A dynamical system with a continuous time formulation and deterministic behaviour is described by a differential equation. If the states \( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}) \in \htmlClass{sdt-0000000038}{\mathcal{X}} \) depend only on time, this is the ordinary differential equation given below. Otherwise, partial derivatives are used.
Note that for continuous time systems, the derivative encodes information about the time evolution of the system, and \( \htmlClass{sdt-0000000027}{T} \) does not give the full next state.
| \( t \) | This symbol represents time. It is often measured by its SI unit seconds (\( \htmlClass{udt-0000000002}{s} \)). |
| \( T \) | This is the symbol for a dynamical system's update operator. |
| \( \mathbf{x} \) | This symbol represents a state of the dynamical system at some time point. |
In polar coordinates, points are described by \( (\htmlClass{sdt-0000000063}{r}, \htmlClass{sdt-0000000123}{\phi}) \): the distance \(\htmlClass{sdt-0000000063}{r}\) to the origin and the angle \(\htmlClass{sdt-0000000123}{\phi}\) measured from the positive \(x\)-axis.
A very simple example of a continuous-time system law is given by the ODE system:
\[ \begin{cases} \dot{\htmlClass{sdt-0000000063}{r}} = 0 \\ \dot{\htmlClass{sdt-0000000123}{\phi}} = 1 \end{cases}\]
This represents a rotation around the origin. Since \( \dot{\htmlClass{sdt-0000000063}{r}} = 0 \), the radial distance never changes, and since \( \dot{\htmlClass{sdt-0000000123}{\phi}} = 1 \) the angle increases at constant velocity of \(1\). This system will result in trajectories tracing out circles centered at the origin in anti-clockwise orientation.