TY - UNPB
T1 - Boundary crisis bifurcation in two parameters
AU - Osinga, Hinke M
N1 - Sponsorship: EPSRC Advanced Research Fellowship grant
PY - 2006/2
Y1 - 2006/2
N2 - The boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the strange attractor touches its own basin of attraction. Here we follow this codimension-one bifurcation in two parameters. One might expect that this leads to a smooth curve in the two-parameter plane. Mathematically, a boundary crisis is effectively a homoclinic or heteroclinic tangency, the locus of which is a well-defined smooth curve in a two-parameter system. However, instead of a boundary crisis, the transition through this tangency curve may lead to a basin boundary metamorphosis or an interior crisis bifurcation, in which the attractor persists. This phenomenon is again well known: at the
point where the type of transition changes, the boundary crisis switches to another branch of homoclinic or heteroclinic tangencies, associated with manifolds of a periodic point with a different period
than before. The curve of boundary crisis bifurcations is not differentiable at such points.
In this paper we show that the curve of boundary crisis bifurcations is, in fact, not even well defined as a piecewise-smooth curve. We illustrate that there are infinitely many gaps in much the same way as
the one-parameter bifurcation diagram of the attractor contains infinitely many windows where the attractor is periodic and not
strange or chaotic. Throughout, we use the Henon map to illustrate our findings.
AB - The boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the strange attractor touches its own basin of attraction. Here we follow this codimension-one bifurcation in two parameters. One might expect that this leads to a smooth curve in the two-parameter plane. Mathematically, a boundary crisis is effectively a homoclinic or heteroclinic tangency, the locus of which is a well-defined smooth curve in a two-parameter system. However, instead of a boundary crisis, the transition through this tangency curve may lead to a basin boundary metamorphosis or an interior crisis bifurcation, in which the attractor persists. This phenomenon is again well known: at the
point where the type of transition changes, the boundary crisis switches to another branch of homoclinic or heteroclinic tangencies, associated with manifolds of a periodic point with a different period
than before. The curve of boundary crisis bifurcations is not differentiable at such points.
In this paper we show that the curve of boundary crisis bifurcations is, in fact, not even well defined as a piecewise-smooth curve. We illustrate that there are infinitely many gaps in much the same way as
the one-parameter bifurcation diagram of the attractor contains infinitely many windows where the attractor is periodic and not
strange or chaotic. Throughout, we use the Henon map to illustrate our findings.
KW - boundary crisis
KW - heteroclinic tangency
M3 - Working paper
BT - Boundary crisis bifurcation in two parameters
ER -