Abstract

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.

Notes

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Graduation Date

2022

Semester

Summer

Advisor

Choudhury, Sudipto

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Identifier

CFE0009249; DP0026853

URL

https://purls.library.ucf.edu/go/DP0026853

Language

English

Release Date

August 2022

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

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Included in

Mathematics Commons

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