Hi! I’m trying a simulate an orbit in the chaotic 3 body dynamics, and perform two different simulations: one for full time (0, nT) where n is an integer and T is the time period of the orbit. On the other hand, I also perform it by breaking it into chunks: (0,\delta t), (\delta t, 2\delta t) and so on until it reaches nT. The final solution from the previous step is the initial condition for the next step. Theoretically I assume this should mean similar solution (except maybe some numerical inconsistencies.) However, the results are very different for long simulations: such as 50 orbits. I understand the dynamics I’m simulating are chaotic, but I need to work with the second way: integrating by chunks and I’m not sure if that is the correct way here. I’ve attached the code below:

```
using DifferentialEquations, LinearAlgebra
const lstar = 238529; # in kms
const tstar = 18913; # in seconds
const μ = 1.901109735892602e-7;
# ODE function
function cr3bp_propagator!(dxdt,x,p,t)
d=sqrt((x[1]+μ)^2 + x[2]^2+ x[3]^2);
r=sqrt((x[1]-1+μ)^2 + x[2]^2 + x[3]^2);
dxdt[1]= x[4];
dxdt[2]= x[5];
dxdt[3]= x[6];
dxdt[4]= 2*x[5] + x[1] - (1-μ)*(x[1]+μ)/(d^3) - μ*(x[1]-1+μ)/(r^3);
dxdt[5]= -2*x[4] +x[2] - (1-μ)*(x[2])/(d^3) - μ*(x[2])/(r^3);
dxdt[6]= -(1-μ)*(x[3])/(d^3) - μ*(x[3])/(r^3);
end
function compare_full_chunks(N,nT)
nrho = [ 1.0025067861477979
0
4.8938916019621225e-3
0
-5.3520383247404298e-3
0 ];
x0 = -[50/lstar;0/lstar;50/lstar;0*tstar/lstar;0*tstar/lstar;1e-6*tstar/lstar];
TP = 2.2460348880483676;
δt= TP/N; t=0;
ic = nrho+x0;
traj=[]; time=[];tol=1e-12;
# solving in full at once
probref=ODEProblem(cr3bp_propagator!,nrho,(0.0,nT*TP));
solref=solve(probref,TsitPap8(),reltol=1e-12,abstol=1e-12);
# solving in chunks of δt time steps
for i in 1:nT*N
tspan = (t,t+δt);
prob=ODEProblem(cr3bp_propagator!,ic,tspan);
sol =solve(prob,TsitPap8(),reltol=tol,abstol=tol);
append!(traj,sol.u);
append!(time,sol.t);
ic = sol.u[end];
t+=δt;
end
return solref.u[end]-traj[end]
end
```

When I call` compare_full_chunks(50,50)`

, the returned value which is the difference between solution end points for the two methods is not close to zero and I’m confused why that is the case. Thanks for your help!