91Ó°ÊÓ

8\. Find the fixed points of the following systems. Linearize the system about each fixed point and determine the nature and stability in the neighborhood of each fixed point, when possible. Verify your findings by plotting phase portraits using a computer. a. $$ \begin{aligned} &x^{\prime}=x(100-x-2 y) \\ &y^{\prime}=y(150-x-6 y) \end{aligned} $$ b. $$ \begin{aligned} x^{\prime} &=x+x^{3} \\ y^{\prime} &=y+y^{3} \end{aligned} $$ C. $$ \begin{aligned} &x^{\prime}=x-x^{2}+x y \\ &y^{\prime}=2 y-x y-6 y^{2} \end{aligned} $$ d. $$ \begin{aligned} &x^{\prime}=-2 x y \\ &y^{\prime}=-x+y+x y-y^{3} \end{aligned} $$.

The Michaelis-Menten kinetics reaction is given by $$ E+S \frac{k_{3}}{k_{1}}+E S \underset{k_{2}}{ } E+P $$ The resulting system of equations for the chemical concentrations is $$ \begin{aligned} \frac{d[S]}{d t} &=-k_{1}[E][S]+k_{3}[E S] \\ \frac{d[E]}{d t} &=-k_{1}[E][S]+\left(k_{2}+k_{2}\right)[E S] \\ \frac{d[E S]}{d t} &=k_{1}[E][S]-\left(k_{2}+k_{2}\right)[E S] \\ \frac{d[P]}{d t} &=k_{3}[E S] \end{aligned} $$ In chemical kinetics, one seeks to determine the rate of product formation \(\left(v=d[P] / d t=k_{3}[E S]\right)\). Assuming that \([E S]\) is a constant, find \(v\) as a function of \([S]\) and the total enzyme concentration \(\left[E_{T}\right]=[E]+[E S] .\) As a nonlinear dynamical system, what are the equilibrium points?

An undamped, unforced Duffing Equation, \(\ddot{x}+\omega^{2} x+\epsilon x^{3}=0\), can be solved exactly in terms of elliptic functions. Using the results of Example \(4.18\), determine the solution of this equation and determine if there are any restrictions on the parameters.