1 Department of Physics, Technical University of Denmark2 Department of Structural Engineering and Materials, Technical University of Denmark
A successful mathematical description of the renal processes requires an understanding of the mechanisms through which these pressures take place. Part of the present thesis addresses the hypothesis that increased coupling between neighboring nephrons and increased strength of the tubuloglomerular feedback process can explain the experimentally observed irregular oscillations in the nephron pressures and flows. The hypothesis is put to test by calculating Lyapunov exponents of a high level mechanism-based model of a nephron and a similar model of two vascular coupled nephrons. Synchronization between oscillating period-doubling systems is the topic of the larger part of the study. Since synchronization is a fundamental phenomenon in all sciences, it is treated from a general viewpoint by analyzing one of the most simple dynamical systems, the R¨ossler system, both in an externally forced version and in the form of two mutually coupled oscillators. The bifurcational mechanism to resonant dynamics and chaotic phase synchronization is described in detail. The transition from synchronized to non-synchronized dynamics is known to take place at a dense set of saddlenode bifurcations that run along the edge of the resonance tongue and appear also to be related to the formation of multilayered tori and torus-doubling bifurcations. A cyclic behavior of sub- and supercriticality of the period doublings in the neighborhood of the contact between period doubling and saddle-node bifurcations cause a set of torus bifurcations that take place at a very small range of parameters. In coupled R¨ossler systems, the same torus bifurcations take a more global role. While a complete, but now folded, period-doubling cascade evolves, a cascade of torus bifurcations emerge from all the period doublings and run along side with three (due to the folding of the period doubling) sets of saddle-node bifurcations at the edge of the tongue. Through homoclinic bifurcations of tori with different periodicity, a second mechanism to phase synchronization is found to occur. Similar bifurcation structures are shown to exist in an externally forced nephron model and in a model of two vascular coupled nephrons, underlining that the discussed phenomena are of a common nature to forced and coupled period-doubling systems.