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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.

enter image description here

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 $S(0)<S_{mid}$, 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.

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