As the perimeter continues to change, disturbances become more pronounced and although the system is still attracted back to stable it takes longer to get there jumping around the equilibrium instead of converging right in.
As the perimeter continues to grow suddenly we notice a change. The system is not settling back to stable. It is alternating between two perimeters on each side of where we thought the stable point was. We realize that what we thought was the attractor is now actually unstable and has become a repeller. If it is used as an initial state it quickly diverges and settles back to the attractors on either side. What has happened is we have just passed a bifurcation point. Now the attractor has divided and we see it is a strange attractor. as we continue to let our perimeter rise the difference between the oscillations increases until suddenly there are four oscillations instead of two. This means we have reched another bifurcation point. As our perimeter continues to rise the bifurcations increase faster and faster until they seem to become infinite and the systems seems totally unstable oscillating around within the given area seemingly at random. As se continue through this area suddenly we see a period of three appear. This soon doubles to period six, twelve, twenty four, etc. until we soon have chaos again. In a famous paper titled Period Three Implies Chaos it is proven that any system which exhibits a period of three among it's bifurcations will also include an area of chaos within it's system for a proper set or perimeters.