Skip to main content

Table 1 The distribution of equilibrium of system (1)

From: Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua’s system

α β γ b c Distribution of equilibrium
= 0 ≠0 ≠0    Plane \((\frac{(\beta+\gamma)y}{\gamma },y,-\frac{\beta y}{\gamma})\)
= 0 = 0 ≠0    Plane (x,x,0)
= 0 ≠0 = 0    Plane (x,0,−x)
= 0 = 0 = 0    Plane (x,y,−x + y)
≠0 = 0 ≠0 \(4-4b+c^{2}\geqslant0\) \(E_{0}=(0, 0, 0)\) and \(E^{1}_{\pm}\)
≠0 = 0 ≠0 \(4-4b+c^{2}<0\) Unique \(E_{0}=(0, 0, 0)\)
≠0 = 0 = 0    Surface \((x, x(b+cx+x^{2}), x(-1+b+cx+x^{2}))\)
≠0 ≠0 = 0 \(-4b+c^{2}\geqslant0\) Unique \(E_{0}=(0, 0, 0)\) and \(E^{2}_{\pm}\)
≠0 ≠0 = 0 \(-4b+c^{2}<0\) \(E_{0}=(0, 0, 0)\)
≠0 ≠0 ≠0 \(A_{0}\geqslant0\) Unique \(E_{0}=(0, 0, 0)\)
≠0 ≠0 ≠0 \(A_{0}<0\) \(E_{0}=(0, 0, 0)\) and \(E_{\pm }\)