Chapter 7.5, Problem 7E

Interpretation Introduction

Interpretation:

To sketch the nullclines for the system.

To determine ${\text{c}}_{\text{1}}{\text{ and c}}_{\text{2}}$ and show that the system exhibits relaxation oscillations for ${\text{c}}_{\text{1}}{\text{< c <c}}_{\text{2}}$.

To show that the system is excitable if $\text{c}$ is slightly less than ${\text{c}}_{\text{1}}$

Concept Introduction:

Nullclinesare a set of points in the phase plane where $\dot{\text{x}}\text{ = }\dot{\text{y}}\text{ = 0}$.

The set of points in a phase plane where $\dot{\text{x}}\text{ = 0}$ is known as x-nullclines. The geometrical meaning of x-nullclines is the points where the vectors are pointed either straight upward or downward.

The set of points in a phase plane where $\dot{\text{y}}\text{ = 0}$ is known as y-nullclines. The geometrical meaning of y-nullclines is the points where the vectors are pointed horizontally.

To find the equation for x-nullclines and y-nullclines, put $\dot{\text{x}}\text{ = }\dot{\text{y}}\text{ = 0}$ and solve for

$\text{x = f(x,y) and y = f(x,y)}$

The Taylor’s series expansion of a function $\sqrt{1-x}$ for $x\ll 1$ is

$\sqrt{1-\text{x}}=1-\frac{1}{2}\text{x}-\frac{1}{8}{\text{x}}^2-\frac{1}{16}{\text{x}}^3\mathrm{............}$

Here, higher order terms can be neglected because $x\ll 1$.