Mean Values at Hopf Points and Oscillation-Induced Gain Modulation

TL;DR

This work proves a mean-value deviation theorem for Hopf bifurcations: near a codimension-one Hopf point, the cycle mean $\langle x \rangle_\alpha$ deviates from the equilibrium $x_0(\alpha)$ by $K(\alpha)\mu(\alpha)$ plus higher-order terms, with the leading coefficient $K(\alpha)$ determined by low-order tensorial derivatives of the vector field. This yields oscillation-induced gain modulation (OIGM), a discontinuity in the mean response slope with respect to the bifurcation parameter, observable across 2D and 3D models. The paper provides a detailed 2D Hopf Mean Value Theorem, extends it to $n$-dimensions, and illustrates OIGM through Predator-Prey, Brusselator, and Wilson-Cowan examples, including a 3D feedback-control system. The results connect classical Hopf theory with practical consequences for mean-field quantities in oscillatory regimes, and offer a framework to predict mean-trace shifts in diverse scientific models.

Abstract

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 $\dot{x} = f(x; α)$ with parameter $α$ is characterized by the emergence of a limit cycle with an amplitude increasing from zero, coinciding with a stability change of an equilibrium $x_0(α)$ when $α$ passes a critical value $α^*$. This bifurcation is associated with the real part of a single eigenpair $λ= μ(α) \pm i ω(α)$ of the linearized system crossing zero: $μ(α^*) = 0$, $μ'(α^*) \neq 0$. We establish a result concerning the temporal mean of the oscillation cycle over the period $T$ of oscillation: $\langle x \rangle_α = \frac{1}{T} \int_0^{T} x(t; α) dt $. We set the mean to be $\langle x \rangle_α = x_0(α)$ 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_α - x_0(α) = K μ(α) + \mathcal{O}(μ(α)^2)$. That is, the mean value deviates from the equilibrium's location in proportion to $μ(α)$, with a mean deviation determined by the vector quantity $K(α) μ(α) $ that depends on the tensors of $f$ up to third-order. If we consider $α$ to be an input to the model, and the mean $\langle x \rangle_α $ as the output, then the mean deviation $K μ(α)$ introduces a discontinuity to the cycle mean gain $\frac{d \langle x \rangle_α}{dα}$ 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.

Figures (5)

• Figure 1: The predator-prey model, consisting of coupled prey ($x_1$) and predator ($x_2$) sizes, illustrates cycle mean deviation from equilibrium. (A) Simulations yield either stable equilibria or stable oscillatory orbits depending on the bifurcation parameter $\alpha$ value relative to the bifurcation point $\alpha^*$. (B) For $\alpha < \alpha^*$ the mean value is the stable equilibrium, but for $\alpha > \alpha^*$ stable cycles emerge (examples shown in black). As $\alpha$ increases beyond $\alpha^*$, the cycle mean (blue) deviates from the unstable equilibria (red). (C) A close-up at the cycle mean and unstable equilibria in phase space, while analytical estimates (cyan) accurately predict numerically computed cycle means (dark blue) near the bifurcation point (see Section \ref{['meanOrbitsThm']}).
• Figure 2: An example of oscillation-induced gain modulation (OIGM). (A) As $\alpha$ increases but remains below $\alpha^*$, the prey ($x_1$) stable equilibria (yellow) decrease. As $\alpha$ increases above $\alpha^*$, the numerically approximated cycle mean prey values (pink) decrease at the same rate as the unstable prey equilibria (orange x's), while the cycle mean analytic estimate (see Theorem 1) is tangent to the aforementioned numerical estimates near the bifurcation but looses accuracy thereafter. (B) As $\alpha$ increases but remains below $\alpha^*$, predator ($x_2$) equilibria (dark blue) first increase, then decrease. Above $\alpha^*$, the unstable equilibria (cyan) continue on the same trajectory as the stable equilibria; however, the cycle mean numerical approximation, and the analytic estimate (lavender and purple, respectively), show that the mean predator values decrease at a faster rate than the unstable equilibria as a function of $\alpha$. Hence, the oscillation induces an abrupt downward shift in the slope of the mean predators, which we term OIGM.
• Figure 3: The Brusselator model with the mean of $x_2$ as a function of $\alpha$, over various $A$-values. (A) For $A < 1$, both the analytically estimated and numerically approximated cycle means (lavender and purple circles) modulate below the unstable equilibrium (cyan). (B) For $A = 1$, the cycle mean coincides with the unstable equilibrium because $g_{11} = 0$. (C,D) For $A > 1$, both the analytically estimated and numerically approximated $x_2$ mean deviates are above its unstable equilibrium value.
• Figure 4: The Wilson-Cowan model exhibits OGIM in both the e- and i-cell populations, but in opposite directions. (A) As input increases beyond the bifurcation point, oscillations emerge with progressively larger amplitudes, show in phase space. The cycle mean also exhibits a trajectory shift at the bifurcation. (B) a close-up of panel A, showing oscillation-induced cycle mean trajectory shifts for both numerically and analytically computed cycle mean estimates. (C) e-cell exhibits oscillation-induced reduction in cycle mean gain as a function of $I$. (D) The opposite occurs for i-cells.
• Figure 5: Mean deviation in an $n=3$-dimensional feedback control model. (A-D): Each panel shows the 3-dimensional phase-space, including the quadratic approximation of the $W^u_{loc}(x_0)$ surface, the numerical and analytic mean deviations, and numerical and analytic limit cycle orbits. A-D show these objects over four $\alpha$-values, each progressively further from the bifurcation point. Note the difference in scales between panels. (E): A close up of the numerically computed and analytically computed mean deviation $K(\alpha) \mu(\alpha)$ over a larger sampling of $\alpha$-values past the bifurcation point. Naturally, nearest to the bifurcation, when $\mu(\alpha)$ is near zero, the analytical approximation is tangent to the numerically computed mean deviation.