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| Models of geophysical fluid flows typically depend on characteristic parameters such as a Reynolds number. At fixed critical values of the parameter(s)
the solution may undergo bifurcations which change the nature of its behavior. However, in practice, shifts and uncertainties in physical or experimental conditions
make it natural to investigate the effect of slow drifts and stochastic perturbations of such parameters on this behavior.
In order to simplify the analysis and increase our understanding of these models we consider instead the more manageable case of a (normalized) damped Duffing oscillator
\[ \ddot{x}+2\gamma\dot{x}+k(x) = f(\theta), \]
where $\gamma$ represents a (small) friction coefficient, $k(x)$ a (typically nonlinear) spring force, and $f(\theta)$ a $2\pi$-periodic (possibly stochastic)
forcing depending on an angle $\theta$. The periodicity of $f$ leads to sustained oscillations in $x$, whose amplitude depends on the specific choice of
$\gamma$, $k(\cdot)$, and $f(\cdot)$.
In the case $k(x)=x$ (linear spring) and $\theta=\omega t$ (static angular frequency $\omega$), resonance effects typically occur
(e.g. for $f(\theta)=\sin\theta$) when $\omega$ matches the natural frequency $(1-\gamma^2)^{1/2}\approx1$ of the mass-spring system. In the case
$k(x)=x-\eta x^3$ (softening nonlinear spring), the resonance amplitude undergoes a saddle-node bifurcation as $\eta$ increases.
In this work we investigate the effect of slow, stochastic, variations in the time evolution of $\theta$, specifically
\[ \dot{\theta} = \omega(\varepsilon t) + \sigma(t) \dot{W}_t, \]
where $\varepsilon\ll1$, $W_t$ is a Wiener process, and $\sigma^2(t)$ represents the variance of the (state-independent) stochastic contribution,
on the system response. We focus in particular on how the resonance amplitude and the saddle-node bifurcation are modified from the static, deterministic, situation.
This work is conducted in collaboration with Prof. Juan M. Lopez (Mathematics, ASU), Jason Yalim (graduate student, ASU), and Stephanie Taylor (undergraduate student, ASU). |
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