A plane autonomous system where there exists two ODEs, d x d t = X ( x , y ) , d y d t = Y ( x , y ). We then usually draw trajectories on the phase plane to indicate the solutions on the plane.If a trajectory does contain a critical point, wouldn't this trajectory just stay at that critical point rather than passing through that critical point since at that point X = Y = 0 and so just becomes stationary?

Question

A plane autonomous system where there exists two ODEs,             d      x              d      t        =  X  (  x  ,  y  )  ,            d      y              d      t        =  Y  (  x  ,  y  ). We then usually draw trajectories on the phase plane to indicate the solutions on the plane.If a trajectory does contain a critical point, wouldn&#039;t this trajectory just stay at that critical point rather than passing through that critical point since at that point   X  =  Y  =  0 and so just becomes stationary?

kuthiwenihca · Accepted Answer

The simplest example is                    d        x                    d        t              =  x  ,                      d        y                    d        t              =  −  y     .When       r    (  0  )  =  (      x    0    ,  0  ),         x    0    ≠  0, then      r    (  t  )  =  (  0  ,      y    0        e          −      t        )    (  −  ∞  &amp;lt;  t  &amp;lt;  ∞  )     ,hence the half   y-axis containing       y    0   is a solution curve. But note that this curve reaches   (  0  ,  0  ) only in the limit   t  →  ∞. This means that the traveling point needs infinite time to reach the origin. And of course the initial condition   t  →  ∞ leads to       r    (  t  )  =  (  0  ,  0  ) for all       r    (  t  )  =  (  0  ,  0  ).Therefore no trajectory &quot;traverses&quot; the stationary point   (  0  ,  0  ).

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