Instabilities of the harmonic oscillator with fluctuating damping

We investigate the instabilities of a linear damped oscillator due to fluctuations of the damping parameter. The fluctuations are driven either by Gaussian white noise or Poisson white noise (white shot noise). We consider three notions of stability. The first two are the well-known notions of stability in the mean and stability in the mean square. We introduce the concept of thermodynamic stability, corresponding to a nonpositive rate of energy dissipation at all times. We derive analytical results for the various instability thresholds, confirm the validity of our approach for white shot noise by numerical simulations, and obtain the unexpected result that mean-square and thermodynamic stability coincide for the two types of white noise.

Figure 1

Results from numerical stochastic simulations to verify Eqs. (4.15) and (4.16). (a) $\langle x\rangle$ and $\langle \zeta x\rangle$ versus time. (b) Left-hand side (circles) and right-hand side (solid line) of Eq. (4.15). We have performed averages over ${10}^6$ realizations, $\delta t=0.005$, $w_0=0.2$, $\omega =3$, $b=1$, and $\beta =5$.

Figure 2

Results from numerical stochastic simulations to verify Eqs. (4.29), (4.30), and (4.31). (a) $\langle x^2\rangle$ and $\langle \zeta x^2\rangle$ versus time. (b) Left-hand side (circles) and right-hand side (solid line) of Eq. (4.30). (c) Left-hand side (circles) and right-hand side (solid line) of Eq. (4.29). We have performed averages over ${10}^6$ realizations, $\delta t=0.005$, $w_0=0.2$, $\omega =3$, $b=1$, and $\beta =5$.

Figure 3

Stability diagram in the $\Delta$–$b/\beta$ plane for the damped oscillator under white shot noise for $\omega /\beta =0.1$. We represent the critical curves provided by the conditions ${\text{C}}_0=0$ and ${\text{C}}_1=0$. In the upper (white) region, the oscillator is unstable in the mean and in the mean square, $\langle x\rangle \to \infty$ and $\langle x^2\rangle \to \infty$ as $t\to \infty$. In the intermediate (yellow) region, it is stable in the mean but unstable in the mean square, $\langle x\rangle \to 0$ and $\langle x^2\rangle \to \infty$ as $t\to \infty$. In the lower (blue) region, it is stable both in the mean and in the mean square, $\langle x\rangle \to 0$ and $\langle x^2\rangle \to 0$ as $t\to \infty$.