a deterministic process in which a variable 's value changes over time according to a rule that is defined in terms of the variable's current value

a deterministic process in which a variable's value changes over time according to a well-defined rule which only involves the variable's current value

a mapping of a metric space into itself

a mathematical description of the time evolution of something, e

a mathematical model of a system that changes over time

a model of a process at developing time, described through for example iterations of functions or systems of differential equations

an abstraction in which the system is separated from its environment or background

a physical setting together with rules for how the setting changes or evolves from one moment of time to the next

a set of quantities called the states of the system, and a rule for mapping each state forward in time to other states

a system in motion (a ball rolling, a star evolving, a sound wave propagating I'm sure you get the idea)

Mathematical concept in which a fixed rule describes the time dependence of a point in geometrical space.

The term dynamical system is used to describe a set of rules, ``the dynamic'', that specify how a system evolves over time. Since a hallmark of nonlinear systems is a sensitivity to initial conditions, it is often imperative to include the initial state or starting condition of the system.

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. The concept unifies very different types of such "rules" in mathematics: the different choices made for how time is measured and the special properties the ambient space may give an idea of the vastity of the the class of objects described by this concept. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, loosing the memory of its physical origin, and the ambient space may be simply a set, without the need of a smooth space-time structure defined on it.