## Q. 1.10.2

Calculate the equilibrium position for the nonlinear system defined by $\ddot{x}$ + x – β²x³ = 0, or in state equation form, letting $x_{1}$ = x as before,

$\dot{x}_{1}$ = $x_{2}$

$\dot{x}_{2}$ = $x_{1}$(β²$x²_{1}$ – 1)

## Verified Solution

These equations represent the vibration of a “soft spring” and correspond to an approximation of the pendulum problem of Example 1.4.2, where sin x ≈ x – x³/6. The equations for the equilibrium position are

$x_{2}$ = 0

$x_{1}(β²x²_{1} – 1)$ = 0

There are three solutions to this set of algebraic equations corresponding to the three equilibrium positions of the soft spring. They are

$x_{e}$ = $\left[\begin{matrix} 0 \\ 0 \end{matrix}\right] , \left[\begin{matrix} \frac{1}{β} \\ 0 \end{matrix}\right] , \left[\begin{matrix} -\frac{1}{β} \\ 0 \end{matrix}\right]$