Science:Math Exam Resources/Courses/MATH103/April 2013/Question 05 (a)/Solution 1

The steady states are given by solving

$\displaystyle x=4x^{3}-12x^{2}+8x$

Subtracting x from both sides yields

$\displaystyle 0=4x^{3}-12x^{2}+7x=x(4x^{2}-12x+7)$

Using the quadratic formula gives

$x={\frac {12\pm {\sqrt {144-4(4)(7)}}}{8}}={\frac {12\pm {\sqrt {32}}}{8}}={\frac {3\pm {\sqrt {2}}}{2}}$

Analytically to check stability, we would need to take the derivative of our function and then plug in the fixed points and check that the absolute value of the derivatives in the fixed points is less than 1 (this would show that the fixed point is stable). It's easier to notice that graphically, none of the fixed points are the limit of an iterative sequence and so none of the fixed points are stable.