Chapter 8.1, Problem 15E

Interpretation Introduction

Interpretation:

To interpret and justify the form of various terms in the equations ${\dot{\text{n}}}_{\text{A}}\text{ = }\left({\text{p + n}}_{\text{A}}\right){\text{n}}_{\text{AB}}{\text{ - n}}_{\text{A}}{\text{n}}_{\text{B}}$, ${\dot{\text{n}}}_{\text{B}}{\text{ = n}}_{\text{B}}{\text{n}}_{\text{AB}}\text{ - }\left({\text{p + n}}_{\text{A}}\right){\text{n}}_{\text{B}}$, where ${\text{n}}_{\text{AB}}\text{ = 1 - }\left({\text{p + n}}_{\text{A}}\right){\text{ - n}}_{\text{B}}$. If ${\text{n}}_{\text{B}}\left(0\right)\text{ = 1 - p}$, and ${\text{n}}_{\text{A}}\left(\text{0}\right){\text{ = n}}_{\text{AB}}\left(\text{0}\right)\text{ = 0}$, to numerically integrate the system until it reaches equilibrium. To show that the final state changes discontinuously as a function of $\text{p}$. To show analytically that ${\text{p}}_{\text{c}}\text{ = 1 - }\frac{\sqrt{\text{3}}}{\text{2}}\approx \text{0}\text{.134}$. To find the type of bifurcation occurs at ${\text{p}}_{\text{c}}$.

Concept Introduction:

• Fixed points occur where ${\dot{\text{n}}}_{\text{A}}\text{ = 0}$, and ${\dot{\text{n}}}_{\text{B}}\text{ = 0}$.

• Linearization

  $\text{A = }\left(\begin{array}{cc}\frac{\partial {\dot{\text{n}}}_{\text{A}}}{\partial {\text{n}}_{\text{A}}}& \frac{\partial {\dot{\text{n}}}_{\text{B}}}{\partial {\text{n}}_{\text{A}}}\\ \frac{\partial {\dot{\text{n}}}_{\text{A}}}{\partial {\text{n}}_{\text{B}}}& \frac{\partial {\dot{\text{n}}}_{\text{B}}}{\partial {\text{n}}_{\text{B}}}\end{array}\right)$

• The Eigenvalue $\text{λ}$ can be calculated using the characteristic equation

  $\left|\left(\text{A - λI}\right)\right|\text{ = 0}$