Destruction of islands of stability

We describe the changes and the destruction of islands of stability in four dynamical systems: (a) the standard map, (b) a Hamiltonian with a cubic nonlinearity, (c) a Hamiltonian with a quartic nonlinearity and (d) the Sitnikov problem. As the perturbation increases the size of the island increases and then decreases abruptly. This decrease is due to the joining of an outer and an inner chaotic domain. The island disappears after a direct (supercritical) or an inverse (subcritical) bifurcation of its central periodic orbit C. In the first case, when C becomes unstable, a chaotic domain is formed near C. This domain is initially separated from the outer 'chaotic sea' by KAM curves. But as the perturbation increases the inner chaotic domain grows outwards, while the outer 'chaotic sea' progresses Inwards. The last KAM curve is destroyed by forming a cantorus and the two chaotic domains join. But even then the escape of orbits through the cantorus takes a long time (stickiness effect). In the inverse bifurcation case the island around the central orbit is limited by two equal period unstable orbits. As,the perturbation changes these two orbits approach and join the central orbit, that becomes unstable. Then the island disappears but no cantori are formed. In this case the stickiness is due to the delay of deviation of an orbit from the unstable periodic orbit when its eigenvalue is not much larger than 1.