Characterization of bifurcations in the Zeeman catastrophe machine using singular value decomposition

Abstract

Stable limit cycles of the dynamic Zeeman Catastrophe Machine are calculated and classified. As the geometry of the device is changed, they evolve into different types at discrete bifurcation points. The process is normally presented as a bifurcation diagram using a stroboscopic map, but this does not provide enough information to uniquely identify the bifurcation event. Three types of bifurcation are common: period-doubling, symmetry-breaking, and turning points. A method is developed to distinguish between these types by calculating the response to a change in device geometry. Assuming that the limit cycle has already been calculated, the response to a change in geometry is defined as the solution of a linear second-order differential equation. The equation is solved in finite-difference form by defining a matrix which contains the second-order response to a change in the limit cycle, as well as an augmented matrix which contains the response to a change in device geometry. At a bifurcation, the solution can be analyzed using singular value decomposition of these two matrices. The resulting singular values contain enough information to uniquely identify, and characterize, each type of event. The corresponding singular vectors can also be used to uniquely characterize the symmetry changes that occur in the response to change at each event.