Multiple mechanisms for stochastic resonance are inherent to sinusoidally driven noisy Hopf oscillators

To ensure their sensitivity to weak periodic signals, some physical systems likely operate near a Hopf bifurcation. Many systems operating near such a bifurcation exhibit stochastic resonance, but it is unclear which mechanisms for resonance are inherent to the bifurcation. To address this question, we study the sinusoidally forced dynamics of noisy supercritical and subcritical Hopf oscillators. We find four qualitatively different mechanisms for stochastic resonance and determine the conditions for each type of resonance.

Figure 1

Steady-state distributions of probability density are shown for stochastic Hopf oscillators in a frame rotating at the forcing frequency for different values of the control parameter $\mu$ in unforced $f=0$ and forced $f={10}^{-2}$ scenarios [Eq. (4)]. Forcing is at the Hopf frequency $\omega ={\omega }_0$ and the noise level $d={10}^{-2}$. In the absence of forcing, the distributions are radially symmetric. Forcing introduces radial asymmetry and a most probable phase of 0 radians. In these cases, the corresponding distributions in the original frame of reference rotate in a counterclockwise direction around the origin at the frequency of driving. [(a)–(f)] Supercritical Hopf oscillator (sup). [(g)–(l)] Subcritical Hopf oscillator (sub). For all figures in this manuscript the Hopf frequency ${\omega }_0=1, b=-1$, and $c=0$ in the supercritical case or $b=1$ and $c=-1$ in the subcritical case.

Figure 2

The state diagram for a deterministic subcritical Hopf oscillator subjected to resonant forcing is shown. Global bifurcations are not shown. There exist two stable responses to tuned forcing with amplitude $f$ for operating points within the green region demarcated by the red saddle-node bifurcation lines $[f_1(\mu ,b,c)<f<f_2(\mu ,b,c)]$ that intersect at a cusp bifurcation $[\mu =9b^2/20c$ and $f=\sqrt{-54b^5/3125c^3}$, orange dot]. Two unstable fixed points arise through a third saddle-node bifurcation line [gray dashed, $f_3(\mu ,b,c)]$ that intersects one of the red lines at $\mu =b^2/5c$ and $f=\sqrt{-4b^5/3125c^3}$ (purple dot) and asymptotically meets the other red line at $\mu =b^2/4c$ and $f=0$. Stochastic resonance in the phase-locked amplitude and the vector strength occurs for operating points in the region shaded dark green, in which ${\alpha }_1<{\alpha }_2$. The phase-locked amplitude also peaks as a function of the noise level for points in the region shaded orange $[f<f_1(\mu ,b,c)$ and $9b^2/20c<\mu <b^2/4c$ or $f<f_4(\mu ,b,c)$ and $\mu <9b^2/20c]$.

Figure 3

Steady-state distributions of probability density for stochastic Hopf bifurcations are shown for a frame rotating at the forcing frequency for different values of the control parameter $\mu$. Forcing is at a frequency, $\omega =0.9{\omega }_0$, lower than the Hopf frequency ${\omega }_0$, and has an amplitude $f={10}^{-2}$; the noise level $d={10}^{-2}$. The distributions are rotated counterclockwise and are less skewed than the distributions resulting from forcing at the Hopf frequency (Fig. 1). [(a)–(c)] Supercritical Hopf oscillator (sup). [(d)–(f)] Subcritical Hopf oscillator (sub).

Figure 4

The phase-locked amplitude and vector strength are shown as functions of the noise level for forcing at the Hopf frequency near supercritical [(a) and (b)] and subcritical [(c) and (d)] Hopf bifurcations. [(a) and (b)] The values of the control parameter are $\mu =1$ (red), $\mu =0.2$ (orange), $\mu =0$ (green), and $\mu =-1$ (blue). [(c) and (d)] The values of the control parameter are $\mu =1$ (red), $\mu =0$ (green), $\mu =-0.15$ (purple), $\mu =-0.22$ (orange), $\mu =-0.3$ (cyan), and $\mu =-1$ (blue). The forcing amplitude $f={10}^{-2}$.

