Quantitative studies on the expression of rinderpest virus genes in acute and persistent infections.

Abstract

This chapter is addressed to some of the unresolved issues concerning the response of the Van der Pol oscillator to whitenoise excitation. The stationary response in this case is distinctly nongaussian. This is due to the fact that the response pdf has a bimodal character that arises from the interaction between noise effects and the stable limit cycle of the system. Consequently, the usual linearization procedures are inadequate for dealing with this kind of problem. In this chapter, different methods based on Gaussianclosure techniques, equivalent nonlinearization, and higherorder stochastic averaging are developed, which lead to reasonably good approximations for the stationary response. In the method based on Gaussian closure, the response is split into a random periodic component (due to the limit cycle) and a Gaussian random component (due to external noise). The assumed form of the solution ensures that the response is nongaussian and has the desired bimodal character. It also incorporates the influence of initial conditions on the stationary response, especially when noise levels are low, where limitcycle effects dominate the response. In the equivalent nonlinearization method, the given Van der Pol oscillator is optimally replaced by another selfexcited system for which the exact stationary response can be evaluated. This replacement automatically retains the qualitative features of randomly excited selfexcited systems. An open question regarding the random response of the Van der Pol oscillator has been the nature of the phase distribution as system nonlinearity increases. In the present work, this issue is examined using a higherorder analysis that combines stochastic averaging and equivalent nonlinearization. This new analytical method is applied for the first time in randomvibration studies. It successfully predicts the nonuniform distribution of the phase process, which the other methods fail to predict. In addition to limitcycle behaviour, the presence of a selfexcitation mechanism leads to other interesting nonlinear phenomena. One such phenomenon is the entrainment of limitcycle frequency when the system is subjected to harmonic excitation. The influence of random noise on this effect in the Van der Pol oscillator is investigated in the next chapter.

The behaviour of the Van der Pol oscillator under periodic excitation is well known. The effect of adding noise to the excitation has been studied in this chapter. Three different solutions are obtained based on:

the Gaussianclosure technique, the method of equivalent linearization, and the combined averaging and nonlinearization technique.

These solutions display qualitatively identical behaviour and compare well with numerical simulations. Key findings include:

At resonance, the periodic term dominates, producing higher mean amplitudes and low variance. Away from resonance, noise effects dominate: the mean decreases and variance increases. For a fixed harmonic excitation, increasing the noise level reduces the mean response but increases variance.

Multiple steady states are typical in nonlinear systems under deterministic excitation. Here, the Gaussianclosure and equivalentlinearization solutions-identical in stationary state-show multivalued moments. These results were interpreted using a stochasticstability analysis, which also identifies regions in parameter space where the assumed form of the solution is valid. Although approximate, the predictions agree well with numerical simulations. Limitations of the linearization approach include:

Gaussian estimates for the response, Loss of validity as P0P \to 0P0.

These limitations are overcome using the combined averaging and nonlinearization method, whose responsemoment estimates compare favourably with digital simulations. The power spectrum of a combination of a periodic function and white noise contains:

a discrete peak at the periodic frequency, and a constant noise background.

At low noise levels, the spectrum is dominated by the single frequency-similar to narrowband random processes. Thus, in data analysis, these two types of signals can be confused. However, unlike periodicplusnoise excitation, a narrowband process is a zeromean stationary random process. Hence, it is natural to ask how the Van der Pol oscillator responds to a narrowband excitation. An attempt to address this is made in the next chapter.