Study on the Fine Structure of the Period-doubling Bifurcation in Friction System;

Simulations of models with different parameters show that the increasing of moment of inertia will result in the appearance of period-doubling and chaotic gaits.

45543; the cyclic fold curve gets closer and closer to the Hopf curve and finally intersects the Hopf curve at a singular point; the period doubling points also get closer and final.

The result of the calculation shows that may the undergo period doubling or Hopf bifurcations,in accordance with Floquet multipliers,the transition boundaries of the system are obtained and the effect of the variations of lubric.

Ithasbeen found thatthe periodic change and period dubling of the gross nationalproduct have taken place under the regulation of grow ing rate w hen the grow ing rateleads to a certain value,chaosbehaviorin the gross nationalproduct willappear.

The logistic equation with one-dimension is taken as tool to prove the chaotic mechanism that a kind of non-linear Lur e system realized through cycle-times in the paper.

The dynamic complexities of these models include Hopf bifurcation,Hopf bifurcation reversal,period-doubling bifurcation,period-halving,attractor crises,chaotic bands with narrow or wide periodic windows,intermittent chaos,and supertransient behavior.