Bernhard Riemann's Gesammelte mathematische Werke und by Weber H., Dedekind R. (eds.)

By Weber H., Dedekind R. (eds.)

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36 L. Hong et al. A collision with a chaotic saddle in a fractal boundary is the typical mechanism by which hyerchaotic attractors can be suddenly destroyed. In the hyperchaotic crises, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary.

The boundary is shown in (b) of figures. The color coding and symbol hold throughout the chapter. As a increases, the chaotic saddle in the basin interior becomes bigger and closer to the boundary. 0:1648; 0:1649/. In the case, the chaotic saddle is touching the saddle in the boundary when a D 0:1648, creating a chaotic saddle in a fractal boundary when a D 0:1649. The chaotic saddle in the fractal boundary has a complicated structure and plays an extremely important role in an upcoming hyperchaotic crisis.

The following statements hold. N / ˇ ˇ ˇ 1. xkCN /) are stable. N / ˇ ˇ ˇ 2. xkCN /) are unstable. ˇ ˇ ˇ ˇ ˇ < 1 (or 1 and ˇ C 1 and ˇ 2 ˇ < 1), 3. xkCN /) occurs. ˇ ˇ ˇ ˇ C ˇ ˇ < 1 and ˇ 1), then 4. xkCN /) occurs. ˇ ˇ ˇ ˇ ˇ ˇ ˇ 5. xkCN /) occurs. 3 Illustrations A numerical prediction of the periodic solutions of the Henon map is presented with varying parameter b for a D 0:85, as shown in Fig. 1. The dashed vertical lines give the bifurcation points. 5 −2 −1 0 1 2 Parameter b Fig. J. Luo and Y.

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