Three Problems Of Resonance In Coupled Or Driven Oscillator Systems
The behaviors of three oscillator systems with various types of coupling or driving terms are discussed, with particular regard to their resonance patterns. In our first problem, a pair of phase-only oscillators are coupled to each other and driven by a third oscillator. By considering the phase differences between each oscillator and the driver, we find the region of parameter space where the system is entrained to the frequency of the driver. A complete analytical equation and multiple approximations for the boundary of this region are discussed. Additionally, the region where the oscillators "drift" relative to the driver is explored by using numerical methods. We find various m:n resonances between the oscillators within this region. Our second and third problems involve a first-order delay differential equation where the system is found to oscillate provided the delay is greater than a critical value. In the second problem, two identical delay limit cycle oscillators are coupled via instantaneous linear terms. This system has two invariant manifolds corresponding to in-phase and out-of-phase motions. Behavior on each manifold is expressed as a single delay limit cycle oscillator with an instantaneous selffeedback term. The strength of this self-feedback term changes the critical delay value needed for the system to oscillate. If strong enough, it also creates additional equilibrium solutions and can even prevent the system from oscillating in a stable way for any amount of delay. The third problem explores parametric excitation of the delay limit cycle oscillator, by adding a sinusoidal time-varying driving term as a perturbation to the delay value. For most parameter values, this term is non-resonant and causes the system to exhibit quasiperiodic behavior. However, using a twovariable expansion perturbation method, we find a 2:1 resonance between the frequency of the driving term and the natural frequency of the unperturbed oscillator. Expanding about this resonance by a combination of analytical and numerical methods reveals a variety of local and global bifurcations forming the transition between resonant and non-resonant behaviors. The corresponding regions of parameter space are found to hold multiple stable and unstable steady-states.
coupled limit cycle oscillators; delay differential equation; parametric excitation
Vladimirsky,Alexander B.; Strogatz,Steven H
Theoretical and Applied Mechanics
Ph. D., Theoretical and Applied Mechanics
Doctor of Philosophy
dissertation or thesis