Jakob L. Laugesen

Border-collision bifurcations in a dynamic management game

Managerial systems often involve piecewise-linear relations associated with non-negativity conditions or other forms of restrictions to the decision variables. The bifurcations that take place in such systems differ qualitatively from the bifurcations that one can observe in smooth dynamical systems. The present paper provides an analysis of the bifurcation structure of the so-called BEER game, a dynamic management game developed at the Sloan School of Management to illustrate how the structure of a system determines its behavior. We show how the BEER model displays abrupt period-doubling bifurcations, truncated bifurcation cascades, and a variety of the so-called border-collision bifurcations. Particular emphasis is paid to the resonance structure observed in one- and two-sector versions of the model.

Coupling-induced complexity in nephron models of renal blood flow regulation.

Tubular pressure and nephron blood flow time series display two interacting oscillations in rats with normal blood pressure. Tubuloglomerular feedback (TGF) senses NaCl concentration in tubular fluid at the macula densa, adjusts vascular resistance of the nephrons afferent arteriole, and generates the slower, larger-amplitude oscillations (0.02-0.04 Hz). The faster smaller oscillations (0.1-0.2 Hz) result from spontaneous contractions of vascular smooth muscle triggered by cyclic variations in membrane electrical potential. The two mechanisms interact in each nephron and combine to act as a high-pass filter, adjusting diameter of the afferent arteriole to limit changes of glomerular pressure caused by fluctuations of blood pressure. The oscillations become irregular in animals with chronic high blood pressure. TGF feedback gain is increased in hypertensive rats, leading to a stronger interaction between the two mechanisms. With a mathematical model that simulates tubular and arteriolar dynamics, we tested whether an increase in the interaction between TGF and the myogenic mechanism can cause the transition from periodic to irregular dynamics. A one-dimensional bifurcation analysis, using the coefficient that couples TGF and the myogenic mechanism as a bifurcation parameter, shows some regions with chaotic dynamics. With two nephrons coupled electrotonically, the chaotic regions become larger. The results support the hypothesis that increased oscillator interactions contribute to the transition to irregular fluctuations, especially when neighboring nephrons are coupled, which is the case in vivo.

Synchronization of period-doubling oscillations in vascular coupled nephrons

The mechanisms by which the individual functional unit (nephron) of the kidney regulates the incoming blood flow give rise to a number of nonlinear dynamic phenomena, including period-doubling bifurcations and intra-nephron synchronization between two different oscillatory modes. Interaction between the nephrons produces complicated and time-dependent inter-nephron synchronization patterns. In order to understand the processes by which a pair of vascular coupled nephrons synchronize, the paper presents a detailed analysis of the bifurcations that occur at the threshold of synchronization. We show that, besides infinite cascades of saddle-node bifurcations, these transitions involve mutually connected cascades of torus and homoclinic bifurcations. To illustrate the broader range of occurrence of this bifurcation structure for coupled period-doubling systems, we show that a similar structure arises in a system of two coupled, non-identical Rössler oscillators.

C-type period-doubling transition in nephron autoregulation.

The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period-doubling bifurcations, mode-locking and other nonlinear dynamic phenomena in the tubular pressures and flows. Using a simplified nephron model, the paper examines how the regulatory mechanisms react to an external periodic variation in arterial pressure near a region of resonance with one of the internally generated mode-locked cycles. We show how the stable and unstable resonance cycles generated in this response undergo interconnected cascades of period-doubling bifurcations and how each period doubling leads to the formation of a new pair of saddle-node bifurcation curves along the edges of the resonance zone. We also show how period doubling of the resonance cycles is accompanied by a torus-doubling process in the quasiperiodic regime that exists outside of the resonance zone.

Anomalous Statistics for Type-III Intermittency

AbstractThe statistics for the distribution of laminar phases in type-IIIintermittency is examined for the map

Oscillator Suppression in the Blood Flow Regulation of Interacting, Non- Identical Nephrons

x_{n + 1} = - ((1 + \mu )x_n + x_n^3 )e^{ - bx_n^2 }

Concepts in Mechanism Based Modeling

. Due to a strongly nonuniform reinjectionprocess, characteristic deviations from the normal statistics are observed.There is an enhancement of relatively long laminar phases followed by anabrupt cut-off of laminar phases beyond a certain length. The paper alsoexamines the bifurcation structure of two symmetrically coupled maps, eachdisplaying a subcritical period-doubling bifurcation. The conditions forsuch a pair of coupled maps to exhibit type-II intermittency are discussed.

Modeling in Biomedical Research and Health Care

Besides their systems nature, as described in the preceding chapters, the single most characteristic feature of a living organism is the self-sustained activity it displays in the form of a wide variety of different oscillatory processes [25, 9, 22, 23]. The respiratory cycle and the beating of the heart are generally recognized as internally generated oscillatory processes that first of all serve to pump oxygen from the atmosphere to the various tissues and cells of the body. The circulating blood, of course, also serves to supply the cells with the nutrients they need, to remove carbon dioxide and other metabolic bi-products, and to maintain hormonal communication between the various organs. The beating of the heart and the ventilation are directly related to our distinction between the living and the dead. We check for the pulse and we check for breathing.