conditions

topology types

$\frac{2e-2}{\delta r}<{\mu }_i<\frac{2e+2}{\delta r}$ ,

$\max \left\{0,\delta r{\mu }_i-2e+d^2-1\right\}<{\omega }_i\le \frac{1}{4}{\left(\delta r{\mu }_i-2e\right)}^2+d^2$

stable node

or ${\mu }_i<\frac{4\left(1-d^2\right)+2e^2}{\delta r}$ ,

$\max \left\{d^2,\frac{1}{4}{\left(\delta r{\mu }_i-2e\right)}^2+d^2\right\}<{\omega }_i\le 1+d^2$

${\mu }_i<\frac{2e-2}{\delta r}$ , $1+d^2<{\omega }_i\le \frac{1}{4}{\left(\delta r{\mu }_i-2e\right)}^2+d^2$

unstable node

or ${\mu }_i>\frac{2e+2}{\delta r}$ , $1+d^2<{\omega }_i\le \frac{1}{4}{\left(\delta r{\mu }_i-2e\right)}^2+d^2$

${\mu }_i>\frac{2e+1-d^2}{\delta r}$ , $0<{\omega }_i<\delta r{\mu }_i-2e+d^2-1$

saddle

${\omega }_i>\max \left\{1+d^2,\frac{1}{4}\left(\delta r{\mu }_i-2e^2\right)+d^2\right\}$

unstable focus

${\omega }_i=\delta r{\mu }_i-2e+d^2-1$

non-hyperbolic

$\frac{2e-2}{\delta r}<{\mu }_i<\frac{2e+2}{\delta r}{\omega }_i=d^2+1$