Dept. of Applied Mathematics, Masaryk University
Janackovo nam. 2a, 602 00 Brno, Czech Republic
email: pribylova@math.muni.cz
Lenka Pribylova
Abstract:
In this paper a 3-dimensional system of autonomous differential equations
is studied. It can be interpreted as an idealized macroeconomic model
with foreign capital investment introduced in [9] or
an idealized model of the firm profit introduced in [3].
The system has three endogenous variables with only one non-linear term
and can be also interpreted as an extended van der Pol's equation.
It's shown that this simple system covers several types of bifurcations:
both supercritical and subcritical Hopf bifurcation and generalized
Hopf bifurcation as well, the limit cycle exhibits period-doubling
bifurcation as a route to chaos. Some results are analytical and
those connected with chaotic motion are computed numerically
with continuation programs Content, Xppaut and Maple. We present conditions
for stability of the cycles, hysteresis, explore period doubling and
using Poincare mapping show a three period cycle that implies chaos.
Submitted April 21, 2008. Published April 17, 2009.
Math Subject Classifications: 70K50, 37D45.
Key Words: Hopf bifurcation; period doubling; chaos.
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Lenka Pribylova Dept. of Applied Mathematics, Masaryk University Janackovo nam. 2a, 602 00 Brno, Czech Republic email: pribylova@math.muni.cz |
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