Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, \text{ and } a_3 > 0 $$ (a) Show that this equation has three equilibrium points (2a) $$ A(t) \equiv A_e \text{ such that } f(A_e) = 0 $$ satisfying either (2b) $$ (i)\ \ A_e = 0 \text{ or } (ii,iii)\ \ 2a_3 A^2_e = \pm\sqrt{a_1^2 + 4a_3\sigma} - a_1 > 0 $$ and find the $\sigma$-range over which each of them exists.
(b) Seeking a solution of (1) of the form (3a) $$ A(t) = A_e + \epsilon_1A_1(t) + O(\epsilon_1^2) \text{ where } |\epsilon_1|<<1 $$ and employing the two-term Taylor polynomial expansion (3b) $$ f(A) = f(A_e) + f'(A_e)(A-A_e) + O(A-A_e)^2 $$ obtain (3c) $$ \frac{dA_1}{dt} = f'(A_e)A_1 $$
(c) Particularizing the result of (3c) sequentially to the critical points of (2b), show that (i) is linearly stable for $\sigma < 0$ and (ii) & (iii) are linearly stable and unstable, respectively, fr the $\sigma$-range over which they are defined.
(d) Determine the global stability behavior of these equilibrium points upon multiplying (1) by $A(t)$, rewriting the resulting equation in the form $(1/2)(dA^2/dt) = F(A^2)$, and plotting $F$ versus $A^2$ for $\sigma < \sigma_{-1} = -a_1^2/(4a_3)$, $\sigma_{-1}<\sigma < 0$, and $\sigma > 0$, respectively.
(e) Defining $\epsilon^2 = |a_1|$, assuming $a_3 = O(1)$ as $\epsilon \to 0$, and considering $\sigma = O(\epsilon^4) > 0$, first demonstrate that $A_e^+ = O(\epsilon)$. Now introducing the rescaled variables $\tau = \sigma t$, $A'(\tau) = A(t)/A_e^+$ where $A', dA'/d\tau = O(1)$ as $\epsilon \to 0$ into the equation $dA/dt = f(A) + O(A^7)$, and taking the limit as $\epsilon \to 0$, deduce that (1) is a valid truncation for that equation.
I've solved parts (a) and (b), but have no idea where to start for the rest of them. Any ideas?