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Suppose, I have a system of simultaneous nonlinear differential equations as follows:

x'=x-x^3-x*y^2-a*x+h
y'=y-y^3-y*x^2+a*y

where, x & y are variable ; x' & y' represents time derivative of x and y ; a & h are parameters.

Now to solve this, one can use DSolve or by taking x'=0 & y'=0 (for fixed point calculation) , i.e. take the polynomial and by using FindRoot or Ruduce or other options.

My quarry is that what is the easiest way to get fixed points (solutions) in terms of a & h .

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  • $\begingroup$ Could use In[14]:= Solve[{x - x^3 - x*y^2 - a*x + h, y - y^3 - y*x^2 + a*y} == 0, {x, y}, Cubics -> False] to get parametrized solutions, some using Root objects. They evaluate numerically when given values for the parameters. $\endgroup$
    – Daniel Lichtblau
    Oct 22 '17 at 17:02
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In this case, you can obtain the fixed points with Solve:

Solve[{x - x^3 - x*y^2 - a*x + h == 0, y - y^3 - y*x^2 + a*y == 0}, {x, y}]

However, this will not work in general.

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  • $\begingroup$ then for general purpose what command should I use @Henrik $\endgroup$
    – Bapi Saha
    Oct 22 '17 at 17:27
  • $\begingroup$ FindRoot. But you can use a long-time solution from NDSolve as starting value. $\endgroup$
    – Henrik Schumacher
    Oct 22 '17 at 17:33

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