3 August, 2026

Notes

2026-10-30

To do!

• find myself back into my
• look into the two “electrons each in boxes” 1D version
• write about the classification on periodic orbits in theory
• finish implementing the self written code for finding periodic orbits
• read and work toward @HenonHeiles_toBifrication_periodic_times (see downloads)
• monodromy matrix

Henon Heiles

on finding periodic orbits

• using @pal2024nonlinearsolve I can use a julia precoded from NonlinearSolve.jl
• also I wrote my own code on finding periodic orbits, the classification seemd important to understand what happend
• I was findng unstable points, though why? – I dont know
• symplectic dynamical systems \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})\) have \(\Delta=det(\mathbf{Df}|_{x=x_0})=1\), such that their phase volume does not change

Two electrons

I can see the following trajectories, and phase map as well as offset of the total energy of the system. I tried to fix it using a projection method that projects the state onto the mannifold defined by \(H(\mathbb{x})=E\). The integration method is a Runge-Kutta order 9.

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), \(T=10^5\).

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), T=\(10^6\).

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), T=\(10^6\). Further option Vern9(lazy=Val{false}())

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), T=\(10^6\) with projection onto manningfold.

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), T=\(10^6\) with projection onto manningfold and further option Vern9(lazy=Val{false}()).

The option setting lazy integration to false should be used “If p is changed in a ContinuousCallback, or in a DiscreteCallback and the continuous solution is used after the full solution, then set lazy=Val{false}().”.

I tried to methods, that predict the time of collision with the walls

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), T=\(10^5\)

and used the leap frog method:

\(x_1=-1.5\), \(x_2= 1.3\), \(p_1=0.4\), \(p_2=-0.2\), T=\(10^5\)

The takeaway is: The precision is high enough, what matters more is the computation time. I should aim to get a qualtitative understanding of the dynamics capable of my system.

Poincarés reoccurence theorem

We want to make a statment on chaotic behaviour. It takes up from measuretheory. Meaning we have a measureable set \(\mathcal{C}\) equipped with a measure \(\mu:\mathcal{C}\rightarrow \mathbb{R}\). Then we would like to describe the concept of maps \(T:\mathcal{C}\rightarrow \mathcal{C}\) such that \[\mu(T\mathcal{C})=\mu(\mathcal{C})\] which captures the concept of an invariant measure under transformation induced by the map \(T\).

The goal is to try to understand the set that describes the notion of escaping the measurable set eventually \[\mathcal{C}_0=\{x\in\mathcal{C}:T^nx \not \in\mathcal{C}| n\in\N\}.\]

We hope to get an understanding by defining a set that are in the measurable set \(\mathcal{C}\) at first, but \[D_n=\{\}\] ### Lost Wonderers Thoughts: