Chapter 6.1, Problem 14E

Interpretation Introduction

Interpretation:

To find the equation of the stable manifold of the system ${\dot{\text{x}}\text{ = x + e}}^{\text{- y}}$, $\dot{\text{y}}\text{ = -y}$. Also, check if an analytical result produces a curve with the same shape as the stable manifold shown in Figure $\text{6}\text{.1}\text{.4}$.

Concept Introduction:

The equation for stable manifold can be found by introducing the new variable $\text{u = x + 1}$ in the given system.

The equation for stable manifold is given by ${\text{y = a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)$.

${\text{e}}^{\text{- y}}\text{= 1 - y + }\frac{{\text{y}}^{\text{2}}}{\text{2}}$

Answer to Problem 14E

Solution:

a) The equation for stable manifold is $\text{y = 2u +}\left(\frac{\text{4}}{3}\right){\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)$.

b) With the same shape as the stable manifold shown in Figure $\text{6}\text{.1}\text{.4}$, the analytical result produces a curve.

Explanation of Solution

a) The system is given as:

${\dot{\text{x}}\text{ = x + e}}^{\text{- y}}$, $\dot{\text{y}}\text{ = -y}$

The given equations can be rewritten as:

$\frac{\text{dx}}{\text{dt}}{\text{ = x + e}}^{\text{- y}}$, $\frac{\text{dy}}{\text{dt}}\text{ = -y}$

It is given that the system has one fixed saddle point at $\left(\text{-1,0}\right)$.

Introducing the new variable $\text{u = x + 1}$ or $\text{x = u - 1}$.

Substituting $\text{x = u - 1}$ in the equation $\frac{\text{dx}}{\text{dt}}{\text{ = x + e}}^{\text{- y}}$,

$\frac{\text{du}}{\text{dt}}{\text{ = u - 1 + e}}^{\text{- y}}$

Dividing the equation $\frac{\text{dy}}{\text{dt}}\text{ = -y}$ by the above equation,

$\left(\frac{\text{dy}}{\text{dt}}\right)\left(\frac{\text{dt}}{\text{du}}\right)\text{ = }\frac{\text{-y}}{{\text{u - 1 + e}}^{\text{- y}}}$

$\frac{\text{dy}}{\text{du}}\text{ = }\frac{\text{-y}}{{\text{u - 1 + e}}^{\text{- y}}}\mathrm{.............................................}\text{ (1)}$

The point $\left(\text{x, y}\right)$ lies on the stable manifold.

The equation for stable manifold is ${\text{y = a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)$.

Differentiating both sides with respect to $\text{u}$,

$\frac{\text{dy}}{\text{du}}{\text{ = a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u }\mathrm{................................................}\text{ (2)}$

From equations $\text{(1)}$ and $\text{(2)}$,

$\frac{\text{-y}}{{\text{u - 1 + e}}^{\text{- y}}}{\text{ = a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}$

Since ${\text{e}}^{\text{- y}}\text{= 1 - y + }\frac{{\text{y}}^{\text{2}}}{\text{2}}$, therefore,

$\frac{\text{-y}}{\text{u - 1 + 1 - y + }\frac{{\text{y}}^{\text{2}}}{\text{2}}}{\text{ = a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}$

$\frac{\text{-y}}{\text{u - y + }\frac{{\text{y}}^{\text{2}}}{\text{2}}}{\text{ = a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}$

Substituting ${\text{y = a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)$ in the above equation,

$\frac{\text{-}\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)\right)}{\text{u - }\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)\right)\text{ + }\frac{{\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)\right)}^{\text{2}}}{\text{2}}}{\text{ = a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}$

Eliminating all the terms higher than the second order term in $\text{u}$,

$\frac{\text{-}\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\right)}{\text{u - }\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\right)\text{ + }\frac{{\left({\text{a}}_{\text{1}}\text{u}\right)}^{\text{2}}}{\text{2}}}{\text{ = a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}$

$\text{-}\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\right)\text{ = }\left({\text{a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}\right)\left(\text{u - }\left({\text{a}}_{\text{1}}{\text{u + a}}_{\text{2}}{\text{u}}^{\text{2}}\right)\text{ + }\frac{{\left({\text{a}}_{\text{1}}\text{u}\right)}^{\text{2}}}{\text{2}}\right)$

${\text{- a}}_{\text{1}}{\text{u - a}}_{\text{2}}{\text{u}}^{\text{2}}\text{ = }\left({\text{a}}_{\text{1}}{\text{ + 2 a}}_{\text{2}}\text{u}\right)\left({\text{u - a}}_{\text{1}}{\text{u - a}}_{\text{2}}{\text{u}}^{\text{2}}\text{ + }\frac{{\text{a}}_{\text{1}}{}^2{\text{u}}^2}{\text{2}}\right)$

