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Unfolding a Chaotic Bifurcation
O. E. Rossler, H. B. Stewart and K. Wiesenfeld
Proceedings: Mathematical and Physical Sciences
Vol. 431, No. 1882 (Nov. 8, 1990), pp. 371-383
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/79976
Page Count: 13
You can always find the topics here!Topics: Pendulums, Disasters, Dynamical systems, Mathematical cusps, Simulations, Oscillators, Symmetry, Phase portrait, Damping, Chaos theory
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The differential equation of the sinusoidally forced pendulum is studied by digital simulation in a regime where two simple, symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased. By introducing a small constant bias in the forcing to break the symmetry, two discontinuous bifurcations unfold from the single merging event. Considering both forcing amplitude and bias together as controls, a codimension two bifurcation of chaotic attractors is defined, whose geometric structure in control-phase space is closely related to the elementary cusp catastrophe. The chaotic bifurcations are explained in terms of homoclinic structures (Smale cycles) in the Poincare map.
Proceedings: Mathematical and Physical Sciences © 1990 Royal Society