Chaos
A modular knowledge base
Glossary
| Term | Definition |
|---|---|
| homeomorphism | A continuous one-to-one map \(f:A\righarow B\) meaning f is injective and surjective, thus \(f\) is bijective and \(f^{-1}\) exists. |
| diffeomorphism | A continuously differentiable map \(f: N \to M\) between two manifolds, such that \(f^{-1}: M \rightarrow N\) is also differentiable. If \(f\) is \(r\)-times differentiable it is called a \(C^r\)-diffeomorphism. |
| manifold | A topological space that is locally homeomorphic to Euclidean space. A manifold can be equipped with additional structure, such as a differentiable structure, which allows for the definition of smooth functions and calculus on the manifold. |
| stable manifold | A stable manifold is a set of points in the phase space of a dynamical system that asymptotically approach a stable equilibrium point as time goes to infinity. It represents the long-term behavior of trajectories that start near the equilibrium. |
| unstable manifold | An unstable manifold is a set of points in the phase space of a dynamical system that approach a steady state solution of a stable orbit as one follows the orbit backward in time to minus-infinity. Vice-versa, it represents the long-term behavior of trajectories that start near the equilibrium a diverge with increasing time. |
| hyper surfaces | Coming from the analogy of a surface in 3D space is a 2D object. An geometric object that lives in a \(n\)–dimensional space, though itself is a \(n-1\)–dimensinal object. |
| isolating integral | |
| non isolating integral | |
| dynamical system | Or sometimes called a Flow, is a set of \(N\) first order differential equations in \(N\) variables $=(x_1, x_2, , x_N) which solve the first order differential equation \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}_0,t)\) with \(\mathbf{f}\in \phi_t:\mathbb{R}^N \rightarrow \mathbb{R}^N\). |
| inital value problem | |
| trajectory | A mathematical 1 dimensional object, that describes a dynamics of a initial point of phase space. |
| ergodic trajectory | A trajectory that will fill th whole volume of the accessible phase space. Thus \(\textbf{x}(\textbf{x}_0,t_1)=\textbf{x}(\textbf{x}_0,t_2) \Rightarrow t_1=t_2\). |
| Hamiltonian Flow | |
| integrable | |
| critical point | A point in state-space, for which a continuous transition between two states exists. |
| Hamiltonian | The total energy function of a system, made of the sum of the kinetic and potential energy. |
| kinetic energy | Is the energy that is in the form of motion of a massive objecmants. It is defined by \(\hat{T}=\sum_i\tfrac{p_i^2}{2m}\) for all particles labeled \(i\) |
| potential energy | Is the energy that is stored in the interaction that the particles obey. It carries the potential to transform it into kinetic energy i.e. movement. |
| surface of section | A surface cutting through the phase space. This registers the trajectory that passes through it. This can have constraints i.e. in a 4D phase space \(x=0\) and \(p_x\geq 0\)- |
| configuration space | The space of all possible positions \(\vec{x}_i \in \mathbb{R}^3\), for all \(N\) particles \(1\leq i \leq N\). I am wondering, what about time? |
| phase space | The space of all possible dynamical degrees of freedom \(N\in\mathbb{N}\), a point of this space defined by a vector \(\mathbf{x}\in\mathbb{R}^N\). For a Hamiltonian system, these correspond to canonical variables \((\textbf{p},\textbf{q})\) with \(\{q_i | 1\leq i \leq N\} = \textbf{q}\) and momentum \(\{p_i | 1\leq i \leq N\} = \textbf{p}\). |
| state | Describes a system dialectically. One state can become another. Often ‘state’ can be used to describe synonymously to ‘mode’ or a ‘function-type’ of a system. The state is determined by boundary–, including initial-conditions. |
| orbit | Similar to a state (i.e. in QM). Only that this is referred to trajectories. Or when the system is a dynamical system itself, (i.e. configuration space, phase space). |
| Hénon-Heiles potential | Two-dimensional potential: \(U = \tfrac{m \omega^2}{2}(x^2+y^2)+\alpha(x^2y-\tfrac{1}{3}y^3)\). It is a model describing stellar motion in a galaxy and is used to study chaos. |
| Complex Numbers | |
| Irrational Numbers | |
| Natural Numbers |
Latest: 21 October, 2026
To do!
- So now that I have a general idea, how the phase space is changing in between \(E \in (-0.5, -0.3)\) I am now curious to see how the indiviual trajectories for the interesting periodic orbits behave
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naming of periodic orbits
So I am currently studying two specific periodic orbits. I call these orbit ‘A’. There is one which is there from \(E = -\infty\) and has a period of \(p=1\), in the sense of poincaré maps, \(M:\Sigma -> \Sigma\). Hence the name of that orbit \(A1\).
Then there is a periodic orbit, with period \(p=3\). Its origin is yet unknown, but seems to come into existance, due to the ‘detection’ of one outer wall.