Classifying Fixed Points in a Second-Order Differential Equation

[coverband]

Homework Statement

Write the second-order differential equation
$$\ddot{x} + 2\epsilon \dot{x} + sin x =0,\epsilon \geq 0,$$
as a pair of coupled first-order equations.Find all its fixedpoints, and determine
how the classification of these fixed points changes with $$\epsilon$$

Homework Equations

The Attempt at a Solution

Let y = dx/dt … (1)
Original equation becomes
$$\dot{y}+ 2\epsilon y + sin x =0$$ ... (2)

Fixed points occur at (0,0) and $$(n\pi,0)$$

Just the last bit: determine how the classification of these fixed points changes with $$\epsilon$$ ...
The way I've done it, it looks like epsilon has no bearing on classification of fixed points

 

[lippyka]

, since they are always at (0,0) and (n\pi,0) regardless of epsilon. Is this correct? Or am I missing something?