Bifurcations in piecewise-smooth, continuous systems

This thesis presents a variety of new results regarding bifurcations of piecewise-smooth, continuous, autonomous systems of ordinary differential equations (ODEs) and maps. Codimension-one, discontinuous bifurcations are classified for planar ODE systems. Particular emphasis is placed on the creation of periodic orbits at these bifurcations. The codimension-two simultaneous occurrences of a discontinuous bifurcation with a saddle-node bifurcation and then with an Andronov-Hopf bifurcation are unfolded for systems of arbitrary dimension. In the latter case a locus of Andronov-Hopf bifurcations emanates from the codimension-two point and the Hopf cycle undergoes grazing that may be very shortly followed by a saddle-node bifurcation of the orbit. A discontinuous Bautin-like bifurcation is also unfolded in two dimensions.;These theoretical unfoldings are applied to an eight-dimensional, piecewise-smooth, continuous ODE model of the growth dynamics of Saccharomyces cerevisiae. A numerical bifurcation analysis reveals a plethora of complex dynamical phenomena. Stable oscillations arise via Andronov-Hopf bifurcations and exist for intermediate values of the dilution rate as has been noted from experiments previously. Oscillatory behavior is also investigated on a two-dimensional slow manifold.;For discrete-time systems the codimension-two simultaneous occurrences of a border-collision bifurcation with a saddle-node bifurcation and then with a period-doubling bifurcation are unfolded for systems of arbitrary dimension. Detailed results are obtained for the latter case in one dimension. A symbolic description is given for a class of "rotational" periodic solutions that display lens-chain structures for a general N-dimensional map. By utilizing the symbolic framework an unfolding of so-called "shrinking points" is obtained. A number of codimension-one bifurcation curves are found to emanate from shrinking points and those that form resonance tongue boundaries are determined.;Border-collision bifurcations are studied in two dimensions in the case that the multipliers of a fixed point are complex valued and "jump" from inside to outside the unit circle at the bifurcation. The resulting dynamics is sometimes similar to the Neimark-Sacker bifurcation of a smooth map in which an attracting periodic or quasiperiodic orbit is created as the fixed point loses stability. However, the bifurcation is often much more complex, exhibiting chaotic dynamics and creating multiple coexisting attractors.