Richard H. Rand

The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model

We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey.A simplified mathematical model of a limit cycle oscillator is presented which consists of only a single dependent variable, the phase θ(t). By coupling N such oscillators together we are able to generate stable phase locked motions which correspond to traveling waves in the spinal cord, thus simulating “fictive swimming”. We are also able to generate irregular “drifting” motions which are compared to the experimental data obtained from cords with selective surgical lesions.

Perturbation methods, bifurcation theory and computer algebra

1 Lindstedts Method.- 2 Center Manifolds.- 3 Normal Forms.- 4 Two Variable Expansion Method.- 5 Averaging.- 6 Lie Transforms.- 7 Liapunov-Schmidt Reduction.- Appendix Introduction to MACSYMA.- References.

Bifurcation of periodic motions in two weakly coupled van der Pol oscillators

Abstract We study a pair of weakly coupled van der Pol oscillators and investigate the bifurcations of phase-locked periodic motions which occur as the coupling coefficients are varied. Perturbation methods are used and their relation to the topological structure of solutions in the four dimensional phase space is discussed. While the problem is formulated for general linear coupling, the case of detuning plus diffusive coupling via displacement and velocity is discussed in more detail. It is shown that up to four phase-locked periodic motions can exist in this case.

The Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling

We investigate the dynamics of a system of twovan der Pol oscillators with delayed velocity coupling.We use the method of averaging to reduce the problem to the studyof a slow-flow in three dimensions.We study the steady state solutions of this slow-flow, with specialattention given to the bifurcations accompanying their change innumber and stability. We compare these stability results with numericalintegration of the original equations and show that the two sets of resultsare in excellent agreement under certain parameter restrictions.Our interest in this system is due to its relevance to coupled laseroscillators.

The transition to chaos in a simple mechanical system

Abstract A simple mechanical device and its response to periodic excitation is considered. The system consists of an inverted pendulum with rigid barriers which limit the amplitude variation from the unstable upright position. The static stable rest positions correspond to the pendulum leaning against one of the barriers. When subjected to periodic excitation the system response can be quite complicated and may include one or several stable subhannonics and/or chaotic motions. The analysis presented here is based on a piecewise linear model which allows explicit analytic expressions to be determined for many bifurcation conditions including: the appearance of certain types of subharmonics by saddle-node bifurcations, the secondary bifurcations of these subharmonics, and a global bifurcation which results in the creation of horseshoes.

Spinup Dynamics of Axial Dual-Spin Spacecraft

We consider spinup dynamics of axial dual-spin spacecraft composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor parallel to a principal axis of the platform. The system is free of external torques, and spinup of the rotor is effected by a small constant internal axial torque. The dynamics are described by four first-order differential equations. Conservation of angular momentum and the method of averaging are used to reduce the problem to a single first-order differential equation which is studied numerically. This reduction has a geometric counterpart that we use to simplify the investigation of spinup dynamics. In particular, a resonance condition due to platform asymmetry and associated with an instantaneous separatrix crossing is clearly identified using our approach.

Dynamics of two strongly coupled van der pol oscillators

Abstract A perturbation method is used to study the steady state behavior of two Van der Pol oscillators with strong linear diffusive coupling. It is shown that a bifurcation occurs which results in a transition from phase-locked periodic motions to quasi-periodic motions as the coupling is decreased or the detuning is increased. The analytical results are compared with a numerically generated solution.

Dynamics of spinup through resonance

Abstract This work concerns the phenomenon of resonant capture, i.e. the failure of a rotating mechanical system to be spunup to a desired terminal state, due to its resonant interaction with another system or with itself. The phenomenon is important in the dynamics of dual-spin spacecraft. Starting from a simple mechanical system consisting of an unbalanced rotor attached to an elastic support and driven by a constant torque, we derive an abstract model of resonant capture. The model is investigated by using perturbation theory and elliptic functions. For a given system, the analysis predicts which initial conditions lead to capture. These predictions are shown to compare reasonably with the results of numerical integration.

The transition from phase locking to drift in a system of two weakly coupled van der pol oscillators

Abstract We investigate the slow flow resulting from the application of the two variable expansion perturbation method to a system of two linearly coupled van der Pol oscillators. The slow flow consists of three non-linear coupled odes on the amplitudes and phase difference of the oscillators. We obtain regions in parameter space which correspond to phase locking, phase entrainment and phase drift of the coupled oscillators. In the slow flow, these states correspond respectively to a stable equilibrium, a stable limit cycle and a stable libration orbit. Phase entrainment, in which the phase difference between the oscillators varies periodically, is seen as an intermediate state between phase locking and phase drift. In the slow flow, the transitions between these states are shown to be associated with Hopf and saddle-connection bifurcations.

Limit cycle oscillations in CW laser-driven NEMS

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 discussed as are experimental results demonstrating the onset of limit cycle oscillations as continuous wave (CW) laser power is increased. A model describing the motion and heating of the devices is derived and analyzed. Conditions for the onset of limit cycle oscillations are computed as are conditions for these oscillations to be either hysteretic or nonhysteretic. An example simulation of a particular device is discussed and compared with experimental results.