3.2 Bifurcations of Equilibria

The dynamical system described by Eq. (7) has three equilibria if $(2c-a)b > 0$:

• $O = (0,0,0)$
• $C^{+}= \left(\sqrt{(2c-a)b}\,,\,\sqrt{(2c-a)b}\,,\,2c-a\,\right)$
• $C^{-}= \left(-\sqrt{(2c-a)b}\,,\,-\sqrt{(2c-a)b}\,,\,2c-a\right)$

By considering a Jacobian matrix for one of these equilibria and calculating their eigenvalues, we can investigate the stability of each equilibrium based on the roots of the system characteristic equation.

Here, we first observe some bifurcations of the equilibria existing in Eq. (7). We gradually change the values of $a$. When $a>a_n$, the origin $O$ is the only equilibrium and it is a sink, where

$$a_n = \frac{(3+ 2\sqrt{3})c}{3}$$ (14)

is the parameter value at which the sink changes to a node. Then, a pitchfork bifurcation for $O$ emerges at the value of

$$a_{p} = 2c$$ (15)

At this moment, $O$ changes to a one-dimensionally unstable saddle and, at the same time, two symmetric sinks are generated, so we have three equilibria when $a< a_p$. The parameter $a_p$ is also the tangent/saddle-node bifurcation value for both $C^{+}$ and $C^{-}$. Note that these bifurcations do not depend on parameter $b$.

Hopf bifurcations emerges from these sinks, at the value of

$$a_h = \frac{3c + \sqrt{-8bc + 17c^2}}{4}$$ (16)

where the complex conjugate eigenvalues are $\mu=\pm i\sqrt{bc}$. When $a_h < a < a_p$, $C^{+}$ and $C^{-}$ are both stable sinks. At $a = a_h$, however, they change to two two-dimensional unstable saddles, and two isolated limit cycles are generated around them, respectively. These simulation results are combined into the figures shown in the next section.