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Bifurcation Sequence in a Simple Model of
Migratory Dynamics
by
Jeppe Sturis and Erik Mosekilde
Technical University of Denmark
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Abstract
A bifurcation sequence in the Waycross model
is studied by means of Poincaré section tech-
niques. The bifurcation parameter B is gradu-
ally reduced from 2.00 to 1.50. This parame-
ter measures the inclination of one type of
minority families (Lomanians) to move into
districts with many families of another type
of minority population (Itrachians). Because
of symmetry the attractors in this 4-dimen-
sional migratory model occur in pairs with
opposite directions of cyclic population
movements. A pair of simple limit cycle
attractors are found to remain stable under
formation of a pair of period-2 attractors.
In a certain parameter range, the model thus
contains four entangled attractors. We follow
how the period-2 attractors become chaotic
through formation and subsequent destabiliza-
tion of 2-dimensional tori. On the way, regu-
lar period-14, period-18 and period-4 attrac-
tors are produced through frequency-locking.
We thereafter observe a case of type III
intermittency when the two period-1 orbits
become unstable, and finally the two chaotic
attractors merge with each other.
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Waycross model
Dynamical Hypothesis
Unstable behavior in migratory systems arises
from positive feed-back mechanisms which
cause people to cluster in neighborhoods that
already house relatively many families of
similar characteristics or origin. The Way-
cross model considers two such subpopula-
tions: Lomanians and Itrachians which can
migrate between three districts of a town. A
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rotating behavior is introduced by assuming
that while Lomanians are attracted to areas
populated by Itrachians, the Itrachians do
not particularly appreciate Lomanians. Once
Lomanian families start to move into Itra-
chian areas, the Itrachians prefer to move
out. The non-linear restraints required to
keep the system within a finite volume of
phase space derive from the logical condition
that all populations must remain non-nega-
tive. Finally, the conservation of families
in the migratory process ensures a relatively
low dissipation. This facilitates the deve-
lopment of complicated non-linear phenomena
such as deterministic chaos.
Since both the total Lomanian and the
total Itrachian populations are assumed to be
constant, the model only contains four inde-
pendent variables: the number of Lomanian fa-
milies in district 1 (L,) and district 2 (L,)
and the number of Itrachian families in the
same two districts (I, and I,, respectively).
Because of symmetry, the attractors
occur in pairs with opposite directions of
cyclic population movements. Our investiga-
tion has revealed that a pair of simple limit
cycle attractors which exist for B = 2.0
remain stable under the formation of a pair
of stable period-2 attractors.
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uf
Poincaré section ~ B = 2.0
To investigate the model, we use Poincaré
sections: The intersections between the tra-
jectories in the four-dimensional phase space
and a three-dimensional hyperplane are recor-
ded for the trajectories passing through the
plane in a particular direction. In this way,
the system is reduced by one dimension. The
figure illustrates the construction of a
Poincaré section using a three-dimensional
phase space and a regular two-dimensional
plane.
~8Hi~
Four periodic attractors - B = 1.8
oe
/ se, p?
A gael
Cy "
Cc, a
i<—@
c3—
N-12
L1-L2
<—eP
4
qh oP
1 cP
po os .. | 2
A two-dimensional projection of the Poincaré
sections for B = 1.8 reveals four co-existing
attractors. The three-dimensional cutting-
plane passes through the unstable equilibrium
point (L, =lL, = Ll, = 1, = 1, = 1;) witha
normal vector of (1,1,1,1). Transient ap-
proaches to the four attractors are seen be-
ginning at P,, P,, Q, and Q,, respectively.
The corresponding stable solutions are deno-
ted c#?, cP, c3 and co.
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rc
a
(\
Le (\\
Four periodic attractors - B = 1.8
es
Three-dimensional representations of the four
co-existing attractors for B = 1.8.
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Original stable orbit
Two tori
and
_ if
two limit cycles A
B= 1.73
won?
A two-dimensional projection of the Poincaré
section for B = 1.73. The two symmetric pe-
riod-1 orbits remain stable, while the two
symmetric period-2 limit cycles each undergo
a torus-bifurcation when B is reduced below
1.75. This means that the trajectories are
located on the surface of a torus (or a
doughnut). In this case the oscillations
never repeat themselves. The solution is said
to be quasi-periodic.
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|
SF
L1
Frequency-locking - B = 1.72
Sometimes the trajectories on the surface of
the doughnut describe a closed curve. This is
the case when B = 1.72, and it is a result of
frequency-locking between the two frequencies
associated with the two unstable modes that
together produce the torus. For this para-
meter value the solution is of period 4.
Other parameter combinations give different
solutions, for example period 14 and 18. We
are here observing part of a so-called
devil’s staircase.
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*
Two chaotic fs
- “42
and
o
two limit cycle “ i
e
” {
attractors :
i
B = 1.70 !
f
on”
. ™.
f ws,
i ae
« .
i] “
i e
i *
moe
L +e
. Mey a mene ty
j as # .
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When B is reduced to 1.7, the tori become
unstable, and the Poincaré sections exhibit
folding. The trajectories can no longer be
described as being located on the surface of
a doughnut. Now these two attractors are
chaotic. It is important to note that the two
original period-1 orbits continue to be
stable.
A
aOR ol
For B = 1.67, another qualitative change has
taken place. By now the two period-1 orbits
have turned unstable. As a result, type III
intermittency is observed. In the Poincaré
section, this can be seen as a merging of the
two halves of each attractor, the merging be-
ing at the location of the unstable periodic
orbits. In the time-plot, the intermittency
is revealed as an alternating increase and
decrease in amplitude of the oscillations.
Type III
intermittency -
B= 1.67 !
Unstable orbit
| cl oe
| mn i a i i i
Le i i LGA
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Merging of attractors
With even further
reduction of B,
the two chaotic
attractors conti-
nue to expand,
until they finally
merge with each
other. The first
two pictures here
show the two chao-
- tic solutions when
B = 1.7. In the
last figure, qua-
litative features ~
of the first two
pictures can be
seen, as a con-
sequence of the
merging. In this
case B= 1.5.
1
L2
