I have an Euler method that has this form: I ^ ( t n + 1 ) = I ^ ( t n ) + h β I ^ ( t n ) [ 1 − I ^ ( t n ) N ]which can also be written like I ^ ( t n + 1 ) = ϕ ( I ^ ( t n ) )where ϕ ( x ) is the iteration function down below: ϕ ( x ) = x + h β x ( 1 − x N )I use h = 6 in this method but if I use a h which is a little bit bigger (for example h = 20), I have an absolute instability error. I want to find the value of h from which this absolute error is shown?

Question

I have an Euler method that has this form:            I      ^        (      t          n      +      1        )  =            I      ^        (      t          n        )  +  h  β            I      ^        (      t          n        )  [  1  −                              I          ^                    (              t                  n                    )        N    ]which can also be written like            I      ^        (      t          n      +      1        )  =  ϕ  (            I      ^        (      t          n        )  )where   ϕ  (  x  ) is the iteration function down below:  ϕ  (  x  )  =  x  +  h  β  x  (  1  −      x    N    )I use   h  =  6 in this method but if I use a   h which is a little bit bigger (for example   h  =  20), I have an absolute instability error. I want to find the value of   h from which this absolute error is shown?

sellk9o · Accepted Answer

Divide the equation by   N to get a new equation in   x  =  I      /    N that does not contain   N. Replace   k  =  h  β, so that what remains is the more simple equation      x          n      +      1        =      x    n    +  k      x    n    (  1  −      x    n    )  =      x    n    (  1  +  k  −  k      x    n    )Now finally divide by   (  1  +  k  ) and set   y  =  k  x      /    (  1  +  k  ) to get the discrete logistic map      y          n      +      1        =  (  1  +  k  )      y    n    (  1  −      y    n    )You can read off the stability of that iteration from any plot of the Feigenbaum diagram.In summary, for   k  ∈  [  0  ,  2  ] you get convergent behavior, after that periodic solutions, looking increasingly chaotic after   k  =  2.82..

vedentst9i · Accepted Answer

If   ϕ  (  x  ) is a contraction in      x          k      +      1        =  ϕ  (      x    k    )then following with      x          k      +      1        −      x    k    =  ϕ  (      x    k    )  −  ϕ  (      x          k      −      1        )  =      ϕ    ′    (  ξ  )  (      x    k    −      x          k      −      1        )to have a fixed point is sufficient that       |        ϕ    ′    (  ξ  )      |    &amp;lt;  1 or  −  1  &amp;lt;  1  −  h            β              x        k              N    +  h  β      (    1    −                  x        k            N        )    &amp;lt;  1

I have an Euler method that has this form: hat I(t_(n+1))=hat I(t_n)+h beta hat I(t_n)[1−hat I(t_n)/N]

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