Assume that X is a non empty Set with elements called States. Assume a general transformation: T : X \to X It is possible to interpret X as a state space and T as the evolution between states. Adding different structures on T and on X allows to model different properties of the dynamical system. It is possible to model time evolution: \hat{T} can be a semigroup with one parameter t called
time that will also belong to a semi-group such as N (t>0) in the discrete time case, R^{+} (t>0) in the continuous time case. A semigroup structure introduces associativity \hat{T_1}(\hat{T_2}\hat{T_3})=(\hat{T_1}\hat{T_2})\hat{T_3} which implies a composition law between different time evolutions: \hat{T}(t_1+t_2)=\hat{T}(t_1)\hat{T}(t_2) this is also ultimately a
homomorphism. It is possible to define an origin of time t=0 adding an identity to the semi-group \hat{T}(0)=\mathbf{1} and it is finally possible also to model reversible time evolution: T can be a group such as \mathbf{Z} or R, and being a group this in fact has a definition of inverse transformations: \exists! \hat{T}^{-1}: \hat{T}^{-1} = \hat{T}(-t), \hat{T}(-t)\hat{T}(t) = \mathbf{1} More commonly there are multiple classes of definitions for a dynamical system: a first one is motivated by
ordinary differential equations and is geometrical in flavor, there is an additional
differentiability structure; a second one is motivated by
ergodic theory and is
measure theoretical in flavor, there is an additional
topological structure and a last one which is motivated by
Category theory and is more
abstract in flavour.
Geometrical definition In the geometrical definition, a dynamical system is the tuple \langle \mathcal{T}, \mathcal{M}, f\rangle . \mathcal{T} is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. \mathcal{M} is a
manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a
graph.
f is an evolution rule
t →
f t (with t\in\mathcal{T}) such that
f t is a
diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain \mathcal{T} into the space of diffeomorphisms of the manifold to itself. In other terms,
f(
t) is a diffeomorphism, for every time
t in the domain \mathcal{T}.
Algebraic dynamical systems An important class of systems from a mathematical perspective is when the map f is algebraic or in general when the map is implicitly defined by a set of algebraic equations and the manifold \mathcal{M} is ideally defined on a generic field.
Real dynamical system A
real dynamical system,
real-time dynamical system,
continuous time dynamical system, or
flow is a tuple (
T,
M, Φ) with
T an
open interval in the
real numbers
R,
M a
manifold typically but not necessarily locally
homeomorphic to a
Banach space, and Φ a
continuous function.
Differentiability If Φ is
continuously differentiable the system is called a
differentiable dynamical system. The
manifold M is then typically but not necessarily locally
diffeomorphic to a
Banach space.
Dimensionality If the manifold
M is locally diffeomorphic to
Rn, the dynamical system is
finite-dimensional; if not, the dynamical system is
infinite-dimensional.
Flows When
T is taken to be the reals, the dynamical system is called
global or a
flow; and if
T is restricted to the non-negative reals, then the dynamical system is a
semi-flow.
Classical definition The modern geometrical definition assumes a
map that provides an explicit description of the dynamical system, this is motivated by
ergodic theory, by
partial differential equations and by mathematical techniques that go beyond differential equations. An explicit description is often not available, the classical geometrical definition is implicit, rooted in classical mechanics, and based on a standard set of
ordinary differential equations and a finite set of
degrees of freedom: The totality of states of motion may be set into one-to-one correspondence with the points, P, of a closed n-dimensional manifold, M, in such wise that for suitable coordinates x_1,...,x_n the differential equations of motion may be written: \frac{dx_i}{dt} = u_i(x_1,...,x_n,t);(i=1,...,n) There can be different regularity conditions to the functions u_i such as being
differentiable or
analytic. This definition implies the
existence and uniqueness of solutions of such equations.
Lagrangian Dynamical system It is also possible to cast the geometrical definition in terms of a
variational principle: Let M be a
differentiable manifold, TM its
tangent bundle, and L: TM\to \mathbb{R} a
differentiable function. A
map \gamma: \mathbb{R}\to M is called a motion in the Lagrangian system, with configuration
manifold M and
Lagrangian L, if \gamma is an extremal of the functional: \Phi(\gamma)=\int_{t_0}^{t_1} L(\gamma,\dot{\gamma}) dt where \dot{\gamma}\in TM_{\gamma(t)} is called velocity vector.
Hamiltonian Dynamical system Dually to the Lagrangian it is possible to use a
Hamiltonian formulation which includes a
Symplectic or
Poisson manifold structure on the
phase space.
Non integrable systems To be complete there are also systems that are typically not
integrable systems such as
dissipative systems,
nonholonomic systems and systems that have a
contact manifold structure for example systems that have a no slip boundary condition (i.e. some constraint on the velocity on the boundary).
