Chapter 8.2, Problem 14E

Interpretation Introduction

Interpretation:

To identify if bifurcation is subcritical or supercritical for the system ${\dot{\text{x}}\text{ = μx + y - x}}^{\text{3}}$, ${\dot{\text{y}}\text{ = - x + μy + 2y}}^3$ when $\text{μ = 0}$.

Concept Introduction:

Any system undergoing a Hopf bifurcation can be put into following form by suitable changes in variables,

$\begin{array}{l}\dot{\text{x}}\text{ = - ωy + f}\left(\text{x,y}\right)\\ \dot{\text{y}}\text{ = ωx + g}\left(\text{x,y}\right)\end{array}$

Here, f and g contain only higher order terms that vanish at the origin.

To decide whether the bifurcation is supercritical orsubcritical, find the sign of quantity ${\text{16a = f}}_{\text{xxx}}{\text{+f}}_{\text{xyy}}{\text{+g}}_{\text{xxy}}{\text{+g}}_{\text{yyy}}\text{+}\frac{\text{1}}{\text{ω}}\left({\text{f}}_{\text{xy}}\left({\text{f}}_{\text{xx}}{\text{+f}}_{\text{yy}}\right){\text{-g}}_{\text{xy}}\left({\text{g}}_{\text{xx}}{\text{+g}}_{\text{yy}}\right){\text{-f}}_{\text{xx}}{\text{g}}_{\text{xx}}{\text{+f}}_{\text{yy}}{\text{g}}_{\text{yy}}\right)$.

If $\text{a > 0}$, then bifurcation is subcritical.

If $\text{a < 0}$, then bifurcation is supercritical.