Chapter 3.6, Problem 6E

Interpretation Introduction

Interpretation:

To show that the study state amplitude $\text{A(t)}$ dependent on $\text{ε}$ according to the power law $\text{A*}\propto {\text{ε}}^{\text{β}}$, where $\text{β = 0}\text{.50 ± 0}\text{.01}$ in context the equation $\tau \dot{A}\text{ = }\epsilon {\text{A - gA}}^3$. Determine the landau equation prediction. Find the relation between $\text{A*}$ and $\text{ε}$ for the equation ${\text{τ}\dot{\text{A}}\text{ = εA - gA}}^{\text{3}}{\text{-kA}}^{\text{5}}$, when $\text{g = 0}$. Sketch the bifurcation diagram of $\text{A*}$ vs $\text{ε}$ for the equation ${\text{τ}\dot{\text{A}}\text{ = h + εA - gA}}^{\text{3}}{\text{-kA}}^{\text{5}}$ in the cases $\text{g > 0}$, $\text{g = 0}$ and $\text{g < 0}$. Explain why the given curve has a strange shape.

Concept Introduction:

Express the equation in dimension less form using dimensionless parameter.

Analyze the fixed points for Landau equation and its prediction.

Draw the bifurcation diagram for various conditions.