Locus of boundary crisis: Expect infinitely many gaps

Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied. In a two-parameter setting the locus of the boundary crisis is associated with curves of homoclinic or heteroclinic bifurcations of periodic saddle points. It is known that this locus has nondifferentiable points. We show here that the locus of boundary crisis is far more complicated than previously reported. It actually contains infinitely many gaps, corresponding to regions (of positive measure) where attractors exist.

Figure 1

Manifolds of the Hénon map (1) for $(\alpha ,\beta )=(1.4269212,0.3)$. The unstable manifold $W^u\left(p_0\right)$ of the fixed point $p_0$ is tangent to the stable manifold $W^s\left(p_1\right)$ of the fixed point $p_1$.

Figure 2

Bifurcations along the curve $T_1$ of tangency between $W^u\left(p_0\right)$ and $W^s\left(p_1\right)$ from the double-crisis vertices $V_1$ to $V_0$. The chaotic attractor that exists just to the left of $T_1$ contains periodic windows (a), each of which is initiated by saddle-node bifurcations that cross $T_1$ transversely (b). The labels in (a) indicate the periods of the periodic windows and the corresponding bifurcating periodic orbits are indicated in (b).

Figure 3

Basins of attraction with the attractors of the Hénon map (1) for $(\alpha ,\beta )$ near $({\alpha }_5,{\beta }_5)=(1.49017,0.26553)$. The phase portraits are for $({\alpha }_5,{\beta }_5-0.005)$ (a), $({\alpha }_5-0.01,{\beta }_5)$ (b), and $({\alpha }_5,{\beta }_5+0.005)$ (c).