267
and we wish to study the same questions as for (1) but with periodic orbits in C+(M)
replacing the stationary solutions. Some additional care is required, since considering
the polar night one has to relax the hypotheses of Q being positive to Q(t,x) ?: q(t)
with q a nonnegative function not identically zero.
How can the concept of S-shapedness be formulated in this framework? It is convenient to make use of the time-I-map (Poincare map) IT = IT(/1,19), which associates
u(I"j /1, 19) with each initial condition 19 E C+(M). As before, u = u( t, Xj /J, 19) denotes
the solution of (7) satisfying u(O,·) = 19 and M is an oriented compact Riemannian manifold without boundary, the Euclidean unit-sphere 8 2 e.g. in the climatological context.
The unique solvability of (7) yields that u( t, x; /1, 19) is I-periodic in t, iff IT(/1, 19) = 19,
i.e. there is a 1-1 correspondence between the I-periodic orbits in C+(M) and the
fixed points of IT = IT(/J,')' In fact, even more is true. A periodic solution of (7) is
orbitally asymptotically stable (unstable), iff the corresponding fixed point is asymptotically stable (unstable) under the time-I-map (cf. [Hess (1991)]). Therefore we can
adapt the concept of S-shapedness by considering the principal branch of the fixed point
set ~ := ((/1,19) E R+ x C+(M) : IT(/1, 19) = 19} in place of the stationary solution set (5
employed in Section 4.
It should be mentioned that one meets a serious conceptional problem, though, with
the notion of S-shapedness in the quasiperiodic case, which arises from additionally
accounting for orbital forcing.
We can follow the line of reasoning described in Section 4. for deriving S-shapedness
provided that the structural hypotheses (H4) and (H5) are adapted appropriately. The
issue of asymptotic convergence of solution trajectories to a periodic orbit is more delicate than previously; in fact, Liapunov's method is not available any longer, and one has
to settle with much weaker results. Finally, the fact that one has to admit zeroes of Q
in view of the polar night, creates a technical obstacle, which results in more restrictive
assumptions on the asymptotic growth of g at infinity.
6. An ODE Setting.
In order to illustrate some of the features of the time-I-map, we first turn again to an
ordinary differential equation. Unfortunately, we cannot interpret ate this equation as
a global model, since there is no seasonal variation of the incoming solar radiation flux
and we wish to study the same questions as for (1) but with periodic orbits in C+(M)
replacing the stationary solutions. Some additional care is required, since considering
the polar night one has to relax the hypotheses of Q being positive to Q(t,x) ?: q(t)
with q a nonnegative function not identically zero.
How can the concept of S-shapedness be formulated in this framework? It is convenient to make use of the time-I-map (Poincare map) IT = IT(/1,19), which associates
u(I"j /1, 19) with each initial condition 19 E C+(M). As before, u = u( t, Xj /J, 19) denotes
the solution of (7) satisfying u(O,·) = 19 and M is an oriented compact Riemannian manifold without boundary, the Euclidean unit-sphere 8 2 e.g. in the climatological context.
The unique solvability of (7) yields that u( t, x; /1, 19) is I-periodic in t, iff IT(/1, 19) = 19,
i.e. there is a 1-1 correspondence between the I-periodic orbits in C+(M) and the
fixed points of IT = IT(/J,')' In fact, even more is true. A periodic solution of (7) is
orbitally asymptotically stable (unstable), iff the corresponding fixed point is asymptotically stable (unstable) under the time-I-map (cf. [Hess (1991)]). Therefore we can
adapt the concept of S-shapedness by considering the principal branch of the fixed point
set ~ := ((/1,19) E R+ x C+(M) : IT(/1, 19) = 19} in place of the stationary solution set (5
employed in Section 4.
It should be mentioned that one meets a serious conceptional problem, though, with
the notion of S-shapedness in the quasiperiodic case, which arises from additionally
accounting for orbital forcing.
We can follow the line of reasoning described in Section 4. for deriving S-shapedness
provided that the structural hypotheses (H4) and (H5) are adapted appropriately. The
issue of asymptotic convergence of solution trajectories to a periodic orbit is more delicate than previously; in fact, Liapunov's method is not available any longer, and one has
to settle with much weaker results. Finally, the fact that one has to admit zeroes of Q
in view of the polar night, creates a technical obstacle, which results in more restrictive
assumptions on the asymptotic growth of g at infinity.
6. An ODE Setting.
In order to illustrate some of the features of the time-I-map, we first turn again to an
ordinary differential equation. Unfortunately, we cannot interpret ate this equation as
a global model, since there is no seasonal variation of the incoming solar radiation flux