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. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.
A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.
Three examples of different geometries: Euclidean, elliptical, and hyperbolic geometry
In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object. For example, a line may be studied in isolation, or it may be studied as an object in two-dimensional space — in which case the ambient space is the plane, or as an object in three-dimensional space — in which case the ambient space is three-dimensional. To see why this makes a difference, consider the statement "Lines that never meet are necessarily parallel." This is true if the ambient space is two-dimensional, but false if the ambient space is three-dimensional, because in the latter case the lines could be skew lines, rather than parallel.
A formal system is, broadly defined as any well-defined system of abstract thought based on the model of mathematics. Euclid's Elements is often held to be t
e first formal system and displays the characteristic of a formal system. The entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory. A formal system need not be mathematical as such, Spinoza's Ethics for example imitates the form of Euclid's Elements.
Each formal system has a formal language, which is composed by primitive symbols. These symbols act on certain rules of formation and are developed by inference from a set of axioms. The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.
Formal systems in mathematics consist of the following elements:
A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. It is usually required that there be a decision procedure for deciding whether a formula is well formed or not.
A set of axioms or axiom schemata: each axiom must be a wff.
A set of inference rules.
A formal system is said to be recursive (i.e. effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, according to context.
Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac's bra-ket notation.