environment|depend on mutation rate and the presence or absence of selection. Systematic exploration of mutation rates reveals a bifurcation into qualitatively di erent classes of diversity dynamics, whether or not selection is present. Class I: At lowmutation rates, diversity dynamics exhibit \punctuated

We discuss the bifurcation structure in multimodal maps and construct explicitly the bifurcation diagrams for short periodic orbits in few-modal maps using symbolic dynamics. We conjecture this topological picture to be valid for all n-modal maps and perform numerical experiments for polynomial

We construct a series of n-modal approximations to maps of the H'enon type and utilize the associated symbolic dynamics to describe the possible bifurcation structures for such maps. We construct the bifurcation surfaces of the short periodic orbits in the topological parameter space

We present a detailed bifurcation study of iterated renormalization process driven by softmax transformation parametrized by a temperature parameter. For each emerging equilibrium we give exact characterization of stable/unstable manifolds of the linearized dynamics. As the system cools down, new

Abstraet. We have studied the bifurcation structure oC the logistic map with a time dependant control parameter. By introducing a specific nonlinear variation for the parameter, we show that the bifurcation structure is modified qualitatively as well as quantitatively from the first bifurcation

mechanisms giving rise to each scenario are investigated. We find that parameter shifts that make ice thicker or open ocean warmer under a given climate forcing make models less prone to stable seasonally ice-free conditions and more prone to bistability and hence bifurcations. The results are used

We expand upon a general framework for studying the bifurcation diagrams of localized spatially oscillatory structures. Building on work by Beck et al., the present work provides an analytical explanation for the numerical results of Houghton and Knobloch on symmetry breaking in systems with one

to account for the glacial cycles. Here an empirical model of the nonlinear response is presented. From the model it is possible to assess the role of stochastic noise in comparison to the deterministic orbital forcing of the ice ages. The model is based on the bifurcation structure derived from the climate

curves regarding order 2 periodic orbits in detail. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order 2 cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling

the bifurcation structure. We ascertain the existence of bifurcating branches, address their stability, and compare the stability to optimality in the constrained problem. These theoretical results are used to explain numerical results obtained from an annealing problem used to cluster data. 1.