Repository: DepositOnce – institutional repository for research data and publications of TU Berlin https://depositonce.tu-berlin.de
TY - THES
AU - Ruschel, Stefan
PY - 2020
TI - Multiple time-scale delay systems in mathematical biology and laser dynamics
T2 - Technische Universität Berlin
DO - 10.14279/depositonce-9876
UR - http://dx.doi.org/10.14279/depositonce-9876
PB - Technische Universität Berlin
M3 - Doctoral Thesis
CY - Berlin
LA - en
AB - This thesis addresses the dynamical behavior of Delay Differential Equations (DDEs) with a multiple time-scale structure as a consequence of large delays, or additional small parameters. More specifically, it gives attention to the effects of delay-induced instabilities of equilibrium solutions as well as families of equilibrium solutions and discusses the corresponding nonlinear dynamical phenomena that are not direct consequences of finite-dimensional geometric theory for Ordinary Differential Equations. When studying such systems of DDEs close to equilibrium, one is first
concerned with the spectral properties of a corresponding linearized system. Using an asymptotic approach, a rigorous description of the spectrum of linear DDEs with multiple hierarchical large delays is provided. It is shown that the spectrum splits into two distinct parts: the strong spectrum and the pseudo-continuous (or weak) spectrum reflecting the hierarchical structure of the delays. It is shown that a generic destabilization, a so-called weak instability, is mediated by a subset of the pseudo-continuous spectrum crossing the imaginary axis, that is associated with the largest delay in the system.
On the basis of three specific examples motivated from Mathematical Biology as well as Laser Dynamics, and applying the available invariant manifolds theory for semiflows, these results are then used to illuminate the nonlinear dynamical behavior of DDEs close to families of weakly unstable equilibrium solutions. In order to demonstrate how this local information can be lifted to understand the global dynamics, a specific 2-delay epidemiological model is analyzed. Here, the dynamics away from the manifold can be studied independently of the center direction, and the case of weak-instability is studied in detail. The obtained results are then interpreted in the biological context and allow for valuable insight into consequences of delays and imperfect implementation of isolation in infectious disease management. Thereafter, the focus changes towards DDEs with a single large delay and an additional small parameter. Here, the obtained results are used to study reduced systems, where the slow time-scale has been eliminated and the problem is reduced to the local study of families of equilibrium solutions. In particular, it is shown how the weak instability of the “laser off” state of the Lang-Kobayashi laser model translates into weakly chaotic solutions bearing some analogy to the weak-strong splitting of the spectrum of the autonomous linear system. Finally, it is shown that geometric singular perturbation theory of DDEs has aspects that are fundamentally different from singular geometric perturbation theory for Ordinary Differential Equations.
A minimal model is studied, where delay induces switched states. Necessary conditions for their existence are derived, and the case of weak-instability of the corresponding family of equilibrium solutions is studied in detail.
KW - delay differential equations
KW - nonlinear dynamics
KW - multiple delays
KW - multiple time scales
KW - large delay
KW - Delay-Differentialgleichung
KW - nichtlineare Dynamik
KW - mehrfache Delays
KW - Mehrskalenanalyse
KW - großes Delay
ER -