We examine a graph of versus the forcing amplitude at , =1 , =0.6 , = -1, = 0.837 in order to detect bifurcations. The bifurcation diagram is shown in Figure 7. In this numerically constructed bifurcation diagram, the discrete points on the Poincare’ section of the attractor realized at each value e are displayed.
Obviously, from Figure 7, we can observe the sequence of period-doubling bifurcations. First, with one of values in the interval (4.7 , 4.7645125), the Poincare’ section consists of five points (five dark points in Fig.8), so there exist the subharmonic motions with the period equaling five times of the period of the external force (Figure 8). At the value 4.7645125, the first period-doubling bifurcation occurs. After the bifurcation, with the values which is in the right vicinity of the value 4.7645125, the subharmonic motions with the period equaling twice the period of the previous motions appear, the Poincare’ sections consist of ten points (Fig. 9), and so on. The chaotic attractor realized at e = 4.8042 appearing after a sequence of period-doubling bifurcations is shown in Figure 10, and Figure 11 is corresponding attractor. The largest Liapunov exponent evaluated is positive ( defines sensitivity to initial conditions on the chaotic attractor.
in many dynamical systems. It is particularly interesting because it may be characterized by a certain universal constant (called the Feigenbaum’s constant) that do not depend on nature of the concrete systems. This constant is considered as a specify criterion to determine if a system becomes chaotic by observing the prechaotic periodic behavior.