4.4. Fractal Boundaries of Basins of Attraction and Transient Chaos in the Region of Principal Resonance

4.4. Fractal Boundaries of Basins of Attraction and Transient Chaos in the Region of Principal Resonance

Let us now turn back to the region of the principal resonance, i.e. to the range of system parameters F, ? where two T-periodic oscillating attractors, namely the resonant attractor S _r and the nonresonant one S _n, exist in both potential wells. In Figure 4.5, this triangle-like zone is bounded by the bifurcation curves snA, snB and pd.

In the case when four attractors coexist, the diagram of their basins of attraction appears to be evidently more complex than that presented in Figure 4.6. To illustrate this fact, first we show basins of attraction for the region of control parameters F, ? where the system behaves regularly, i.e. below the Melnikov criterion, F _M.

In Figure 4.7, the basins of attraction of the attractors existing in the left potential well are filled with green and yellow colors, while the basins of the attractors related to the right well with blue and red colors. The attractors are denoted by the points , , , , and the hilltop saddle by the point D _H. Then we make a crucial observation that, apart from a larger number of attractors, two additional, new saddle points have appeared, namely and . They constitute the mapping points of the unstable T-periodic solutions (denoted D _n in Figure 4.4), and the stable manifolds of the solutions define the...