${\text{- a}}_{\text{1}}{\text{u - a}}_{\text{2}}{\text{u}}^{\text{2}}{\text{ = a}}_{\text{1}}{\text{u - a}}_{\text{1}}{}^{\text{2}}{\text{u - a}}_{\text{1}}{\text{a}}_{\text{2}}{\text{u}}^{\text{2}}\text{+ }\frac{{\text{a}}_{\text{1}}{}^{\text{3}}{\text{u}}^{\text{2}}}{\text{2}}{\text{ + 2 a}}_{\text{2}}{\text{u}}^{\text{2}}{\text{- 2 a}}_{\text{1}}{\text{a}}_{\text{2}}{\text{u}}^2{\text{ - 2 a}}_{\text{2}}{}^{\text{2}}{\text{u}}^{\text{3}}{\text{ + a}}_{\text{1}}{}^{\text{2}}{\text{a}}_{\text{2}}{\text{u}}^{\text{3}}\mathrm{..................}\text{ (3)}$

Dividing both sides by $\text{u}$,

${\text{- a}}_{\text{1}}{\text{ - a}}_{\text{2}}{\text{u = a}}_{\text{1}}{\text{ - a}}_{\text{1}}{}^{\text{2}}{\text{ - a}}_{\text{1}}{\text{a}}_{\text{2}}\text{u+ }\frac{{\text{a}}_{\text{1}}{}^{\text{3}}\text{u}}{\text{2}}{\text{ + 2 a}}_{\text{2}}\text{u}{\text{- 2 a}}_{\text{1}}{\text{a}}_{\text{2}}{\text{u - 2 a}}_{\text{2}}{}^{\text{2}}{\text{u}}^2{\text{ + a}}_{\text{1}}{}^{\text{2}}{\text{a}}_{\text{2}}{\text{u}}^2$

Simplifying it further,

${\text{- 2 a}}_{\text{1}}{\text{ + a}}_{\text{1}}{}^{\text{2}}{\text{ = 3 a}}_{\text{2}}{\text{u - 3 a}}_{\text{1}}{\text{a}}_{\text{2}}\text{u + }\frac{{\text{a}}_{\text{1}}{}^{\text{3}}\text{u}}{\text{2}}{\text{ - 2 a}}_{\text{2}}{}^{\text{2}}{\text{u}}^2{\text{ + a}}_{\text{1}}{}^{\text{2}}{\text{a}}_{\text{2}}{\text{u}}^2$

${\text{- 2 a}}_{\text{1}}{\text{ + a}}_{\text{1}}{}^{\text{2}}\text{ = u }\left({\text{3 a}}_{\text{2}}{\text{ - 3 a}}_{\text{1}}{\text{a}}_{\text{2}}\text{ + }\frac{{\text{a}}_{\text{1}}{}^{\text{3}}}{\text{2}}{\text{ - 2 a}}_{\text{2}}{}^{\text{2}}{\text{u + a}}_{\text{1}}{}^{\text{2}}{\text{a}}_{\text{2}}\text{u}\right)$

To solve for equilibrium point, substituting $\text{u = 0}$,

${\text{- 2 a}}_{\text{1}}{\text{ + a}}_{\text{1}}{}^{\text{2}}\text{ = 0}$

${\text{a}}_{\text{1}}\left({\text{a}}_{\text{1}}\text{- 2}\right)\text{ = 0}$

${\text{a}}_{\text{1}}\text{ = 0, 2}$

Since ${\text{a}}_{\text{1}}\text{ = 0}$ gives the unstable manifold, considering ${\text{a}}_{\text{1}}\text{ = 2}$.

Comparing the coefficients of ${\text{u}}^{\text{2}}$ from equation $\left(\text{3}\right)$,

${\text{- a}}_{\text{2}}{\text{ = - a}}_{\text{1}}{\text{a}}_{\text{2}}\text{+ }\frac{{\text{a}}_{\text{1}}{}^{\text{3}}}{\text{2}}{\text{ + 2 a}}_{\text{2}}{\text{- 2 a}}_{\text{1}}{\text{a}}_{\text{2}}$

Substituting ${\text{a}}_{\text{1}}\text{= 2}$,

${\text{- a}}_{\text{2}}{\text{ = - 2a}}_{\text{2}}\text{+ }\frac{{\left(\text{2}\right)}^{\text{3}}}{\text{2}}{\text{ + 2 a}}_{\text{2}}\text{- 2}\left(\text{2}\right){\text{a}}_{\text{2}}$

${\text{- a}}_{\text{2}}{\text{ = 4 - 4a}}_{\text{2}}$

${\text{3a}}_{\text{2}}\text{ = 4}$

${\text{a}}_{\text{2}}\text{ = }\frac{\text{4}}{3}$

Substituting ${\text{a}}_{\text{1}}\text{= 2}$ and ${\text{a}}_{\text{2}}\text{ = }\frac{\text{4}}{3}$ in the equation for stable manifold,

$\text{y = 2u +}\left(\frac{\text{4}}{3}\right){\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)$

Thus, the equation for stable manifold for the given system is $\text{y = 2u +}\left(\frac{\text{4}}{3}\right){\text{u}}^{\text{2}}\text{+ O}\left({\text{u}}^{\text{3}}\right)$.

b) The graph of the stable manifold on the phase portrait is shown below.

From the above graph, it is clear that with the same shape as the stable manifold shown in Figure $\text{6}\text{.1}\text{.4}$ the analytical result produces a curve.