Phase-locked Solutions Research Articles

On a periodically forced, weakly damped pendulum. Part 1: Applied torque

AbstractCoplanar forced oscillations of a mechanical system such as a seismometer or a fluid in a tank are modelled by the coplanar motion of periodically forced, weakly damped pendulum. We consider the phase-locked solutions of the differential equation governing planar motion of a weakly damped pendulum driven by a periodic torque. Sinusoidal approximations previously obtained for downward and inverted oscillations at small values of the dimensionless driving amplitude ε are continued into numerical solutions at larger values of ε. Resonance curves and stability boundaries are presented for downward and inverted oscillations of periods T, 2T, and 4T, where T(≡ 2π/ω) is the dimensionless forcing period. The symmetry-breaking, period-doubling sequences of oscillatory motion are found to occur in bands on the (ω, ε) plane, with the amplitudes of stable oscillations in one band differing by multiples of about π from those in the other bands, a structure similar to that of energy levels in wave mechanics. The sinusoidal approximations for symmetric T-periodic oscillations prove to be surprisingly accurate at the larger values of ε, the banded structure being related to the periodicity of the J0 Bessel function.

Amplitude response of coupled oscillators

We investigate the interaction of a pair of weakly nonlinear oscillators (eg. each near a Hopf bifurcation) when the coupling strength is comparable to the attraction of the limit cycles. Changes in amplitude cannot then be ignored, and there are new phenomena. We show that even for simple forms of these equations, there are parameter regimes in which the interaction causes the system to stop oscillating, and the rest state at zero, stabilized by the interaction, is the only stable solution. When the uncoupled oscillators have a local frequency that is independent of amplitude (zero shear), and the coupling is scalar, we give a complete description of the behavior. We then analyze more complicated equations to show the effects of amplitude-dependent frequency in the uncoupled equations (nonzero shear) and nonscalar coupling. For example, when there is shear, there can be bistability between a phase-locked solution and an unlocked “drift” solution, in which the oscillators go at different average frequencies. There can also be bistability between a phase locked solution and an equilibrium, nonoscillatory, solution. The equations for the zero shear case have a pair of degenerate bifurcation points, and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities. When the coupling is nonscalar, there are a variety of new behaviors, including bistability of an in-phase and an anti-phase solution for two identical oscillators, parameters for which only the anti-phase solution is stable, and “phase-trapping,” i.e. nonlocked solutions in which the oscillators still have the same average frequency. It is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system.

The Uniqueness and Stability of the Rest State for Strongly Coupled Oscillators

In this paper it is shown that an unstable steady state can be stabilized in the presence of sufficient inhomogeneity and strong diffusion for a continuum and discrete model. These results are applied to a pair of coupled oscillators and to an oscillatory reaction-diffusion system. It is also shown that for the reaction-diffusion system, the trivial rest state is the unique phase-locked solution. Some additional numerical results are presented that illustrate the nature of this bifurcation for a realistic model oscillator.

Symmetry and phaselocking in chains of weakly coupled oscillators

AbstractWeakly coupled chains of oscillators with nearest‐neighbor interactions are analyzed for phaselocked solutions. It is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound. It is also shown that qualitative behavior of these solutions depends on whether the coupling is “diffusive” or “I synaptic”. terms defined in the paper. The methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two‐point (continuum) boundary value problem that is easier to analyze; the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law. An application to the neurophysiology of motor behavior is discussed briefly.

Correlation functions and variability in a periodically forced oscillatory climate model

Abstract The effect of fluctuations in a periodically forced climate model involving the coupling between sea ice and mean ocean temperature is considered. The transitions between phase-locked (periodic) and quasi-periodic solutions are studied by a singular perturbation method of solution of the stochastic differential equations, as well as by numerical simulations. The variances and covariances of the principal climatic variables are computed.

Subharmonic frequency locking in the resistive Josephson thermometer

Phase-locked oscillatory solutions are examined as a basis for the dc impedance of the resistive superconducting quantum-interference device Josephson thermometer. The calculations are based on the resistively shunted junction model in the limit 2..pi..L/sub s/I/sub c//Phi/sub 0/> or =1, where L/sub s/ is the loop inductance and I/sub c/ is the junction critical current, and for a junction resistance large compared with the external shunt resistance. An algorithm for representing frequency entrainment in (kappa,..omega..) space (drive amplitude, frequency) leads to zones with rotation number p/q having the form of leaf-shaped regions joined and overlapping at their tips. High-resonance zones are very thin and locally similar. No chaotic behavior has been observed. The model can simulate the ''rising'' curves of dc impedance as a function of drive amplitude.

N:m Phase-locking of weakly coupled oscillators

Two weakly coupled oscillators are studied and the existence of n:m phase-locked solutions is shown. With the use of a slow time scale, the problem is reduced to a two-dimensional system on an invariant attracting torus. This system is further reduced to a one-dimensional dynamical system. Fixed points of this system correspond to n:m phase-locked solutions. The method is applied to a forced oscillator, linearly coupled λ-ω systems, and a pair of integrate and fire neuron models.

Phase-locked solutions

Phase-locked solutions