This thesis is concerned with two different interesting phenomena which can occur when a second parameter is introduced to a one parameter system of equations which exhibits a period-doubling cascade.

The first situation we consider is when the second parameter is introduced to control the coupling strength in a system of coupled maps with dihedral symmetry which undergoes a period-doubling bifurcation. We first analyse the codimension one bifurcations which can occur in this setting - namely the period-doubling bifurcation and the symmetry-breaking bifurcation(s) which are guaranteed to exist by the Equivariant Branching Lemma - and then continue to investigate the mode interaction which occurs when the period-doubling and symmetry-breaking bifurcations coalesce.

We then investigate the local solution structure in a neighbourhood of the mode interaction point for each of the possible combinations of period-doubling and symmetry-breaking bifurcations. We take a generic map and provide low order expansions for the solution branches, find parameter values at which primary and secondary bifurcations occur, investigate the existence of paths of limit points in a neighbourhood of the mode interaction, and provide bifurcation diagrams to illustrate the analysis for specific examples.

The second setting we study is the transition of a (parameter-dependent) supercritical period-doubling cascade to a subcritical period-doubling cascade as a second parameter is varied. We investigate and classify the different possible supercritical period-doubling cascades and subcritical period-doubling cascades which can occur in a class of two dimensional maps. We then describe how an analysis of certain codimension 2 bifurcation points can be used to describe the mechanisms by which we might observe a supercritical period-doubling cascade being converted to a subcritical period-doubling cascade.

We show that a new dynamical structure, which we call an alternating period-doubling cascade, can be observed in two dimensional maps with two parameters, and indeed that such structures can be generated as an intermediate step in the transition of a supercritical period-doubling cascade to a subcritical period-doubling cascade. The different possible alternating period-doubling cascades which can be observed in our class of maps are classified, and their dynamical behaviour is studied.

Finally, we show that alternating period-doubling cascades can exhibit universal behaviour. We find two solutions to an appropriate two dimensional renormalisation operator, and obtain universal spatial and parameter scalings corresponding to each solution.