Determine the stability of (x,y)=(0,0): (1) { x ˙ = − 2 x − y + 2 x y 2 − 3 x 3 y ˙ = x 3 − y − x 2 y − 7 y 3 (2) { x ˙ = x − x y 4 y ˙ = y − x 2 y 3

Question

Determine the stability of (x,y)=(0,0):                    (1)                                    {                                                                                                                                  x                  ˙                                =                −                2                x                −                y                +                2                x                                  y                  2                                −                3                                  x                  3                                                                                                                                                                  y                  ˙                                =                                                      x                    3                                                  −                y                −                                  x                  2                                y                −                7                                  y                  3                                                                                                      (2)                                    {                                                                                                                                  x                  ˙                                =                x                −                x                                  y                  4                                                                                                                                                                  y                  ˙                                =                y                −                                  x                  2                                                  y                  3

Taniya Wood · Accepted Answer

For system of equations:                    (2)                                    {                                                                                                                                  x                  ˙                                =                x                −                x                                  y                  4                                                                                                                                                                  y                  ˙                                =                y                −                                  x                  2                                                  y                  3                                                                                  We use the Chetaev theorem to prove this. Let the function   V  (  x  ,  y  ) have the form  V  (  x  ,  y  )  =      x    2    −      y    2  This function is positive definite in the subdomain   V in which the inequality   |  x  |  &amp;gt;  |  y  | holds.We calculate the derivative                     d        V                    d        t             by virtue of the system and define its sign in the subdomain   V.      V    ˙    (  x  ,  y  )  =  2  (      x    2    −      y    2    )  =  2  V  (  x  ,  y  )- It can be seen that the derivative                     d        V                    d        t             is also positive definite in the subdomain   V defined by the relation   |  x  |  &amp;gt;  |  y  |.Besides that, the function   V  (  x  ,  y  ) is zero on the boundary of   V including the point   (  0  ,  0  ).So, all conditions of the Chetaev theorem are fulfilled. Therefore, the zero solution of the system is unstable.

Nubydayclellaumvcd · Accepted Answer

Great expert answer!

Determine the stability of (x,y)=(0,0): <mtable displaystyle="true"> <mlabeledtr> <

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