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Damped rational double-barrier phase portrait

For

\overset{x}{¨} + k \overset{x}{˙} + \frac{2 x ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{3}} = 0,

the mechanical energy is

E (x, y) = \frac{1}{2} y^{2} + \frac{x ^{2}}{( 1 + x ^{2} ) ^{2}}, \dot{E} = - k y^{2} .

The origin is a center for

k = 0

and an asymptotically stable equilibrium for

k > 0

, while

(\pm 1, 0)

are saddles. For positive damping, their stable manifolds form the boundary between the basin of the origin and the two escape regions.

Table of contents 86 1
- Outward escape from the rational potential barrier Damped rational double-barrier phase portrait 34

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