Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 83, pp. 1-16.
Title: Bifurcation diagram of a cubic three-parameter
autonomous system
Authors: Lenka Barakova (Mendel Univ., Brno, Czech Rep.)
Evgenii P. Volokitin (Sobolev Institute of Math. Russia)
Abstract:
In this paper, we study the cubic three-parameter autonomous planar system
$$\displaylines{
\dot x_1 = k_1 + k_2x_1 - x_1^3 - x_2,\cr
\dot x_2 = k_3 x_1 - x_2,
}$$
where $k_2, k_3>0$. Our goal is to obtain a
bifurcation diagram; i.e., to divide the parameter space into
regions within which the system has topologically equivalent phase
portraits and to describe how these portraits are transformed at
the bifurcation boundaries. Results may be applied to the
macroeconomical model IS-LM with Kaldor's assumptions. In this
model existence of a stable limit cycles has already been studied
(Andronov-Hopf bifurcation). We present the whole bifurcation
diagram and among others, we prove existence of more difficult
bifurcations and existence of unstable cycles.
Submitted February 9, 2005. Published July 19, 2005.
Math Subject Classifications: 34C05, 34D45, 34C23.
Key Words: Phase portrait; bifurcation; central manifold;
topological equivalence; structural stability;
bifurcation diagram;limit cycle.