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- Erika Camacho, Richard Rand, Howard Howland
- 2004

In this work we study a system of two van der Pol oscillators, x and y, coupled via a ''bath'' z: € x À ð1 À x 2 Þ_ x þ x ¼ kðz À xÞ € y À ð1 À y 2 Þ_ y þ y ¼ kðz À yÞ _ z ¼ kðx À zÞ þ kðy À zÞ We investigate the existence and stability of the in-phase and out-of-phase modes for parameters > 0 and k > 0. To this end we use Floquet theory and numerical… (More)

- Keith Aubin, Maxim Zalalutdinov, Tuncay Alan, Robert Reichenbach, Richard Rand, Alan Zehnder +2 others
- 2004

— Limit cycle, or self oscillations can occur in a variety of NEMS devices illuminated within an interference field. As the device moves within the field, the quantity of light absorbed and hence the resulting thermal stresses changes, resulting in a feedback loop that can lead to limit cycle oscillations. Examples of devices that exhibit such behavior are… (More)

- Maxim Zalalutdinov, Jeevak Parpia, Keith Aubin, Harold Craighead, Tuncay Alan, Alan Zehnder +1 other
- 2003

Self-sustained mechanical vibrations of a disc-type micro-fabricated resonator were experimentally observed when a continuous wave (CW) laser beam was focused on the periphery of the disc (for a 40 µm diameter resonator, natural frequency 0.89MHz, the laser power above a 250 W threshold was required). A theoretical model for self-oscillatory behavior has… (More)

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can… (More)

- Christoffer Heckman, Jakob Kotas, Richard Rand
- 2013

In this work, a differential delay equation (DDE) with a cubic nonlinearity is analyzed as two parameters are varied by means of a center manifold reduction. This reduction is applied directly to the case where the system undergoes a Hopf-Hopf bifurcation. This procedure replaces the original DDE with four first-order ODEs, an approximation valid in the… (More)

- Stephen Wirkus, Richard Rand
- 1999

We investigate the dynamics of a system of two van der Pol oscillators with delayed velocity coupling. We use the method of averaging to reduce the problem to the study of a slow-flow in three dimensions. We study the steady state solutions of this slow-flow, with special attention given to the bifurcations accompanying their change in number and stability.… (More)

We analyze a model of gene transcription and protein synthesis which has been previously presented in the biological literature. The model takes the form of an ODE (ordinary differential equation) coupled to a DDE (delay differential equation), the state variables being concentrations of messenger RNA and protein. Linear analysis gives a critical time delay… (More)

This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using LindstedtÕs perturbation method.

- Leslie Ng, Richard Rand
- 2002

We investigate the effect of nonlinearites on a parametri-cally excited ordinary differential equation whose linearization exhibits the phenomena of coexistence. The differential equation studied governs the stability era mode of vibration in an un-forced conservative two degree of freedom system used to model the free vibrations of a thin elastica. Using… (More)

- Richard Rand, Jeffrey Wong
- 2007

We study the dynamics of a system of four coupled phase-only oscillators. This system is analyzed using phase difference variables in a phase space that has the topology of a three-dimensional torus. The system is shown to exhibit numerous phase-locked motions. The qualitative dynamics are shown to depend upon a parameter representing coupling strength.… (More)