Chapter 8.1, Problem 14E

Interpretation Introduction

Interpretation:

Consider the system equations are ${\dot{\text{x}}}_{\text{1}}{\text{ = - x}}_{\text{1}}\text{+ F}\left({\text{I - bx}}_{\text{2}}\right)$, ${\dot{\text{x}}}_{\text{2}}{\text{= - x}}_{\text{2}}\text{ + F}\left({\text{I - bx}}_{\text{1}}\right)$ where the gain function is given by $\text{F}\left(\text{x}\right)\text{ = }\frac{\text{1}}{{\text{1 + e}}^{\text{-x}}}$, I is the strength of the input signal, and b is the strength of the mutual antagonism.

To sketch the phase plane for various values of I and b(both positive).

To show that the symmetric fixed point ${\text{x}}_{\text{1}}{}^{\text{*}}{\text{ = x}}_{\text{2}}{}^{\text{*}}{\text{ = x}}^{\text{*}}$ is always a unique solution.

To show that at a sufficiently large value of b, the symmetric solution loses stability at a pitchfork bifurcation and also find type of pitchfork bifurcation.

Concept Introduction:

A phase plane is defined as the graphical representation of the differential equation which represents the limit cycle of the defined system equation.

A phase portrait is defined as the geometrical representation of the trajectories of the dynamical system in the phase plane of the system equation. Every set of the original condition is signified by a different curve or point in the phase plane.