Differential Equations: Problem Set 5

(2.7) 2, 4, 8, 10

For each of the following, find the equilibria, determine their stability and produce a bifurcation diagram. Include appropriate sentences summarizing the various cases.

1. $x^{\prime }=h-x^3$

2. $x^{\prime }=hx-x^2$

3. $x^{\prime }=(1-x)(x^2-h)$

4. $x^{\prime }=(x-\sqrt{h})(3-hx)$

5. $x^{\prime }=-(1+x)(x^2-h^2)$

6. $x^{\prime }=(x-1)(x-h)$

7. $x^{\prime }=x(1-x{)}^2+\alpha$