Analysis of the Lev Ginzburg Equation

A second-order differential equation taking the form $$\ddot x=f(x,\dot x,p)\leqno(*)$$ was derived by Ginzburg$^1$. In this ODE, $\dot x$ and $\ddot x $ represent the first and second derivatives with respect to time; and $p$ stands for four parameters, $p=(p_1,p_2, p_3,p_4)$, where $(p_1,p_2,p_3)$ are non-negative and $p_4$ can be of either sign. The function $f$ is linear in $x$, but quadratic in $\dot x$. Ginzburg's main purpose in constructing this equation was to take into consideration the ``inertial behavior of biological populations." However, this ODE can also be used to model a variety of physical dynamical systems.$^2$ With four parameters, there is a broad range of possible solution behaviors. Our present purpose is to prove that limit-cycle behavior can occur for Eq.~($*$) under the appropriate conditions on the parameters. We demonstrate this result by means of the Hopf bifurcation theory.$^3$ References $^1$L. Ginzburg and M. Colyvan, Ecological Orbits (Oxford, New York, 2004). $^2$S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley; Reading, MA; 1994). $^3$E. Beltrami, Mathematics for Dynamic Modeling (Academic Press, Boston, 1987).