Stability analysis for ODEs with non constant inputs { x ˙ = I ( t ) x + x 2 y ˙ = x where I ( t ) is a random input (for example).

Question

Stability analysis for ODEs with non constant inputs      {                                                      x              ˙                                =          I          (          t          )          x          +                      x            2                                                                          y              ˙                                =          x                        where   I  (  t  ) is a random input (for example).

Annie Wells · Accepted Answer

In the case where   I  (  t  ) is a noisy process, this would be modelled as a stochastic differential equation.Some observations:  x  ,  y  decouple; So there won&#039;t be any periodic orbits in   (  x  ,  y  ).Naively looking at the fixed points of this problem; we have             x      ˙        =  0 which implies   x  =  0 or   x  =  −  I  (  t  ). If we look at a ball about the   x  =  0 solution then your stability is tied to whether   I is positive or negative. When we linearise about the   x  =  −  I  (  t  ) solution, we find that the stability of this is exactly the opposite of the   x  =  0 solution.Essentially,   x with jump from   x  =  0 to   x  =  I  (  t  ) when   I  &amp;gt;  0 and back   x  =  0 when   I  &amp;lt;  0.   y will act like a kind of accumulator and not have any fixed points unless   I  &amp;lt;  0 for all time.

Iyana Jackson · Accepted Answer

They are called non-autonomous differential equations, or non-autonomous dynamic systems, or also time-varying (since &quot;t&quot; explicitly appears in the righthand side) differential equations. Lyapunov stability theory might apply to specific cases to the best of my knowledge.

Stability analysis for ODEs with non constant inputs {x=I(t)x+x^2, y=x

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