Mathematical Sciences: Global Bifurcations of Forced Oscillator Systems

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Families of maps generated by periodically forced oscillators undergo a rich and fascinating set of bifurcations. The investigator proposes to develop new numerical and analytical tools to further investigate the bifurcations of these forced oscillator systems. Current numerical "continuation" routines are able to trace out one-parameter curves of local bifurcations of fixed and periodic points for orbits of low period, typically less than four or five. Proposed work will include development of new routines that are expected to trace out bifurcations curves for higher period orbits. These new "high period" routines will then be used for locating "global bifurcations," where invariant manifolds, typically a stable one and an unstable one, become tangent before crossing. Once developed, these routines will enable a much more automatic and detailed bifurcation study, not only of maps generated by forced oscillators, but also of any two-parameter family of maps. Many processes in science and engineering can be mathematically categorized as "forced oscillator systems." One example of an oscillator is a pendulum, because it swings back and forth, or oscillates. We usually think of a pendulum as having a fixed support (the nail in the wall to which the string with a weight is tied), but if we keep moving the location of the support, we say the pendulum is being "forced;" if we move the support in a repeating pattern, such as back and forth, then we have a "periodically forced oscillator." It turns out that the resulting motion depends critically on how fast the support is moved back and forth (the forcing frequency) and how far it is moved back and forth (the forcing amplitude). Fairly simple mathematical models also exhibit this same behavior, including the sensitivity to changes in "parameters" analogous to forcing frequency and amplitude. Changes in the qualitative behavior of the system (how the pendulum bob moves) that occur as the parameters are varied are called "bifurcations." Much is known about these bifurcations, but much is still unknown. The investigator proposes to fill in some of the missing gaps in this area. Results will help explain observed behavior in any process that can be classified mathematically as a forced oscillator system.