This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by for a certain constant :then at the limit, we would end up with a function that satisfies . This is a Feigenbaum function, which appears in most period-doubling routes to chaos (thus it is an instance of universality). Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant .thumb|480x480px|For the wrong values of scaling factor , the map does not converge to a limit, but when , it converges. thumb|487x487px|At the point of chaos , as we repeat the functional equation iteration with , we find that the map does converge to a limit.The constant can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is , it converges. This is the second Feigenbaum constant.
This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by for a certain constant :then at the limit, we would end up with a function that satisfies . This is a Feigenbaum function, which appears in most period-doubling routes to chaos (thus it is an instance of universality). Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant .thumb|480x480px|For the wrong values of scaling factor , the map does not converge to a limit, but when , it converges. thumb|487x487px|At the point of chaos , as we repeat the functional equation iteration with , we find that the map does converge to a limit.The constant can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is  it converges. This is the second Feigenbaum constant.