In mathematics, physics, engineering and systems theory, a dynamical system is the description of how a system evolves in time. The study of dynamical systems is the focus of dynamical systems theory. It has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept. For example, an astronomer can experimentally record the positions of how the planets move in the sky, and this can be considered a complete enough description of a dynamical system. In the case of planets there is also enough knowledge to codify this information as a set of differential equations with initial conditions, or as a map from the present state to a future state in a predefined state space with a time parameter t, or as an orbit in phase space.
Overview
The concept of a dynamical system has its origins in Newtonian mechanics and more precisely in celestial mechanics. There, as in other natural sciences and engineering disciplines, there is some need to predict the evolution of the system, but maybe also pose other questions such as stability, qualitative or long term behaviour, dependence on parameters, existence of periodic, stochastic or chaotic behaviour. The relation from one state and another is either explicit such as a function in the parameter t predicting position and velocity of a particle or implicit such as a differential equation, difference equation or other time scale. Some times it may not be possible to define such a description, there may not even be a differential equation predicting stock price, or it maybe impossible to build one but still talk stock prices can be considered a dynamical system based on experimental data changing over time. Important properties are existence and uniqueness of solutions, integrability (i.e. the existence of conserved quantities), the possibility to solve the system and be able to compute the state at any point in time. Other properties are whether the system is discrete, continuous, differentiable, smooth, deterministic, ergodic, stochastic or chaotic. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit. Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.
For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:
The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability. The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood. The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid. The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.
Examples Simple 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.
Other classical examples include:
Any mathematical map can be treated as the definition of a dynamical system for example:
History
Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his Dynamical Systems. Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.
Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. In the late 20th century, the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft.
Generalizations The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory, physics such as phase space, quantum state and thermodynamic state, engineering such as system theory, control theory and even information theory.
Mathematical intuition From a mathematics perspective in the most general case the state space X is treated as a generic set of abstract algebra. This space X has a semi-group structure on it (i.e. where only associativity is required) and there is most often a natural choice for an Identity element, which is typically attached to the origin of the chosen reference frame. This semi-group can be intuitively interpreted as the time coordinate t. Time in fact has an addition operation and an origin, the identity, like a group. The action of the semi-group on X is a set of maps from X to itself parametric in the time t, and this is intuitively the time evolution.
Generalizing the state space It is possible to allow different choices of the state space such as a function space (e.g. the pressure, temperature and velocity of a gas in a rocket are a function in the space of solutions of some fluid dynamics PDEs and they may vary over time), a quantum state space (e.g. the state of an atom can be described by a set of functions in an hilbert space and a set of probabilities for these), or a manifold (e.g. the state of a black hole can be described by a metric tensor on a Riemann manifold and its position will be a vector in the same manifold). Other choices can be a phase space, a Configuration space or even a discrete space (e.g. the set of prime numbers or a finite field).
Time as a multidimensional manifold
Time can be generalized too as a generic set of continuous parameters, for example the control parameters of a robot can be a manifold. There is no need that time has a direction, that is smooth or even that it has whatsoever meaning similar to the intuition of time, in fact it can be generalized to even more general algebraic objects. A general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, and for example in image processing. Time typically is considered often an external parameter as in classical and quantum mechanics, and this is typically called time domain representation, and it goes hand in hand with the Hamiltonian mechanics formulation. This is not necessarily always the case: general relativity for example is frame independent, and gravity has an influence over time too, and in quantum electrodynamics the use of the Lagrangian mechanics formulation is more common where time and space are on same footing. In both cases the literature still talks about dynamical systems.
Discrete dynamical system
Time can also be a discrete parameter. When time is generalized to the multi-dimensional case, i.e. as a general set of control or external parameters, this space can be interpreted as a Lattice, i.e. as the discrete points of a manifold or the tics of a stock price. Discrete Time events therefore can be counted by integers, for example like the measurements of the position of the planets in the sky, but this can vastly differ from the intuition of time as a clock that has equispaced time events. One of the tasks is typically to extract some mathematical model from the data.
Not deterministic
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 not deterministic they may allow multiple future states (i.e. the maps are generalized into multivalued functions and not uniquely defined everywhere) and the system can be subject to a bifurcation.
Stochastic
Some systems are also stochastic, either in the input parameters such as an oscillator with a random force, or in the initial conditions, or in the predicted variables as in a Stochastic differential equation. In that random events also affect the evolution of the state variables, and this includes stochastic jump processes which are not continuous, a prototype example of a stochastic dynamical system are stock prices.
Chaotic and Quantum systems
Lastly, there are chaotic systems (i.e. typically deterministic but not predictable) such as:
complex dynamics Hyperbolic dynamics multiplicative chaos non-deterministic chaos And quantum systems (i.e. deterministic until they are measured), or quantum chaotic systems.
