# Poincaré Section from action

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!

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}
] -
Table[
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};
Reap[NDSolve[
{Splice[equationsOfMotion],
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:

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


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?

• 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. Mar 30 at 5:16
• I am glad it helped. Let me know if you get better pictures! Mar 30 at 16:02
• thank you once again. I have one question. How to view the closeup of the above plot, as given in the paper, indicating chaos? Apr 6 at 5:33
• 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? Apr 6 at 8:59
• 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. Apr 6 at 17:49