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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 \(-4 \epsilon ^{2} \beta ^{2}_{\epsilon 0I}\) \(\lambda ^{2}\) \(-4 \epsilon ^{2} \beta ^{2}_{\epsilon 0I}\) \(\lambda ^{2}\)