Question related to the ODEs Stability

When does an ODE become unstable?

Given an ODE, y=f(x). A fixed point is a point where y=0. This requires f(x)=0. So any roots of the function f(x) is a fixed point. A fixed point is stable if, roughly, it taking an initial value that is ‘close’ to the fixed point the trajectory of the solution, under the ODE, will always stay ‘close’ to the fixed point. The fixed point is unstable if, roughly, for any ‘nearer’ to the fixed point, the solution that starts near the fixed point but becomes far from the fixed point in finite time. To be even more, a fixed point is stable if the equation of motion y=f(x) forces a particle to move toward the fixed point if it starts close to the fixed point, and that a fixed point is unstable if the equation of motion forces a particle to move away from the fixed point.

Link: http://people.uleth.ca/~roussel/nld/stability.pdf

How does stability affect engineering simulations?

Engineering simulations are depending on the input boundary conditions and simulating parameters. Ex, Atmospheric stability is a measure of the atmosphere’s tendency to promote or resist vertical mixing. This stability depends on thermal buoyancy effects and thus on the rate of change of temperature with height above the surface.

Can stability condition be derived for all types of problems?

Every problem that we try to solve is based on an expression of some form or another. If it well conditioned, then small changes in the input to the expression will lead to small changes in the results. If small changes in the input lead to large changes in the output, then we call the problem ill-conditioned. The exact cutoff between a well and ill-conditioned depends on the context of the problem and the uses of the results.