Bifurcations of Heterodimensional Cycles

Abstract

Mathematical models of real-world systems can exhibit highly complicated phenomena that organize observed behavior or dynamics. In this thesis, we study one such phenomenon known as a heterodimensional cycle. A heterodimensional cycle consists of two saddle periodic orbits that have unstable manifolds of di erent dimensions|together with connecting orbits from one periodic orbit to the other, and vice versa. A system with a heterodimensional cycle is structurally unstable, meaning its dynamics is sensitive to arbitrarily small parameter changes. Nevertheless, the existence of a heterodimensional cycle can be a \robust" phenomenon, in which case it is known to generate highly complex dynamics, also called wild chaos. Heterodimensional cycles are complicated structures, and all the known examples have been constructed abstractly without a realistic application in mind. However, Zhang, Kirk and Krauskopf (2012) found and computed a codimension-one heterodimensional cycle of the Atri model, which is an explicit, four-dimensional vector eld model of intracellular calcium oscillations. This forms our starting point, and we use advanced numerical methods, including Lin's method, to compute new and more complicated heterodimensional cycles. With the continuation software AUTO, we explore a two-parameter region of the Atri model where heterodimensional cycles are found, and we show how the loci (curves) of these new heterodimensional cycles t together in an overall bifurcation structure, which also involves local and global bifurcations of equilibria and periodic orbits. Speci cally, we discover that the heterodimensional cycle found by Zhang et al. undergoes a sequence of geometrical transformations, which implies new dynamical phenomena, such as a strong homoclinic orbit. Furthermore, we nd two novel codimension-two bifurcations: a heterodimensional cycle at a period-doubling bifurcation, and a resonant heterodimensional cycle. The former bifurcation generates new heterodimensional cycles involving the period-doubled orbit, and the latter gives rise to in nitely many families of codimension-one homoclinic tangencies. Finally, we relate our research back to the theory of structural stability and discuss the existence of a robust heterodimensional cycle of the Atri model. We show how heterodimensional cycles can be abundant in the limit of a period-doubling cascade. Moreover, we identify a codimension-two organizing center (called a 3DL bifurcation), whose existence suggests that heterodimensional cycles can exist throughout a large two-parameter region.