238 R. Schwieger and H. Siebert
ODE-System from the functionfgiven in Example 1 by using one of the methods
explained in [ 15 ]^2. We obtain:
x ̇=f ̃(x)−x
x(0) =x 0withf ̃:[0,1]^4 →[0,1]^4 given by
f ̃(x)=(
1 −x 4 x+0^4. 5 x 1 +0x^1. 5 x 2 x+0^2. 5 ·(
1 −x 4 x+0^4. 5)
1 −x 3 x+0^3. 5)t
.It can easily be checked that the mapf ̃is in the monotonic model ensem-
ble M(Σ). Figure 2 shows the solution of the ODE-System, if we choose
x 0 =
(
0. 60. 60. 60. 6
)t
, and illustrates that it is a reasonable function, thus
belonging to the solution set.
Fig. 2.Trajectories of a solution of an ODE in the model ensemble. Its abstrac-
tion is given by (− 1 ,− 1 ,− 1 ,−1)→(1,− 1 ,− 1 ,−1)→(1, 1 ,− 1 ,−1). The ODE was para-
metrized withx 0 =(0. 6 , 0. 6 , 0. 6 , 0 .6),d =(1, 1 , 1 ,1),θ =(0. 5 , 0. 5 , 0. 5 , 0 .5) and
k=(1, 1 , 1 ,1).
We are now looking for qualitative features of solutions in order to find
properties common to all ODEs in the ensemble. The idea is to obtain a rough
description of solution trajectories by keeping track of the sign changes in the
derivative. In general, to each reasonable solutionx:[0,T]→X,Tfinite, we
can assign a unique ordered, maximal sequence (tj)j∈{ 0 ,...,M},tj∈[0,T],
witht 0 =0,tM=Tandtj∈(0,T) with the vector ̇x(tj)havingazeroentry
forj∈{ 1 ,...,M− 1 }indicating sign jumps of the trajectory. IfTis infinite
we can define a similar sequence not ending inT. In the interval between the
(^2) More specifically speaking, we use a multivariate interpolation and a subsequent
concatenation with Hill Cubes to obtain an ODE-System, which is guaranteed to
have a Jacobi matrix, whose abstraction coincides with the matrixΣon the off
diagonals. Then we choose arbitrarily Hill coefficients and thresholds.