Systematics in bifurcations of exponential growth equations with a non-linear feedback

Bifurcation phenomena may be examined in terms of the class of non-linear differential equations of the first kind in the form image, where α and β are constants and n is an integer. A transformation of variables is applied taking into account the initial values x(0) = x0. The solutions are presented in the form of dimensionless variables and in polar coordinates to visualize limit cycles. The well known logistic and pitchfork equations (n = 1 and 2, respectively) are special cases, as is also the equation for explosive population growth (n = 1, α and β imaginary).