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

Abstract

Bifurcation phenomena may be examined in terms of the class of non-linear differential equations of the first kind in the form x ̇ (t) = α nβx(t) − {x(t)} n+1 , 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) = x 0. 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).