I am given a perturbative action $$\frac{S}{\mathcal{T}}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+7.11 c_0^3+35.3 c_0 c_1^2+4.66 c_0 \dot{c_0}{}^2+1.32 c_0 \dot{c_1}{}^2-7.57 \dot{c_0} c_1 \dot{c_1}$$ where $\omega _0^2=-1.4$ and $\omega _1^2=7.57$, by solving the time evolution based on the above action, we can examine if the system exhibits chaos or not by constructing a Poincaré Section.

How shall I construct such Poincaré Sections defined by $c_1( t)=0$ and $\dot{c_1} (t)>0$ for bound orbits with energy E=9.28 X 10^(-6) and 0<t<8000 ?

I have read this but don't understand how to apply it to my problem. The related paper from which, I am trying to reproduce the results is here(specifically on page 5, figure 4).

Any help in this regard would be truly beneficial!


1 Answer 1


I will give it another go. I understand the previous attempt had some flaws.

Let us obtain the equations of motion again.

\[Omega]sq[0] = -1.4; \[Omega]sq[1] = 7.57;
lagrangian = 
  Sum[c[n]'[t]^2 - c[n][t]^2 \[Omega]sq[n], {n, {0, 1}}] + 
   7.11 c[0][t]^3 + 35.3 c[0][t] c[1][t]^2 + 
   4.66 c[0][t] c[0]'[t]^2 + 1.32 c[0][t] c[1]'[t]^2 - 
   7.57 c[0]'[t] c[1][t] c[1]'[t];
eulerLagrange[lagrangian_, vars_, dvars_] :=
            D[D[lagrangian, dvar], t],
            {dvar, dvars}
            ] -
            D[lagrangian, var],
            {var, vars}
            ]) == ConstantArray[0, Length@vars]];
equationsOfMotion = 
 eulerLagrange[lagrangian, {c[0][t], c[1][t]}, {c[0]'[t], c[1]'[t]}]

And the key is that we have to simultaneously solve both equations of motion. As we do this we collect the values of $c_0, \dot{c}_0$ every time $c_1(t)=0$.

sol = Table[Block[{a, b, \[Chi], d},
        {a, b, \[Chi], d} = {-0.10, c\[Prime], -0.002, 0.002};
            c[0][0] == a, c[0]'[0] == b, c[1][0] == \[Chi], c[1]'[0] == d,
            WhenEvent[c[1][t] == 0,
                    Sow[{c[0][t], c[0]'[t]}]]},
            {c[0][t], c[1][t]},
            {t, 0, 8000}
    ], {c\[Prime], -0.1, 0.1, 0.01}];

Now we get better looking results:

        Table[Flatten[sol[[i]][[2]], 1], {i, Length@sol}],
        PlotTheme -> "Scientific"

enter image description here

Still, room for improvement. In particular I don't know what the initial conditions for the $c_1$ field are. Maybe if we could figure those out the plot would be closer to the one in the paper?

  • $\begingroup$ Thank you for this illuminating answer! It definitely helped me get started. I will try to use this code in my work and edit the question accordingly. $\endgroup$
    – codebpr
    Commented Mar 30, 2022 at 5:16
  • $\begingroup$ I am glad it helped. Let me know if you get better pictures! $\endgroup$
    – Diffycue
    Commented Mar 30, 2022 at 16:02
  • $\begingroup$ thank you once again. I have one question. How to view the closeup of the above plot, as given in the paper, indicating chaos? $\endgroup$
    – codebpr
    Commented Apr 6, 2022 at 5:33
  • $\begingroup$ Also, instead of giving the initial conditions, they have provided the energy of orbits as E=9.28*10^(-6). How to include that in the code? $\endgroup$
    – codebpr
    Commented Apr 6, 2022 at 8:59
  • $\begingroup$ To zoom in just change the plot range. For the energy you'll have to get a formula that expresses energy in terms of the fields then solve for the fields numerically. $\endgroup$
    – Diffycue
    Commented Apr 6, 2022 at 17:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.