# Global stability in a ramsey growth model with multiple equilibria

Let's say that we have a model with multiple equilibria. $$c$$ refers to consumption and $$S$$ refers to natural capital stock.
The dynamics that describe the economy are as follows:

$$\dot{S}\left(t\right)=R\left(S\left(t\right)\right)-c\left(t\right)$$

where $$R(S)$$ is the function for environmental regeneration.

$$\dot{c}=-\frac{u_{c}\left(c\right)}{u_{cc}\left(c\right)}\left[R_{S}\left(S\right)-\rho-\frac{\bar{\psi}h_{S}\left(S\right)}{u_{c}\left(c\right)}\right]$$

is the Keynes-Ramsey equation.

My question is simple. There is always a discussion about the fact that there may be only a unique optimal path to a fixed point among the other equilibria. However, in this phase diagram, when I take into account the directions of arrows, I think there is never a global optimum since for all initial condition $$S(0), the economy either diverges or converges to $$S_{low}$$. Also, for all $$S_{0}>S_{mid}$$, the economy either diverges or converges to $$S_{high}$$.

Then, is it possible to prove the non-existence of global stability in this manner? or is there any formal (and more rigorous) way to prove the non-existence of global stability? Any hint or explanation is appreciated. Thanks in advance.