Mean Values at Hopf Points and Oscillation-Induced Gain Modulation
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2025
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| author | Nesse, William Harold Hutchinson, Cooper John |
| author_facet | Nesse, William Harold Hutchinson, Cooper John |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_03593 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean Values at Hopf Points and Oscillation-Induced Gain Modulation Nesse, William Harold Hutchinson, Cooper John Dynamical Systems 34C23, 37G05, 37G10, 37G35 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. |
| title | Mean Values at Hopf Points and Oscillation-Induced Gain Modulation |
| topic | Dynamical Systems 34C23, 37G05, 37G10, 37G35 |
| url | https://arxiv.org/abs/2510.03593 |