A Surfeit of Cycles

Abstract

Chaotic sets are organized about “skeletons” of periodic orbits in the sense that every point on a chaotic set is arbitrarily close to such an orbit. The orbits have the stability property of saddles: attracting in some directions; repelling in others. This topology has implications for changing climates that evidence pronounced variability on time scales ranging from decades to tens of thousands of years. Among these implications are the following: 1. A wide range of periodicities should be (and are) observed. 2. Periodicities should (and do) shift – often abruptly – as the evolving climatic trajectory sequentially shadows first one periodic orbit and then another. 3. Models that have been “tuned” (parametric adjustment) to fit trajectorial evolution in the vicinity of one periodic orbit are likely to fail when the real system moves to another region of the phase space. 4. In response to secular forcing, chaotic sets simplify via the elimination of periodic orbits. If one accepts the reality of anthropogenic warming, the long-term prediction is loss of intrinsic variability. 5. In response to periodic forcing, nonlinear systems can manifest subharmonic resonance i.e., “cyclic” behavior with periods and rotation numbers rationally related to the period of the forcing. Such cycling has been implicated in millennial and stadial variations in paleoclimatic time series. 6. Generically, the dynamics of system observables, such as climate sensitivity, are qualitatively equivalent to those of the whole. If the climate is chaotic, so too is sensitivity. These considerations receive minimal attention in consensus views of climate change that emphasize essentially one-to-one correspondence between global temperatures and exogenous forcing. Caveat emptor.