Figure 5

Stochastic resonance for a self-oscillating ($\mu =0.2\ge 0$) and detuned supercritical Hopf oscillator. (a) The state of a deterministic oscillator is a function of the forcing amplitude and frequency. Areas of less than perfect entrainment (light purple regions) are separated from those exhibiting perfect entrainment (white regions) by supercritical Hopf (dashed red lines) or a saddle-node (solid red lines) bifurcations, which meet tangentially at Bogdanov-Takens points (red dots). Global bifurcations are not shown. Dots indicate the operating points $(\omega =0.89,f=4\times {10}^{-2})$ (orange), $(\omega =0.85,f=4\times {10}^{-2})$ (cyan), and $(\omega =0.89,f=2\times {10}^{-2})$ (purple). (b) At the orange operating point a stable limit cycle (dark green) surrounds an unstable fixed point (dark blue). The speed on the cycle is a minimum (maximum) at the gray (magenta) point. A circular locus with the same center as the limit cycle and passing through the cycle's point of minimum speed is shown (light blue). (c) Angular probability distributions $P_{\psi }\left(\psi \right)$ for the orange operating point are shown for the noise levels $d={10}^{-3}$ (red), $d={10}^{-2}$ (green), and $d={10}^{-1}$ (blue). The deterministic system resides for the greatest time at the limit cycle's point of minimum speed (gray line). (d) Adler dynamics along the light blue circle in (b) ($A=1.19$) yields the probability distributions $P_{\mathrm{A}}\left(\psi \right)$ for the noise levels $d_{\mathrm{A}}={10}^{-2}$ (red), $d_{\mathrm{A}}={10}^{-1}$ (green), and $d_{\mathrm{A}}={10}^0$ (blue). The Adler system's speed is chosen to be a minimum at the same angle for which the speed on the limit cycle is a minimum (gray line). [(e)–(h)] The phase-locked amplitude and vector strength are shown as functions of the noise level (solid lines) and in the deterministic limit (dashed lines). [(e) and (f)] Responsiveness for operating points indicated in (a). [(g) and (h)] Responsiveness for Adler dynamics on circular loci matched to the operating points indicated in (a); $A=1.19$ (orange), $A=1.65$ (cyan), and $A=2.5$ (purple).

Figure 6

Stochastic resonance for a detuned subcritical Hopf oscillator. [(a)–(d)] State diagrams for a deterministic system as functions of the forcing amplitude and stimulus frequency are depicted similarly to Fig. 5. The forced oscillator is perfectly entrained when it possesses one stable fixed point in the white regions or two stable fixed points in the central green regions. Less than perfect entrainment occurs when the system possesses an unstable fixed point and a stable limit cycle (light purple regions) or a stable fixed point coexisting with a stable and unstable limit cycle (blue regions). (a) The state diagram for $\mu \ge 0$ ($\mu =0$) is illustrated. The Hopf bifurcations are supercritical. (b) The diagram corresponding to $b^2/5c<\mu <0$ ($\mu =-0.1$) is shown. The red (blue) Hopf bifurcation lines are supercritical (subcritical). (c) The state diagram for $b^2/4c<\mu <b^2/5c$ ($\mu =-0.22$) is similar to (b) except that the central red line dips below the central blue line. (d) The state diagram when $9b^2/20c<\mu <b^2/4c$ ($\mu =-0.3$) is depicted. The Hopf bifurcations illustrated are supercritical. [(e) and (f)] The detuned ($\omega =0.9{\omega }_0$) phase-locked amplitude and vector strength are shown as functions of the noise level. The values of the control parameter are $\mu =1$ (red), $\mu =0$ (green, indicated in panel a), $\mu =-0.22$ [orange, indicated in (c)], and $\mu =-1$ (blue). The forcing amplitude $f={10}^{-2}$. (g) At the orange operating point in (c), a stable limit cycle (dark green) surrounds an unstable limit cycle (dashed red line), which encloses a stable fixed point (dark blue dot). The points at which the speed on the stable limit cycle is a minimum (gray) and maximum (magenta) are shown. (h) Amplitude distributions $P_{\rho }\left(\rho \right)$ for the orange point are shown in which the noise level $d={10}^{-3}$ (dark red), $d=2\times {10}^{-3}$ (cyan), $d=5\times {10}^{-3}$ (purple), $d={10}^{-2}$ (light red), and $d={10}^{-1}$ (light blue). The vertical dark blue (gray) line corresponds to the stable fixed point (point of minimum speed on the stable limit cycle) shown in (g).