> [!NOTE] **Definition** > > An autonomous $2\times2$ system of equations is a [[System of First Order Ordinary Differential Equations|system]] of the form $\begin{align} \frac{d}{dt}x_{1}(t) = f_{1}(x_{1}(t),x_{2}(t)) \\ \frac{d}{dt}x_{2}(t) = f_{2}(x_{1}(t),x_{2}(t)) \end{align}$for $t \in (\alpha, \beta)$ with some given functions $f_{1},f_{2}: \mathbb{R}^{2}\to \mathbb{R}, \; i=1,2$. # Properties **Stationary point**: A [[Stable Stationary Point of First Order Autonomous Recurrence Relation|stationary point]] of the above system is a point satisfying $(f_{1}(\bar{x}_{1},\bar{x}_{2}), f_{2}(\bar{x}_{1},\bar{x}_{2}))=(0,0).$ The [[Hartman-Grobman Theorem|Hartman-Grobman theorem]] asserts that the stationary point is stable iff the [[Linearisation near Stationary Point of Autonomous 2 x 2 System of First Order Ordinary Differential Equations|linearisation of the system near the stationary point]] is stable about the origin. # Applications **Examples**: See [[Example of 2-D autonomous system]] and [[Lotka-Volterra System Modelling Predator-Prey Interaction]].