II. A Business cycle Model
A forced van der Pol oscillator model of business cycle was chosen as a prototype
model to study the complex economic dynamics.
(8)
When
, (8) is the famous van der Pol equation. For VDP equation, the origin
in the phase space
fixed point if
is the only equilibrium solution, which is an unstable
.
In the case that the exogenous forcing is not equal to zero, (8) can evolve into a
periodic attractor or an aperiodic solution when we vary the control parameters
and
.
II. A Business cycle Model
Chaotic dynamics has been numerically studied [12].
A rigorous mathematical proof for the existence of chaos is needed.
[12] A. C.-L Chian, F. A. Borotto, E. L. Rempel, and C. Rogers, Attractor merging crisis in chaotic business cycles,
Chaos, Solitons and Fractals, 2005, 24: 869-875.
II. A Business cycle Model
☺Our work:
A rigorous proof for existence of chaos from mathematical
point of view is given.
II. A Business cycle Model
For the convenience of discussion in the following, we rewrite (8) as follows
(9)
which are four coupled first-order differential equations.
II. A Business cycle Model
Limit cycle and Chaos
(a)
(b)
Fig. 16 (a) A limit cycle of period-1 in the state space
(b) A chaotic attractor in the state space
for
for
.
.
A rigorous proof for the existence of topological horseshoe will be given.
II. A Business cycle Model
Firstly, we consider the plane
which is shown in Fig. 17.
Fig. 17. The attractor of business cycle model with
and the Poincaré section
.
II. A Business cycle Model
On this Poincaré section, after many trial numerical simulations we choose a proper
cross section
with its four vertices being
and will study the corresponding Poincaré map
This Poincaré map
is defined as follows: for each point
chosen to be the sixth return intersection point with
cycle model with initial condition
.
,
is
under the flow of business
II. A Business cycle Model
Fig.18. The cross section
and its image
under the sixth return Poincaré map
with
. The drawing that the arrow points at is a magnification of a part
of the original diagram.
Theorem 3. The Poincaré map
corresponding to the cross-sections
has the property that there exists a closed invariant set
for which
is semi-conjugate to the 2-shift map. Therefore,
This implies that for the parameter
system (8) has positive topological entropy.
, the business cycle
II. A Business cycle Model
Proof.
According to the Topological horseshoe Theorem, to prove this statement, we must find
two mutually disjointed subsets of
, such that a -connected family with respect
to them is existed.
Fig. 19 subset a and its image
After many attempts, the first subset is denoted by with
and
be its left and
right edge, respectively, as shown in Fig. 19. Under the sixth return Poincaré map , the
subset is mapped to its image
with
mapped to
and
mapped
to
. It is easy to see that
is on the left side of the edge
, and
is on the
right side of the edge
. In this case, we say that the image
lies wholly across
the quadrangle
with respect to
and
.
II. A Business cycle Model
Fig. 20 subset b and its image
The second subset is shown in Fig. 20, with
and
be its left and
right edge, respectively. Just like the situation for subset , the subset is mapped
to
under the Poincaré map ,
is mapped to
which is on
the left side of the edge
, and
is mapped to
which is on the right
side of the edge
. Thus the image
lies wholly across the quadrangle
with respect to
and
as well.
II. A Business cycle Model
Conclusions:
Given a rigorous computer-assisted proof for the existence
of chaos in this business cycle model
Let enter into the magic chaos
world

Let enter into the magic chaos
world
Thanks for your coming