This dissertation focuses on the effects of distributed delays modeled by 'weak generic kernels' on the collective behavior of coupled nonlinear systems. These distributed delays are introduced into several well-known periodic oscillators such as coupled Landau-Stuart and Van der Pol systems, as well as coupled chaotic Van der Pol-Rayleigh and Sprott systems, for a variety of couplings including diffusive, cyclic, or dynamic ones. The resulting system is then closed via the 'linear chain trick' and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. A variety of dynamical regimes and transitions between them result. As an example, in certain cases the delay produces transitions from amplitude death (AD) or oscillation death (OD) regimes to Hopf bifurcation-induced periodic behavior, where typically we observe the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The conditions for transition between AD parameter regimes and OD parameter regimes are investigated for systems in which OD is possible. Depending on the coupling, these transitions are mediated by pitchfork or transcritical bifurcations. The systems are then investigated numerically, comparing with the predictions from the linear stability analysis and previous work. In several cases the various transitions among AD, OD and periodic domains that we observe are more intricate than the simple AD states, and the rough boundaries of the parameter regimes where they occur, which have been predicted by linear stability analysis and also experimentally verified in earlier work. The final chapter extends these studies by including the effects of periodically amplitude modulated distributed delays in both position and velocity. The existence of quasiperiodic solutions motivates the derivation of a second slow flow, together with a comparison of results and predictions from the second slow flow and the numerical results, as well as using the second slow flow to approximate the radii of the toroidal attractor. Finally, the effects of varying the delay parameter are briefly considered.
If this is your thesis or dissertation, and want to learn how to access it or for more information about readership statistics, contact us at STARS@ucf.edu
Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Roopnarain, Ryan, "Various Dynamical Regimes, and Transitions from Homogeneous to Inhomogeneous Steady States in Nonlinear Systems with Delays and Diverse Couplings" (2022). Electronic Theses and Dissertations, 2020-. 1278.