North-Holland PHYSICS LETTERS A How common is chaos?

The solutions of about 4 × 10 ~ low-dimensional, low-order polynomial maps and ordinary differential equations were classified as either fixed point, limit cycle, chaotic, or unstable. Of those cases for which the solutions are stable, representing candidate models for the real world, typically afew percent were found to be chaotic. It is now widely understood that complex behav-ior in nature may have a simple underlying cause. Several examples of low-dimensional nonlinear maps [ 1,2] and ordinary differential equations [ 3,4] with chaotic solutions have been extensively studied. On the other hand, many familiar processes are known to have regular periodic solutions. It would thus be of interest o quantify the issue of how common chaos is in nature. As a modest first step, this paper quan-tifies the occurrence of chaos in certain simple sys-tems of nonlinear equations that are known to char-acterize a wide range of natural phenomena