Dependence of universal constants upon multiplication period in nonlinear maps

Noninvertible one-dimensional maps with cycle periods undergoing multiplication by a factor <i>N</i>, as a result of (tangent) bifurcation, are governed by map-independent universal constants αN<math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math>,δN<math><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> as the parameter <i>λ</i> of the map approaches the point of accumulation λN∞<math><mrow><msub><mrow><mo>λ</mo></mrow><mrow><mi>N</mi><mi>∞</mi></mrow></msub></mrow></math>. By explicit computation, we have determined the constants for all cycle structures and all values of <i>N</i> up to 7 (and in addition for many cycles up to <i>N=11</i>). We find that the relation between <i>α</i> and <i>δ</i> is roughly independent of the detailed cycle structure and follows quite well the Eckmann-Epstein-Wittwer asymptotic prediction that 3δ=2α2<math><mrow><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mrow></math>. .AE