The Deterministic Circuit Model for Noise Influence
on the Averaged Transient Responses of Large-scale
Nonlinear ICs Analyzed with It6's Stochastic
MODEL FOR NOISE-INDUCED PHENOMENA IN THE EXPECTATION:
CASE OF STOCHASTIC RESONANCE
The advances of the last decade demonstrate the keen interest of many researchers and engineers invarious noise-induced phenomena in expectations of stochastic processes: stochastic resonance, stochastic linearization, stochastic self-oscillations, stochastic phase transitions, stochastic chaos. The
first part of these terms, the word "stochastic", points out that the listed effects are due to the influence of noise upon the expectations. There is a vast literature on the topic (e.g. [9-13] and the references therein).
Stochastic resonance, linearization, self-oscillations, phase transitions, chaos are related to different aspects of stochastic systems. However, they all have something in common. Namely, they are associated with the noise and the nonlinearities of the system. These are the features which are taken into account with expression (2.4) for vector (2.3) in ODE (2.1). Indeed, the nonlinearities in (2.4) are presented by the second derivatives of the entries of drift function g, whereas the noise is accounted with diffusion function H (see (1.6)). As an example, we consider the connection of ODE (2.1) with only stochastic resonance. Stochastic resonance is usually associated with the following features. The stochastic system under consideration is nonlinear. (No stochastic resonance in linear stochastic systems has been found so far.) The system is driven by a deterministic periodic input signal. The nonlinearities and the periodic input cause stochastic resonance in the periodic output signal. Stochastic resonance in the output results in
(e.g. [12, Section 4.1]) an increase in the expectation of the periodic output signal or even the stronger phenomenon that the signalto- noise ratio (SNR) of the periodic output signal achieves a local maximum at a nonzero standard deviation (or root-mean-square value) of the signal.
The improvement in the output-signal amplitude or in the SNR is the most important practical advantage of stochastic resonance.
According to the above features, we shall consider the case when g(t, x) g(v, x), H(t, x) H(v, v-- +v(t), (3.1) where (C) is a vector independent of (t, x) and v(t) is a small periodic signal. In so doing, we analyze the periodic solution of system (2.1) under condition (3.1), i.e. d2x dt2 Og((C) + v(t), x) dr(t) Ov dt + Og((C) + v(t), x) Ox g((C) + v(t), x) + + ,,(t), x). (3.2) The periodic solutions can be studied by means of
the well-known techniques. The corresponding theory can be found, for example, in [22-26]. More practice-oriented analytical treatments are also available (like the finite-equation method by the authors [27, 28]). However, the general theory does not always explain the resonance in a compact form. To fill this gap, we simplify the ODE (3.2). We assume that: signal v(t) is small enough to enable one to
replace ODE (3.2) with its small-signal representation dZx/dt2 log((C), x)/Ov]dv(t)/dt + Og((C),x)/Ox]g((C),x) + ((C), x); (3.3) the autonomous version dZx/dt2 [Og((C),x)/Ox]g((C),x) + ((C),x) (3.4) of the system (3.3) has the unique equilibrium point x 2 such that det [Og(Yc)/Ox] O, number II[Og(Sc)/Ox]-N(sc)]l is much less than the typical values of IIg'(x)II for all x 2 and det{A2I [Og((C),2)/Ox]A-[O[t((C),Yc)/Ox])=0= Re A < 0 where I is the d d identity matrix.
Diego A. Cáceres M. C.I 19235570 Materia: CRF
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