Theory
We should start by stating the mathematical preliminaries.
Linear Algebra
Even though in general we do not have linear equations vector spaces are elemental do understanding dynamical systems and their behavior.
We define a vector space \(\mathcal{V}\) by the object that contains all maps from an alphabet \(\Sigma\) to a field \(\mathbb{F}\), \(v:\mathcal{A}\rightarrow \mathbb{F}\); Thus we write \(v\in \mathbb{F}^\mathcal{A}\).
An alphabet \(\mathcal{A}\) can be an arbitrary set, like \(\Sigma=\{1,2,\dots,n\}_{n\in\mathbb{N}}\) or a set of unordered symbols \(\Sigma=\{a,b,c,d,\dots,z\}\) of the Latin alphabet. All maps \(v,u\in\mathbb{F}^\Sigma\) and all \(a\in\Sigma\) need to satisfy
- completeness of addition, meaning that the vector \(u+v\in\mathbb{F}^\Sigma\) defined by the equation \(u(a)+v(a)=(u+v)(a)\) and
- completeness of scalar multiplication, such that for any \(\alpha\in\mathbb{F}\), \(\alpha v\in\mathbb{F}\) the equation \((\alpha v)(a)=\alpha\, v(a)\) holds.
The dimension of the vector space is given by the number of elements in the alphabet \(|\Sigma| =n\) which defines a one-to-one map, \(f:\{1,2,\dots,n\}_{n\in\mathbb{N}} \rightarrow \Sigma\). The vector space is then commonly referred to as \(\mathbb{F}^n\).
Dynamical systems
A dynamical system defined by a set of \(N\) fist order differential equations
\[\frac{d}{dt}\x\equiv \dot{\x}=\mathbf{f}(\x).\]
The function \(\mathbf{f}:\R^N\rightarrow \R^N\) takes \(\x\in\R^N\) to its tangentspace \(\dot{\x}\in\R^N\) and defines all possible trajectories \(\x(\x_0,t)\) for initial conditions \(\x_0\) parameterized by the time \(t\in I\).