Bifurcation dynamics of three-dimensional systems

Oscillations described by autonomous three-dimensional differential equations display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario the key instability is often an initial bifurcation from a single period oscillation to either its subharmonic of period two, or a symmetry breaking bifurcation. A generalized third-order nonlinear differential equation is developed which embraces the dynamics vicinal to these bifurcation events. Subsequently, an asymptotic averaging formalism is applied which permits an analytic development of the bifurcation dynamics, and, within quantifiable limits, prediction of the instability of the period one orbit in terms of the system control parameters. Illustrative applications of the general formalism, are made to the Rossler equations, Lorenz equations, three-dimensional replicator equations and Chua's circuit equations. The results provide the basis for discussion of the class of systems which fall within the framework of the formalism.