Are equilibrium values of a differential equation uniquely dependent on its constants?

I have a dynamic model which undergoes two distinct stages. The system starts out with certain initial conditions and once a specific point is reached in stage 1, these ending conditions are used as the initial conditions for stage 2, and the remainder of the solution is calculated. I am attempting to solve for the steady-state equilibrium point by taking the derivative of the potential energy with respect to each generalized coordinate, setting equal to zero, and solving. For stage 1:

$$
V_{im} = frac{1}{2}k_1y_1^2 + frac{1}{2}k_2(y_2-y_1)^2 + k_r theta_o^2+m_1gy_1
+m_2g(y_1+y_2)+2m_3g(y_1-l sin{theta_o}) +frac{1}{4}k_{1n}y_1^4+frac{1}{4}k_{2n}(y_2-y_1)^4
$$
and for stage 2:

$$
V_{de} = frac{1}{2}k_1y_1^2+frac{1}{2}k_2(y_2-y_1)^2+k_rtheta^2 + m_1gy_1 nonumber
+ m_2g(y_1+y_2) + 2m_3g(y_1-lsin{theta}) + frac{1}{4}k_{1n}y_1^4
+ frac{1}{4}k_{2n}(y_2-y_1)^4
$$
where $ theta_o$ was a constant while $ theta$ is variable. Still, these reveal the same two derivatives:

$$
frac{partial V}{partial y_1} = k_1y_1+k_2(y_1-y_2)+g(m_1+m_2+2m_3) +k_{1n}y_1^3-k_{2n}(y_2-y_1)^3=0
$$
and

$$
frac{partial V}{partial y_2} = k_2(y_2-y_1)+m_2g +k_{2n}(y_2-y_1)^3= 0
$$

Therefore, I would assume that both stages would have the same steady-state position, but this is not always the case once I run the entire simulation numerically in MATLAB (details not shown). Sometimes the steady-state of stage 1 matches the calculated value and not stage 2, and sometimes it is the opposite way around. What could be causing this discrepancy?

For comparison, stage 1 and stage 2 position graphs as described are below:

The values of the constants are the same throughout the simulations.