Mean Values at Hopf Points and Oscillation-Induced Gain Modulation

We present a result concerning the mean value of orbits emerging from Hopf bifurcations. We then apply this result to identify a new phenomenon termed {\it oscillation-induced gain modulation}. A Hopf bifurcation of a system $\dot{x} = f(x; \alpha)$ with parameter $\alpha$ is characterized by the emergence of a limit cycle with an amplitude increasing from zero, coinciding with a stability change of an equilibrium $x_0(\alpha)$ when $\alpha$ passes a critical value $\alpha^*$. This bifurcation is associated with the real part of a single eigenpair $\lambda = \mu(\alpha) \pm i \omega(\alpha)$ of the linearized system crossing zero: $\mu(\alpha^*) = 0$, $\mu'(\alpha^*) \neq 0$. We establish a result concerning the temporal mean of the oscillation cycle over the period $T$ of oscillation: $\langle x \rangle_{\alpha} = \frac{1}{T} \int_0^{T} x(t; \alpha) dt$. We set the mean to be $\langle x \rangle_{\alpha} = x_0(\alpha)$ when the equilibrium has no surrounding limit cycle. However, when a limit cycle exists, we show that that the deviation of the mean from the equilibrium is expressible as $\langle x \rangle_{\alpha} - x_0(\alpha) = K \mu(\alpha) + \mathcal{O}(\mu(\alpha)^2)$. That is, the mean value deviates from the equilibrium's location in proportion to $\mu(\alpha)$, with a mean deviation determined by the vector quantity $K(\alpha) \mu(\alpha)$ that depends on the tensors of $f$ up to third-order. If we consider $\alpha$ to be an input to the model, and the mean $\langle x \rangle_{\alpha}$ as the output, then the mean deviation $K \mu(\alpha)$ introduces a discontinuity to the cycle mean gain $\frac{d \langle x \rangle_{\alpha}}{d\alpha}$ at the bifurcation, which we term oscillation-induced gain modulation (OIGM). We the cycle mean deviation result for general Hopf points in two-dimensional and $n$-dimensional systems, as well as showcase several examples of OIGM.