NdSolve::ndnum error and Polar Coordinates plot

Questions
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1

Hello,
I’m trying to solve the system of differential equations in polar coordinates and after that plot this solution.
The system is

system_mat.jpg
system_mat.jpg1300×271 46.1 KB

Here is the code

c = 100;
b = 1;
a = 0.99;
w = NDSolve[{r'[t] ==
    r[t]*(Cos[tet[t]]^2 + a*Sin[tet[t]]^2) - 
     c*r[t]^2*Sin[tet[t]]*Cos[tet[t]]*(Cos[tet[t]] - a*Sin[tet[t]]) - 
     r[t]^2*(Cos[tet[t]]^3 + a*b*Sin[tet[t]]^3), 
     tet'[t] == (a - 1)*Sin[tet[t]]*Cos[tet[t]] - 
     c*r[t]*Sin[tet[t]]*Cos[tet[t]]*(a*Cos[tet[t]] - Sin[tet[t]]) - 
     r[t]*Sin[tet[t]]*Cos[tet[t]]*(a*b*Sin[tet[t]] - Cos[tet[t]]), 
     r[0] == 0, tet[0] == 0}, {r[t], tet[t]}, {t, 0, 1000}]

How to plot this solution on polar plot?
I tried use ParametricPlot, PolarPlot and ListPolarPlot - no results

Thanks for help.

Best regards,
Danila

2

Built-in functions in Mathematica start with capital letters. Sin and Cos.

3

Ohh, thanks. My fault.

I tried to use PolarPlot as I wrote above, but no results.
Can you give some advice how to plot this solution?

4

Could you please first edit your post and correct syntax of your code?

5

Of course, sorry.
Post was edited.

6

With the given parameters and initial conditions, r and tet are always zero.

In[1]:= c = 100;
b = 1;
a = 0.99;
w = NDSolve[{r'[t] == 
    r[t]*(Cos[tet[t]]^2 + a*Sin[tet[t]]^2) - 
     c*r[t]^2*Sin[tet[t]]*Cos[tet[t]]*(Cos[tet[t]] - a*Sin[tet[t]]) - 
     r[t]^2*(Cos[tet[t]]^3 + a*b*Sin[tet[t]]^3), 
   tet'[t] == (a - 1)*Sin[tet[t]]*Cos[tet[t]] - 
     c*r[t]*Sin[tet[t]]*Cos[tet[t]]*(a*Cos[tet[t]] - Sin[tet[t]]) - 
     r[t]*Sin[tet[t]]*Cos[tet[t]]*(a*b*Sin[tet[t]] - Cos[tet[t]]), r[0] == 0, 
   tet[0] == 0}, {r[t], tet[t]}, {t, 0, 1000}]

Out[4]= {{r[t] -> InterpolatingFunction[][t], tet[t] -> InterpolatingFunction[][t]}}
 
        (* This samples only every 100th point.  All 1001 were zero.  *) 
In[5]:= Table[ Evaluate[{r[t], tet[t]} /. w[[1]]], {t, 0, 1000, 100}]

Out[5]= {{0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, 
   {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}}

     (*  r' and  tet'  are zero at the beginning, and it looks like they 
          never get the chance to grow.  *) 
In[6]:= Eliminate[{r'[t] == 
    r[t]*(Cos[tet[t]]^2 + a*Sin[tet[t]]^2) - 
     c*r[t]^2*Sin[tet[t]]*Cos[tet[t]]*(Cos[tet[t]] - a*Sin[tet[t]]) - 
     r[t]^2*(Cos[tet[t]]^3 + a*b*Sin[tet[t]]^3), 
   tet'[t] == (a - 1)*Sin[tet[t]]*Cos[tet[t]] - 
     c*r[t]*Sin[tet[t]]*Cos[tet[t]]*(a*Cos[tet[t]] - Sin[tet[t]]) - 
     r[t]*Sin[tet[t]]*Cos[tet[t]]*(a*b*Sin[tet[t]] - Cos[tet[t]]), r[0] == 0, 
   tet[0] == 0} /. t -> 0, {r, tet}]

Out[6]= r[0] == 0. && tet[0] == 0. && r'[0] == 0. && 
   100. tet'[0] == -1. Cos[tet[0]] Sin[tet[0]]

In[7]:= FullSimplify[%]
Out[7]= r[0] == 0 && tet[0] == 0 && r'[0] == 0 && tet'[0] == 0
7

Thank you, yes, I found some mistakes in math model.