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Get a simple form of an equilibrium point of a nonlinear ODEs system?

Posted 15 days ago
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Dear all, I am a beginner user of Mathematica Software. I used nonlinear ODEs system and get enumerate result how to put the results in a simple form.

f[T_, I1_] = \[Alpha]1 T (1 - \[Alpha]2 T) - \[Alpha]4 T I1;
g[T_, I1_] = \[Sigma] - \[Delta] I1 - \[Mu]2 T I1 + (\[Rho]2 T I1)/(
   m2 + T) ;

s = Solve[{f[T, I1] == 0, g[T, I1] == 0}, {T, 
   I1}]; Simplify[s, {m2 > 0, \[Alpha]1 > 0, \[Alpha]2 > 
   0, \[Alpha]4 > 0, \[Delta] > 0, \[Mu]2 > 0, \[Rho]2 > 0, \[Sigma] >
    0}]

the result is still complicated and I can't determine the sign of points. The points should to be positive and reals.

However, I want to do a simulation of the model by Mathematica. can do it?

6 Replies

You can display the solutions in the complex plane:

With[{s = s},
 Manipulate[
  Graphics[{PointSize[Large], Point[ReIm[T]], Red, 
     Point[ReIm[I1]]} /. s,
   Frame -> True],
  {{m2, 1/2}, 0, 1},
  {{\[Alpha]1, 1/2}, 0, 1},
  {{\[Alpha]2, 1/2}, 0, 1},
  {{\[Alpha]4, 1/2}, 0, 1},
  {{\[Delta], 1/2}, 0, 1},
  {{\[Mu]2, 1/2}, 0, 1},
  {{\[Rho]2, 1/2}, 0, 1},
  {{\[Sigma], 1/2}, 0, 1}]]
Posted 14 days ago

how this code is help me to solve my problem.

I want to find the simplest result of the steady-state. if you do run the model you can see how it is enormous. this is my problem.

You have 4 solutions. You need to choose the ones that are real and positive. This plot visualizes the solutions in the complex plane, so that you can choose:

With[{s = s}, 
 Manipulate[
  Graphics[{PointSize[
     Large], {Dashed, Point[ReIm[T]], Text["T", ReIm[T], {0, -2}], 
      Red, Point[ReIm[I1]], Text["I1", ReIm[I1], {0, -2}]} /. s[[i]]},
    Frame -> True, PlotRangePadding -> Scaled[.1]],
  {i, {1, 2, 3, 4}}, {{m2, 1/2}, 0, 1},
  {{\[Alpha]1, 1/2}, 0, 1}, {{\[Alpha]2, 1/2}, 0, 1},
  {{\[Alpha]4, 1/2}, 0, 1}, {{\[Delta], 1/2}, 0, 1},
  {{\[Mu]2, 1/2}, 0, 1}, {{\[Rho]2, 1/2}, 0, 1},
  {{\[Sigma], 1/2}, 0, 1}]]

Solution number 1 has T=0, is that acceptable?

Posted 13 days ago

yes on of them is T=0, there is two points are complex the fourth is real but i can't determine its sign.

It is unclear to me what is wanted. First, what is the system of ODE's? Second, what sort of solution is expected? Of necessity it will depend on parameters, and even the number of real solutions might change based on parameter values.

Posted 14 days ago

the points should to be positive and reals. any point can't writ as positive and real should to delete from study of the stability.

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