Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Dynamical Systems

Field: Differential Equations

Systems that evolve over time according to a fixed rule.

Sequence of Expressions

Typically a tuple (T,M,Φ)(T, M, \Phi) where TT is time, MM is the state space, and \Phi is the evolution function.
In the geometrical definition, a dynamical system is the tuple ⟨T,M,f⟩\langle {\mathcal {T}},{\mathcal {M}},f\rangle . T{\mathcal {T}} is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. M{\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∈Tt\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 T{\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 T{\mathcal {T}} . 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 numbersR, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable the system is called a differentiable dynamical system. If the manifold M is locally diffeomorphic to R^{n}, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. This does not assume a symplectic structure. 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. A discrete dynamical system, 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. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers the system is called a semi-cascade. A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice. Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing. Given a global dynamical system (R, X, Φ) on a locally compact and Hausdorfftopological spaceX, it is often useful to study the continuous extension Φ* of Φ to the one-point compactificationX* of X. Even after losing the differential structure of the original system, there are compactness arguments to analyze the new system (R, X*, Φ*). In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected. - ^Galor, Oded (2010). Discrete Dynamical Systems. Springer.
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 Φ−1σ∈Σ\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 μ(Φ−1σ)=μ(σ)\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 Φn=Φ∘Φ∘⋯∘Φ\Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi for every integer n are studied . For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated. 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.
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.
Definition

Math definition

In the general context of mathematics, it's possible to define the dynamical system as a general discrete map as in the Formal definition. A generic sequence is already per se a discrete dynamical system. Recursion and interation of maps is another such case. A prototype of this is the Logistic map. - ^John Gemmer, Chapter 14, Discrete dynamical systems https://www.dam.brown.edu/people/jgemmer/GreenwellCh14.pdf - ^John Gemmer, Chapter 14, Example 1 - ^John Gemmer, Chapter 14, Example 2 and 3 - ^John Gemmer, Chapter 14, Example 4
From an empirical perspective, all dynamical systems derived from temporal data are discrete, Gauss for example proved that with the measurement of 3 positions and times of Ceres in the sky is possible to fully determine the orbit, therefore be able to compute any possible position and velocity of the asteroid in the past or the future and therefore fully characterize the dynamical system. Typical tasks with experimental data are to derive a mathematical model. - ^https://sites.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html - ^Moore, Samuel A.; Mann, Brian P.; Chen, Boyuan (17 December 2025). "Automated global analysis of experimental dynamics through low-dimensional linear embeddings". npj Complexity. 2 (1) 36. doi:10.1038/s44260-025-00062-y.
Let MM be a smooth manifold (the phase space) and let XX be a smooth vector field on MM. A dynamical system is formally defined by the flow ϕt:U→M\phi_t: U \to M, where U⊂MU \subset M is an open set and t∈Rt \in \mathbb{R} is time, such that ϕt\phi_t is the unique solution to the initial value problem (IVP) generated by XX: \n\n\frac{d}{dt} \big\|_t \big\phi_t(x_0) = X(\big\phi_t(x_0\big)\), \text{ with } \phi_0(x_0) = x_0 \n\nThis flow ϕt\phi_t maps points in UU to points in MM and describes the evolution of the state x(t)=ϕt(x0)x(t) = \phi_t(x_0) over time tt. The system is autonomous if the vector field XX does not explicitly depend on tt.