title: Chaotic dynamics and diffusion in a piecewise linear equation.
abstract: Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
authors.affiliation: Department of Mathematics, Shah Jalal University of Science and Technology, Sylhet-3114, Bangladesh.
authors.forename: Pabel
authors.initials: P
authors.lastname: Shahrear
authors.affiliation: Department of Physiology, 3655 Promenade Sir William Osler, McGill University, Montreal, Quebec H3G 1Y6, Canada.
authors.forename: Leon
authors.initials: L
authors.lastname: Glass
authors.affiliation: Department of Mathematics and Statistics, University of Victoria, P.O. Box 1700 STN CSC, Victoria, British Columbia V8W 2Y2, Canada.
authors.forename: Rod
authors.initials: R
authors.lastname: Edwards
mesh_terms: D000818:Animals; D001683:Biological Clocks / Q000235:genetics*; D003198:Computer Simulation; D004058:Diffusion; D006801:Humans; D016014:Linear Models; D008957:Models, Genetic; D017711:Nonlinear Dynamics; D015398:Signal Transduction / Q000235:genetics*; D014157:Transcription Factors / Q000235:genetics*; D015533:Transcriptional Activation / Q000235:genetics*
chemical_list: D014157:Transcription Factors