6.4. Universality

If a one-dimensional difference equation involves one control parameter A then we represent it by

 

 

Typicaiiy, tor some range of A the output of the system is regular, periodic with period T, but then as A changes further the output becomes increasingly erratic in character. At first the system reproduces itself af:er a time IT until a further special A value is attained and so on. This process of period-doubling appears to the characteristic route by which simple, periodic systems degenerate into complex, aperiodic behaviour and chaos [94]. When A reaches a particular value the doubling will have occurred infinitely often and no periodicity will remain. After some early indications in the work of May and Oster [95],. Feigenbaum [96] has shown that this period-doubling phenomenon possesses remark­able general properties: if we label the parameter value at which the nth period-doubling occurs by A„,

The value of 8 so obtained (and there is very rapid convergence in (6.37)) appears to be universal amongst a family of discrete one-dimensional mappings, F, with a single maximum in [0,1]; its value is found K be S ~ 4.669201609... This property can also be understood theoretically [2,97] and is related to the presence of critical exponents as in current theories of critical phenomena wherein universal quantitative features are determined by the presence of some qualitative features in the system. Further universal pioperties have been found for one-dimensional maps exhibiting period-doubling [96] and the presence of period-doubling has been observed in the transition to turbulence of real systems [98]. It also characterizes chaotic solutions of differential equations and higher dimensional mappings.

The beauty of this theory is that it promise.» to provide a mathematically precise, yet experimentally testaMe theory of turbulent sjstems that does not require a precise model. Any member of the large class of equations exhibiting a transition to chaos via period-doubling will possess the same invariants like (6.37). In particular, a theory of fluid turbulence is quite independent of the Navier-Stokes equations. Also, if a particular process gives rise to a complicated mapping then we can extract infornation about the process in question by studying the simplest member of the mapping's uni­versality class.

If return mappings for the Einstein equations exhibit period-doubling then one could use the Feigenbaum theory to develop a description of gravitational turbulence also. A description that could describe the local behaviour of the general solution vo the Einstein equations in terms of numerical invariants like (6.37). Unfortunately, .ne Mixmaster return mapping T given by (4.20) does not exhibit period-doubling and 170 on [0,1]. However, it is possible that the chaotic evolution of other properties of the Nlixmacter Universe - oscillation amplitudes, oscillation periods or combinations of these quantities - .night display [99] period-doubling and the accompanying universality. Finally, if period-ooubling is in some sense the generic feature of chaot;c or turbulent behaviour then ore might anticipate its presence in the general solution to ine Einstein equations and conclude that the Mixmaster evolution possesses cer tain special constraints not present in the in homogeneous genera! solution.