Literature review ODE Stability

1.When does a ODE become unstable?

if we increases the step size ODE equation becomes unstable

for example use this pythone code

import math
import matplotlib.pyplot as plt

def f(t,y):
	return -1000*y-math.exp(-t)

t_start=0
t_end=0.1
n=50

dt=(t_end-t_start)/(n-1)
t=0;
y_old=0;
y_new=[]
for i in range(0,n):
	y_new.append(y_old+dt*f(t,y_old))
	y_old=y_new[i]
	t=t+dt

plt.plot(y_new)
plt.show()

graph of small step size the equation becomes stable

n=500

graph of larger step size the equation becomes unstable

n=50

2.How does stability affect engineering simulations?

unstability increases the errors in the simulations thereby our results become diverge from the tolerance limits

3.Can stability condition be derived for all types of problems?

yes, stability conditions are derived for all ODE.there are two common stability conditions they are

a)eigen value stability condition

b)von neuman stability conditions