Geodesic
Bifurcation diagram of a cubic three-parameter autonomous system
Electronic journal of differential equations, Tome 2005 (2005)
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 $k_2, k_3$. 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.
Classification : 34C05, 34D45, 34C23
Keywords: phase portrait, bifurcation, central manifold, topological equivalence, structural stability, bifurcation diagram;limit cycle 
@article{EJDE_2005__2005__a60,
     author = {Bar\'akov\'a,  Lenka and Volokitin,  Evgenii P.},
     title = {Bifurcation diagram of a cubic three-parameter autonomous system},
     journal = {Electronic journal of differential equations},
     year = {2005},
     volume = {2005},
     zbl = {1075.34035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a60/}
}
TY  - JOUR
AU  - Baráková,  Lenka
AU  - Volokitin,  Evgenii P.
TI  - Bifurcation diagram of a cubic three-parameter autonomous system
JO  - Electronic journal of differential equations
PY  - 2005
VL  - 2005
UR  - http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a60/
LA  - en
ID  - EJDE_2005__2005__a60
ER  - 
%0 Journal Article
%A Baráková,  Lenka
%A Volokitin,  Evgenii P.
%T Bifurcation diagram of a cubic three-parameter autonomous system
%J Electronic journal of differential equations
%D 2005
%V 2005
%U http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a60/
%G en
%F EJDE_2005__2005__a60
Baráková,  Lenka; Volokitin,  Evgenii P. Bifurcation diagram of a cubic three-parameter autonomous system. Electronic journal of differential equations, Tome 2005 (2005). http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a60/