Noise in oscillators: a review of state space decomposition approaches

Traversa, Fabio L.; Bonnin, Michele; Corinto, Fernando; Bonani, Fabrizio

doi:10.1007/s10825-014-0651-3

Cited by 13 publications

(13 citation statements)

References 45 publications

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“…Thus, the gradient vectors of the phase and amplitudes evaluated on χ are left eigenvectors of the monodromy matrix, which are called the adjoint covariant Lyapunov vectors [45][46][47] .…”

Section: Reduction Framework and A Methods To Calculate The Respomentioning

confidence: 99%

Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems

Shirasaka

Kurebayashi

Nakao

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Phase reduction framework for limit-cycling systems based on isochrons has been used as a powerful tool for analyzing the rhythmic phenomena. Recently, the notion of isostables, which complements the isochrons by characterizing amplitudes of the system state, i.e., deviations from the limit-cycle attractor, has been introduced to describe the transient dynamics around the limit cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we introduce a framework for a reduced phase-amplitude description of transient dynamics of stable limit-cycling systems. In contrast to the preceding study, the isostables are treated in a fully consistent way with the Koopman operator analysis, which enables us to avoid discontinuities of the isostables and to apply the framework to system states far from the limit cycle. We also propose a new, convenient bi-orthogonalization method to obtain the response functions of the amplitudes, which can be interpreted as an extension of the adjoint covariant Lyapunov vector to transient dynamics in limit-cycling systems. We illustrate the utility of the proposed reduction framework by estimating the optimal injection timing of external input that efficiently suppresses deviations of the system state from the limit cycle in a model of a biochemical oscillator.

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Section: Reduction Framework and A Methods To Calculate The Respomentioning

confidence: 99%

Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems

Shirasaka

Kurebayashi

Nakao

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…We assume that (27) admits of an asymptotically stable Tperiodic solution x s (t), represented by a limit cycle in its state space. We shall derive an equivalent description of system (25) (or (26)) in terms of phase and amplitude deviation variables, analogous to the one derived in [18], [26], [27]. The phase function used in our description coincides locally, in the neighborhood of the limit cycle, with the asymptotic phase defined in [6], [13], [28].…”

Section: Oscillators With Colored Noisementioning

confidence: 99%

“…For our purpose it is more convenient to work with the Itô SDE (26). The reason to prefer the Itô over the Stratonovich interpretation is that Itô integrals are adapted processes, i.e.…”

Section: Oscillators With Colored Noisementioning

confidence: 99%

Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the itô-Stratonovich Dilemma

Bonnin

Traversa

Bonani

2019

IEEE Trans. Circuits Syst. I

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We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an OrnsteinUhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a driftdiffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of previous phase reduced models.

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“…white noise Substituting the defined A and B in (21) and simplifying the terms leads to ln τ 2 + 1 2 > ln τ (22) Knowing that ln e 1/2 = 1/2, the condition of (22) is simplified to τ < τ 2 e. Therefore, it can be concluded that in order to calculate the normalised phase variance of the Flicker noise for the known τ 1 and τ 2 , the parameter τ must satisfy τ 1 < τ < τ 2 e. The PSD of the Flicker noise is S 1/ f ( f ) = S ⋅ f −1 . Comparing this PSD with (11), the normalising factor is X 2 = Sln(τ 2 /τ 1 ).…”

Section: Lorentzian Noise ( F −2 )mentioning

confidence: 99%

Phase noise of electrical oscillators due to multi‐Lorentzian noise sources based on the LTV model

Hashemi

2017

IET Microwaves, Antennas & Propagation

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Using the general expression of the phase variance expressed by the linear time variant (LTV) phase noise model, phase variances due to multi‐Lorentzian and white noise sources of electrical oscillators have been extracted and phase noise expressions due to white and Flicker noises have been calculated. Despite the basic LTV model in which the expressed phase noise due to white noise is valid for offset frequencies larger than the −3 dB corner frequency of the output and it tends to infinity for small offset frequencies, in the proposed model, the extracted phase noises due to white and Flicker noises are valid for the entire spectrum and become limited for small offset frequencies. Comparison between the results of the proposed model by those of the harmonic balance technique shows the validity of the model. Also, the effects of the physical parameters of the oscillator circuit on the extracted phase noise can be observed in the expressions. The proposed model is applicable to calculate the phase noise of the oscillators with single perturbation source applied to the intended output node.

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Noise in oscillators: a review of state space decomposition approaches

Cited by 13 publications

References 45 publications

Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems

Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems

Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the itô-Stratonovich Dilemma

Phase noise of electrical oscillators due to multi‐Lorentzian noise sources based on the LTV model

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