THESIS
2020
xxiv, 169 pages : illustrations (some color) ; 30 cm
Abstract
A Hopf bifurcation, where a fixed-point solution loses stability and a limit cycle is
born, is prevalent in many nonlinear dynamical systems. When a system prior to a Hopf
bifurcation is exposed to a sufficient level of noise, its noise-induced dynamics can provide
valuable information about the impending bifurcation and the post-bifurcation dynamics.
In this thesis, we present a system identification (SI) framework that exploits the noise-induced
dynamics prior to a supercritical or subcritical Hopf bifurcation. The framework
is novel in that it is capable of predicting the bifurcation point and the post-bifurcation
(limit-cycle) dynamics using only pre-bifurcation data. Specifically, we present two different
versions of the framework: input-output and output-only. For the inpu...[
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A Hopf bifurcation, where a fixed-point solution loses stability and a limit cycle is
born, is prevalent in many nonlinear dynamical systems. When a system prior to a Hopf
bifurcation is exposed to a sufficient level of noise, its noise-induced dynamics can provide
valuable information about the impending bifurcation and the post-bifurcation dynamics.
In this thesis, we present a system identification (SI) framework that exploits the noise-induced
dynamics prior to a supercritical or subcritical Hopf bifurcation. The framework
is novel in that it is capable of predicting the bifurcation point and the post-bifurcation
(limit-cycle) dynamics using only pre-bifurcation data. Specifically, we present two different
versions of the framework: input-output and output-only. For the input-output
version, the system is forced with additive noise generated by an external actuator, and
its response is measured. For the output-only version, the intrinsic noise of the system
acts as the noise source, so no external actuator is required, and only the output
signal is measured. In both versions, the Fokker–Planck equations, which describe the
probability density function of the fluctuation amplitude, are derived from self-excited
oscillator models. Then, the coefficients of these models are extracted from the experimental
probability density functions characterizing the noise-induced response in the
fixed-point regime, prior to the Hopf point itself. These two versions of the SI framework
are tested on three different experimental systems: a hydrodynamic system (a low-density
jet), a laminar thermoacoustic system (a flame-driven Rijke tube), and a turbulent thermoacoustic
system (a gas-turbine combustor). For these systems, we demonstrate that
the proposed framework can identify the super/subcritical nature of the Hopf bifurcation
and the system’s order of nonlinearity. Moreover, by extrapolating the identified
model coefficients, we are able to forecast the locations of the bifurcation points and the
limit-cycle features after those points. To the best of our knowledge, this is the first time
that SI has been performed using data from only the pre-bifurcation (fixed-point) regime,
without the need for a priori knowledge of the location of the bifurcation point. Given
that such noise-induced dynamics are universal near a Hopf bifurcation, the proposed SI
framework should be applicable to a variety of nonlinear dynamical systems in nature and
engineering.
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