Table 1 Values for the trace (Tr), the determinant (Det) and the discriminant (Δ) of the linearisation of system (13) at the fixed points \(\bar{\mathcal{S}}^{\pm }_{\mathrm{osc}}\) near the curves \(C^{\pm } _{\mathrm{HB}}\) (\(\bar{\alpha }^{\pm }= 0\)) and near to the uncoupled case (\(\bar{\alpha }^{\pm }\geq 0\) and ϵ small)
From: The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability
\(\bar{\mathcal{S}}^{+}_{\mathrm{osc}}\) | \(\bar{\mathcal{S}}^{-}_{\mathrm{osc}}\) | |||
|---|---|---|---|---|
\(\bar{\alpha }^{+} \rightarrow 0^{+}\) | \(\bar{\alpha }^{+} \geq 0\), ϵ→0+ | \(\bar{\alpha }^{-} \rightarrow 0^{+}\) | \(\bar{\alpha }^{-} \geq 0\), ϵ→0+ | |
Tr | \(-4 \epsilon \beta _{\epsilon 0R}\) | −2λ | \(4 \epsilon \beta _{\epsilon 0R}\) | −2λ |
Det | \(4 \epsilon ^{2} (\beta ^{2}_{\epsilon 0I} + \beta ^{2}_{\epsilon 0R})\) | \(4 \epsilon \lambda (C_{\mathrm{det}}+\beta _{\epsilon 0R})\) | \(4 \epsilon ^{2} (\beta ^{2}_{\epsilon 0I} + \beta ^{2}_{\epsilon 0R})\) | \(-4 \epsilon \lambda (C_{\mathrm{det}}+\beta _{\epsilon 0R})\) |
Δ | \(-4 \epsilon ^{2} \beta ^{2}_{\epsilon 0I}\) | \(\lambda ^{2}\) | \(-4 \epsilon ^{2} \beta ^{2}_{\epsilon 0I}\) | \(\lambda ^{2}\) |