Discrete dynamical system A
discrete-time dynamical system is a tuple (
T,
M, Φ), where
M is a
manifold locally diffeomorphic to a
Banach space, and Φ is a function.
T can be taken to be the integers or the non negative integers. The manifold itself can be a
graph or made discrete for example with a
discrete topology.
Measure theoretical definition A dynamical system may be defined formally as a measure-preserving transformation of a
measure space, the triplet (
T, (
X, Σ,
μ), Φ). Here,
T is a monoid (usually the non-negative integers),
X is a
set, and (
X, Σ,
μ) is a
probability space, meaning that Σ is a
sigma-algebra on
X and μ is a finite
measure on (
X, Σ). A map Φ:
X →
X is said to be
Σ-measurable if and only if, for every σ in Σ, one has \Phi^{-1}\sigma \in \Sigma. A map Φ is said to
preserve the measure if and only if, for every
σ in Σ, one has \mu(\Phi^{-1}\sigma ) = \mu(\sigma). Combining the above, a map Φ is said to be a '
measure-preserving transformation of X
', if it is a map from
X to itself, it is Σ-measurable, and is measure-preserving. The triplet (
T, (
X, Σ,
μ), Φ), for such a Φ, is then defined to be a
dynamical system. The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the
iterates \Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi for every integer
n are studied.{{citation needed|reason=iteration and recursive relations are just special cases, each map from t_n to t_{n+1}, can be fully different, iteration is just a special case actually of \Phi = \Phi{t_n} \circ \Phi{t_{n-1}} \circ \dots \circ \Phi_{t_0} |date=February 2026}} For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.
Relation to geometric definition The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the
Krylov–Bogolyubov theorem) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance. Some systems have a natural measure, such as the
Liouville measure in
Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic
dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the
attractor, but attractors have zero
Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution. For hyperbolic dynamical systems, the
Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of
stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
Topological dynamical system A
topological dynamical system is a global dynamical system (
T,
X, Φ) on a
locally compact and
Hausdorff topological space X. T is a
topological isomorphism and therefore a
homeomorphism.
Compactification It is often useful to study the continuous extension Φ* of Φ to the
one-point compactification X* of
X. Even after losing the differential structure of the original system, there are compactness arguments to analyze the new system (
R,
X*, Φ*). This is similar in spirit to
Projective geometry where all limit points to infinity are the same point. Another more general technique is to use
Stone–Čech compactification which is similar in spirit to
affine geometry where all limit points at infinity are considered different.
Relevance In compact dynamical systems the
limit set of any orbit is
non-empty,
compact and
simply connected. As an example in a topological dynamical system the limit orbit of an
attractor is contained within the manifold itself. This is a non trivial statement for multiple reasons: limit orbits may never be reached; limit orbits may have
Lebesgue measure zero; attaching a probability to a limit orbit would be non trivial; an attractor may also have multiple limit orbits and the distinction between different
compactifications may be relevant.
Definition with Category theory Categories vs semi-groups Definition with monoids In the context of category theory, categories are always defined together with an Identity map, therefore these definitions are based on
monoids instead of semi-groups. A
dynamical system is a
tuple (
T,
X, Φ) where
T is a
monoid, written additively,
X is a non-empty
set and Φ is a
function: \Phi: U \subseteq (T \times X) \to X with: \mathrm{proj}_{2}(U) = X (where \mathrm{proj}_{2} is the 2nd
projection map) and for any
x in
X: \Phi(0,x) = x \Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x), for \, t_1,\, t_2 + t_1 \in I(x) and \ t_2 \in I(\Phi(t_1, x)) , where we have defined the set I(x) := \{ t \in T : (t,x) \in U \} for any
x in
X. In particular, in the case that U = T \times X we have for every
x in
X that I(x) = T and thus that Φ defines a
monoid action of
T on
X. The function Φ(
t,
x) is called the
evolution function of the dynamical system: it associates to every point
x in the set
X a unique image, depending on the variable
t, called the
evolution parameter.
X is called
phase space or
state space, while the variable
x represents an
initial state of the system. We often write: \Phi_x(t) \equiv \Phi(t,x) \Phi^t(x) \equiv \Phi(t,x) if we take one of the variables as constant. The function \Phi_x:I(x) \to X is called the
flow through
x and its
graph is called the
trajectory through
x. The set \gamma_x \equiv\{\Phi(t,x) : t \in I(x)\} is called the
orbit through
x. The orbit through
x is the
image of the flow through
x. A subset
S of the state space
X is called Φ-
invariant if for all
x in
S and all
t in
T \Phi(t,x) \in S. Thus, in particular, if
S is Φ-
invariant, I(x) = T for all
x in
S. That is, the flow through
x must be defined for all time for every element of
S. == Construction of dynamical systems ==