Theory and Modern Applications
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 \(E_{\pm }\) | |