# Attempting to perturb ODE when initial condition is equilibrium point does not work

I have the following system of differential equations:

$$x' = ax- cy + e1$$

$$y' = by- dx + e2$$

for variables $$x,y$$ and parameters $$a,b,c,d,e1,e2$$.

I'd like to solve this in python, which is easy enough with odeint.

def f(y, t, params):
weaponsC1, weaponsC2 = y
a, b, c, d, e1, e2 = params
t_0 = 10
t_1 = 11
derivs = [a*x- c*y + e1,
b*y - d*x + e2]
return derivs

#solve
psoln = odeint(f, y0, t, args=(params,), rtol=1e-10)


This returns what you'd expect:

For

$$a = 1\\ b = 1 \\ c = 2 \\ d = 2 \\ e1 = 5 \\ e2 = 5$$

and initial condition: $$(2,2)$$, the code above returns:

Now I'd like to add a forcing function $$g$$ such that I can perturb the ODE by some constant amount at some time. After a bit of a struggle with neater methods, I settled on doing it like this:

def f(y, t, params):
weaponsC1, weaponsC2 = y
a, b, c, d, e1, e2 = params
t_0 = 10
t_1 = 11
derivs = [a*weaponsC1- c*weaponsC2 + e1, + (2*(t > t_0)*(t < t_0+0.5)) + (5*(t > t_1)*(t < t_1+0.5)),
b*weaponsC2 - d*weaponsC1 + e2 + (2*(t > t_0)*(t < t_0+0.5))+ (5*(t > t_1)*(t < t_1+0.5))]
return derivs


A graph of just the forcing function is shown, thought at different time value:

def g(t_0):
return((2*(t > t_0)*(t < t_0+0.5)))

plt.plot(t,(g(3) + g(4)))


when I plot the ODE with this forcing function, it works fine for any initial condition which is not the equilibrium:

However, when I set the equilibrium as the starting condition, try as I may, I can never impact the ODE:

this is the same code that made the above working solution; the only change is the starting condition.

I know ODEint assumes the diffeq is differentiable, and mine is not — could this be the cause of the issue? Or is this the result of some property of the ODE?