Parametric curve in the context of Discrete dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.

At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.

Linear referencing, also called linear reference system or linear referencing system (LRS), is a method of spatial referencing over linear or curvilinear elements, such as roads or rivers. In LRS, the locations of physical features are described parametrically in terms of a single curvilinear coordinate, typically the distance traveled from a fixed point, such as a milestone. It is an alternative to referencing by geographic coordinates, which would involve two coordinates, latitude and longitude.

Point features (e.g. a signpost) are located by a single distance value while linear features (e.g. a no-passing zone) are delimited by two distance values, corresponding to beginning and end. If the subjacent linear referencing element or route is changed, only the linear coordinates of those locations on the changed segment need to be updated.Linear referencing is suitable for management of data related to linear features like roads, railways, oil and gas transmission pipelines, power and data transmission lines, and rivers.It is used in engineering, construction, and utilities management.

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.