Size Ratio Universality in Chaotic Systems

Feigenbaum discovered a pattern in the period doubling route to chaos for nonlinear systems whose first return map is unimodal with a quadratic extremum. This led to the definition of the Feigenbaum universal constant δ. This constant could then be used to predict the parameter value for which such nonlinear systems turn chaotic, that is, the parameter value (P∞) for which the period tends to infinity. He also recognized that each successive period doubling bifurcation is a replica of the bifurcation just before itself. This observation suggested a universal size scaling in the period doubling sequence, which led to the definition of the second Feigenbaum universal constant α [Feigenbaum, M. J. , 1983]. The size of the chaotic attractor at a given parameter value is defined as the difference between the maximum and minimum values of the variable attained by the system for that parameter value. The demonstration of how the Fibonacci sequence appears within the Feigenbaum scaling of the period doubling cascade to chaos hinted at the possibility of a correlation between the size of the attractor (chaotic) at P∞ to the period 2 attractor [Linage et al., 2006]. Our work here lays down an argument for the existence of a universality relating the size of the chaotic attractor at P∞ to that of period 2. This discovery has immense potential in its application for designing experiments in the chaotic regime for systems exhibiting period doubling with the variation in system parameter values that is characteristic of an entire class of nonlinear phenomena.