zeroes. Setting pj := p,( (j) for j=1,2, we conclude that PI is a local
maximum, whereas P2 is a local minimum of p,. Thus, 6 displayed in the p, ( - plane
is S-shaped in the sense that it is trace of a simple (Jordan) curve, which connects
(0,0) and (00,00) and has two (an even number) of turning points, the strict .local
extrema of p,. Moreover, we find asymptotically stable equilibria on the forward bending
segments of the curve and unstable ones on the backward bending segment ( s). The
latter can be seen by invoking the principle of linearized 8tability, which asserts that
( is asymptotically stable (unstable) under the solution flow fJ I--t ejt(()(-; fJ) provided
that -P,(OQiY(O _g'(() < 0 (> 0), which is equivalent to II:(() > 0 « 0) thanks to
P,(OQ[l- a(OJ = g(O· But, 11:(0 > 0 « 0) characterizes forward (backward) bending
curve segments.
What is the climatological significance of the S-shapedness of 6? Computer simulations of various models show (cf. [Ghil/Childress (1987), Hetzer et al. (1989), North
et al. (1981)J and the references therein) that the present climate state should be identified with a solution pair (p,O on the upper stable branch, i.e. (> (2 rather close
to the turning point (P,( (2), (2) and passing through such a turning point (caused by a
reduction in p ) would be a "climate catastrophe". On the other hand, S-shapedness
allows only for this one simple type of catastrophes, the turning point catastrophe, and