Periodic orbit bifurcations and breakup of shearless invariant tori in nontwist systems
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This thesis explores two nontwist systems: the spherical pendulum as an example of a continuous one and the standard nontwist map (SNM) as an example of a discrete one. Whereas the spherical pendulum is a concrete example of a physical system exhibiting nontwist phenomena, the SNM is an abstract, numerically easily accessible model permitting systematic studies of nontwist effects characteristic of a wide range of applications. For the spherical pendulum, a system that has captured physicists' and mathematicians' interest for centuries, the gradual progress in understanding this seemingly simple, but still not fully explored problem is outlined. The known solutions for the unforced (integrable) spherical pendulum are reviewed and approximated by power series. The approximations are then used to analytically calculate, for the vertically forced case, xed points and low-period periodic orbits. These are found to undergo collision phenomena typical for nontwist systems. For the SNM, a detailed cartography of parameter space is developed, based on periodic orbit collision curves and their branching thresholds, hyperbolic manifold reconnection thresholds, and the boundary for the onset of global chaos. This is used to nd meanders, multiple shearless curves, and extended scenarios for periodic orbit reconnection/collision. Based on Greene's residue criterion, the breakup of new types of shearless orbits: meanders, outer shearless tori, and a nonnoble torus is studied in detail within the framework of renormalization theory.