Van Der Pol Relaxation Oscillator Research Articles

Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study

There is evidence that ice age cycles are paced by astronomical forcing, suggesting some kind of synchronisation phenomenon. Here, we identify the type of such synchronisation and explore systematically its uniqueness and robustness using a simple paleoclimate model akin to the van der Pol relaxation oscillator and dynamical system theory. As the insolation is quite a complex quasiperiodic signal involving different frequencies, the traditional concepts used to define synchronisation to periodic forcing are no longer applicable. Instead, we explore a different concept of generalised synchronisation in terms of (coexisting) synchronised solutions for the forced system, their basins of attraction and instabilities. We propose a clustering technique to compute the number of synchronised solutions, each of which corresponds to a different paleoclimate history. In this way, we uncover multistable synchronisation (reminiscent of phase- or frequency-locking to individual periodic components of astronomical forcing) at low forcing strength, and monostable or unique synchronisation at stronger forcing. In the multistable regime, different initial conditions may lead to different paleoclimate histories. To study their robustness, we analyse Lyapunov exponents that quantify the rate of convergence towards each synchronised solution (local stability), and basins of attraction that indicate critical levels of external perturbations (global stability). We find that even though synchronised solutions are stable on a long term, there exist short episodes of desynchronisation where nearby climate trajectories diverge temporarily (for about 50 kyr). As the attracting trajectory can sometimes lie close to the boundary of its basin of attraction, a small perturbation could quite easily make climate to jump between different histories, reducing the predictability. Our study brings new insight into paleoclimate dynamics and reveals a possibility for the climate system to wander throughout different climatic histories related to preferential synchronisation regimes on obliquity, precession or combinations of both, all over the history of the Pleistocene.

STABLE OSCILLATIONS AND DEVIL'S STAIRCASE IN THE VAN DER POL OSCILLATOR

A forced van der Pol relaxation oscillator is studied experimentally in the regime of stable oscillations. The variable parameter is chosen to be the driving frequency. For a range of parameter values, we show that the rotation number varies continuously from 0 to 1. This work provides experimental evidence that period-adding bifurcations to chaos previously reported by Kennedy and Chua are intimately connected to the existence of a regime of stable oscillations where the rotation number shows a Devil's-staircase structure.

Spectral Analysis of Stochastic Phase Lockings and Stochastic Bifurcations in the Sinusoidally Forced van der Pol Oscillator with Additive Noise

Noise effects on the phase lockings and bifurcations in the sinusoidally forced van der Pol relaxation oscillator are investigated. Deterministic (noise-free) one-dimensional Poincare mapping is extended to the iteration of the operator defined by a stochastic kernel function. Stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator. In particular, a new method which uses spectra (eigenvalues and eigenfunctions) of the operator to analyze stochastic bifurcations intensively is proposed.

Parameter estimation by reduced-order linear associative memory (ROLAM).

In a series of papers we have shown that nonlinear parameter estimation by linear association provides accurate estimates of the parameters in complex systems described by nonlinear differential equations even in the presence of additive white noise of considerable power. The technique is based on linearly associating the system's output with a set of parameter values spanning the region of interest. When an actual output is measured, the system's unknown parameters could be estimated by a matrix inversion. The size of the inverted matrix, being equal to the length of the output vector, poses a limiting factor upon the generalization of the technique. In this paper we propose a modification which requires the inversion of a matrix whose dimension equals the number of model parameters. The modified version is called reduced-order associative memory (ROLAM). The technique is applied to two complex lumped-parameter nonlinear models: the Van der Pol relaxation oscillator and the passive neuron model of the granule cells. Results validate ROLAM as a parameter-estimation tool which is especially suited in cases where the number of parameters is large, the number of samples in the observation signal is high, or when on-line parameter estimation is required. It is also shown that ROLAM provides an optimal parameter estimate in the special case of single-parameter nonlinear models.

A note on stochastic phase lockings in a relaxation oscillator forced by a sine input with additive noise

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. Using the probability density functions of the first-passage time, the operator which governs a transition of an input phase density after one cycle of the oscillator is defined. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.

Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. The probability density functions of the first-passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.

Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network

1. The gastric mill central pattern generator (CPG) controls the chewing movements of teeth in the gastric mill of the lobster. This CPG has been extensively studied, but the precise mechanism underlying pattern generation is not well understood. The goal of this research was to develop a simplified model that captures the principle, biologically significant features of this CPG. We introduce a simplified neuron model that embodies approximations of well-known membrane currents, and is able to reproduce several global characteristics of gastric mill neurons. A network built with these neurons, using graded synaptic transmission and having the synaptic connections of the biological circuit, is sufficient to explain much of the network's behavior. 2. The cell model is a generalization and extension of the Van der Pol relaxation oscillator equations. It is described by two differential equations, one for current conservation and one for slow current activation. The model has a fast current that may, by adjusting one parameter, have a region of negative resistance in its current-voltage (I-V) curve. It also has a slow current with a single gain parameter that can be regarded as the combination of slow inward and outward currents. 3. For suitable values of the fast current parameter and the slow current parameter, the isolated model neuron exhibits several different behaviors: plateau potentials, postinhibitory rebound, postburst hyperpolarization, and endogenous oscillations. When the slow current is separated into inward and outward fractions with separately adjustable gain parameters, the model neuron can fire tonically, be quiescent, or generate spontaneous voltage oscillations with varying amounts of depolarization or hyperpolarization. 4. The most common form of synaptic interaction in the gastric CPG is reciprocal inhibition. A pair of identical model cells, connected with reciprocal inhibition, oscillates in antiphase if either the isolated cells are endogenous oscillators, or they are quiescent without plateau potentials, or they have plateau potentials but the synaptic strengths are below a critical level. If the isolated cells have widely differing frequencies (or would have if the cells were made to oscillate by adjusting the fast currents), reciprocal inhibition entrains the cells to oscillate with the same frequency but with phases that are advanced or retarded relative to the phases seen when the cells have the same frequency. The frequency of the entrained pair of cells lies between the frequencies of the original cells. The relative phases can also be modified by using very unequal synaptic strengths.(ABSTRACT TRUNCATED AT 400 WORDS)

Breakdown of Stability in Singularly Perturbed Autonomous Systems II. Estimates for the Solutions and Application

In Nipp, [this Journal, 17 (1986), pp. 512–532] error estimates were derived for the trajectories of a singularly perturbed autonomous system. The results are extensions of a classical result due to A. N. Tikhonov. In this paper we give the transfer of those estimates to the solutions of the autonomous system and an application to a problem in biomathematics. Special cases of our results are treated by Lebovitz and Schaar in Stud. Appl. Math., 54 (1975), pp. 229–260, 56 (1977), pp. 1–50, respectively. In order to transfer their estimates to the solutions, however, they introduce an artificial condition excluding a whole class of problems containing, e.g., the van der Pol relaxation oscillator as well as the example given in this paper.

Singular perturbations and a mapping on an interval for the forced Van der pol relaxation oscillator

This paper deals with the Van der Pol relaxation oscillator with a large sinusoidal forcing term. By using singular perturbation techniques asymptotic solutions of such a system are constructed. These asymptotic approximations are locally valid and may take the form of a two time scale expansion in one region and a boundary layer type of solution in a next region. Integration constants are determined by averaging and matching conditions. From these local solutions a difference equation is constructed. There is an equivalence between solutions of the difference equation being an iterated mapping on a compact interval and solutions of the system itself. This equivalence makes it possible to analyze subharmonics and chaotic type of solutions to the full extent. As a result of this we find domains in the parameter space, where regular subharmonics exist. These domains overlap so that for some parameter values different subharmonics coexist. For parameter values in this range chaotic type of solutions are found as well. They are described by using concepts of symbolic dynamics.

An extension of Tikhonov's theorem in singular perturbations for the planar case

A classical error estimate for singularly perturbed ordinary differential equations due to A. N. Tikhonov is generalized in the neighbourhood of a so-called leaving point where a certain stability assumption ceases to be valid. This is done for the planar case, and an application to the Van der Pol relaxation oscillator is given.

Almost discontinuous oscillations: The generalized multivibrator

The behavior of a general free-running multivibrator circuit is investigated. The circuit contains two three-terminal active devices drawing control current and subject to saturation, and various passive circuit elements, some of which are parasitic. The operation of the multivibrator is described by a system of nonlinear equations which is a higher dimensional generalization of the van der Pol relaxation oscillator equations. The methods of singular perturbation theory are applied to show under what circumstances the multivibrator will, and under what circumstances it will not, oscillate. The period and waveform of the oscillations are also obtained.

Frequency Entrainment of a Forced van der Pol Oscillator

A van der Pol relaxation oscillator that is subjected to external sinusoidal forcing can exhibit stable and unstable periodic and almost periodic responses. For some forcing amplitudes it even happens that two stable subharmonics having different periods may coexist. We investigate here the stable responses of such forced oscillators. By numerically computing the rotation number of stable oscillations for various values of the forcing amplitude and oscillator tuning, we obtain descriptions of regions of phase locking, successive bifurcation of stable subharmonic and almost periodic oscillations, and overlap regions where two distinct stable oscillations can coexist.