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Figure 14 | The Journal of Mathematical Neuroscience

Figure 14

From: The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

Figure 14

Bifurcation diagram with parameters \(\mathcal{P}\) and \(b_{\mathrm{sp}}=-0.03\) in (47) (corresponding to the case \(\beta _{\epsilon 0R} > 0\), \(C_{\mathrm{det}} + \beta _{\epsilon 0R} > 0\) and satisfying \(\beta _{\epsilon 0R} < - C _{\mathrm{det}} + \sqrt{C^{2}_{\mathrm{det}} + \beta ^{2}_{\epsilon 0I}}\) as described in Section 4.1). (A): Two-parameter bifurcation diagram in the \((\lambda , \epsilon )\)-plane. The legend indicates bifurcations of a fixed point (FP) or a limit cycle (LC) giving rise to or involving the \(\varDelta \varphi = 0\) in-phase (IP) or \(\varDelta \varphi = \pi \) anti-phase (AP) solution branches; PD: period doubling; PF: pitchfork; TR: torus bifurcation. Text labels indicate the solutions that are stable in a given region, e.g. ‘IP+AP’ is a region with coexisting, stable IP and AP solutions. (B): One-parameter bifurcation diagram at \(\varepsilon =0.05\) showing the FP branch, IP branch and AP branch; dashed segments are unstable. The IP and AP branches bifurcate from the FP branch in subsequent Hopf bifurcations (bullet) for λ increasing. The IP branch emerges stable and remains stable. For increasing λ, the AP branch is initially unstable, gains stability at a torus bifurcation (star) and loses stability at a pitchfork bifurcation (diamond). (C): Coexisting solutions at \(\lambda \approx 3.05\) and \(\epsilon =0.05\) in the \((E_{1},E_{2})\)-plane. Motion on the diagonal (blue) corresponds to in-phase oscillations. (D): As (C) in the \((E_{1},I_{1})\)-plane for one EI oscillator. (E): As (C) at \(\epsilon =0.5\), where a torus bifurcation (star) is on an unstable branch that gains stability at a fold of limit cycle (square)

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