Formal definition Assume that X is a non empty set with elements called states. Assume a general transformation:
T : X → X {\displaystyle 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: T ^ {\displaystyle {\hat {T}}} can be a semigroup with one parameter t {\displaystyle t} called time that will also belong to a semi-group such as N ( t > 0 ) {\displaystyle N(t>0)} in the discrete time case, R + ( t > 0 ) {\displaystyle R^{+}(t>0)} in the continuous time case. A semigroup structure introduces associativity
T 1 ^ ( T 2 ^ T 3 ^ ) = ( T 1 ^ T 2 ^ ) T 3 ^ {\displaystyle {\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:
T ^ ( t 1 + t 2 ) = T ^ ( t 1 ) T ^ ( t 2 ) {\displaystyle {\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 {\displaystyle t=0} adding an identity to the semi-group
T ^ ( 0 ) = 1 {\displaystyle {\hat {T}}(0)=\mathbf {1} }
and it is finally possible also to model reversible time evolution: T can be a group such as Z {\displaystyle \mathbf {Z} } or R {\displaystyle R} , and being a group this in fact has a definition of inverse transformations:
∃ ! T ^ − 1 : T ^ − 1 = T ^ ( − t ) , T ^ ( − t ) T ^ ( t ) = 1 {\displaystyle \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 ⟨ T , M , f ⟩ {\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle } . T {\displaystyle {\mathcal {T}}} is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. M {\displaystyle {\mathcal {M}}} is a manifold, and typically, but not always, f is an evolution rule t → f t (with t ∈ T {\displaystyle t\in {\mathcal {T}}} ) such that f t is a diffeomorphism of the manifold to itself. Typically there is some extra structure on the manifold:
To be locally a Banach space, this allows to use standard techniques of functional analysis. A symplectic structure which is the typical case for a phase space A Euclidean Space such as configuration space due to holonomic constraints A discrete structure like a graph. A set of diffeomorphisms such as coordinate transformations A metric tensor like in a Riemann manifold
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. Being locally homeomorphic to a Banach space allows to use theorems of existence and uniqueness of solutions for differential equations, and linear operators, and this makes it analogous the classical definition based on a system of differential equations.
Differentiability If Φ is continuously differentiable the system is called a differentiable dynamical system. The function f is therefore a "smooth" mapping of the time-domain T {\displaystyle {\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 {\displaystyle {\mathcal {T}}}
The manifold M is then typically 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:
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:
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).
Algebraic dynamical systems
An important class of systems from a mathematical perspective is when the map f {\displaystyle f} is algebraic or in general when the map is implicitly defined by a set of algebraic equations and the manifold M {\displaystyle {\mathcal {M}}} is ideally defined on a generic field.
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 the triplet (T, M {\displaystyle {\mathfrak {M}}} , Φ), where Φ is a measure-preserving transformation of a measure space M {\displaystyle {\mathfrak {M}}} , Φ is also the action of a semigroup T as the general case. Here the Measure space M {\displaystyle {\mathfrak {M}}} is defined by the triplet (X, Σ, μ), where X is a set, Σ is a sigma-algebra on X and μ is a finite measure on the measurable space (X, Σ). Often M {\displaystyle {\mathfrak {M}}} is a probability space, i.e. a measure space where the total measure of the full space is 1, in other words with a finite and normalized probability measure.
A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ − 1 σ ∈ Σ {\displaystyle \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 − 1 σ ) = μ ( σ ) {\displaystyle \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, M {\displaystyle {\mathfrak {M}}} , Φ), with M = {\displaystyle {\mathfrak {M}}=} (X, Σ, μ) for such a Φ, is then defined to be a measure-preserving dynamical system.
Conventions There are two conventions for the definition of the maps:
Physics and continuous time convention: The map Φ embodies the time evolution of the dynamical system, it is possible to interpret it for discrete time evolution as Φ= Φ n {\displaystyle \Phi _{n}} parametrized by n ∈ Z + {\displaystyle n\in \mathbb {Z} ^{+}} or for a continuous time evolution as Φ= Φ t {\displaystyle \Phi _{t}} where t ∈ R + {\displaystyle t\in \mathbb {R} ^{+}} , in the general case each map is different. Assuming an initial time t 0 {\displaystyle t_{0}} and a state vector x → ( t ) ∈ X {\displaystyle {\vec {\mathbb {x} }}(t)\in X} , the map can be seen as a time translation map x → ( t + t 0 ) = Φ ^ ( t ) [ x → ( t 0 ) ] {\displaystyle {\vec {\mathbb {x} }}(t+t_{0})={\hat {\Phi }}(t)[{\vec {\mathbb {x} }}(t_{0})]} . In the discrete case this means that the total map Φ ^ n = Φ n ∘ Φ n − 1 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 {\displaystyle {\hat {\Phi }}_{n}=\Phi _{n}\circ \Phi _{n-1}\circ \dots \circ \Phi _{1}\circ \Phi _{0}} can be seen as composition of individual maps. This convention is also related to the study of limit points and Cauchy sequences. Mathematics and discrete convention: all maps are defined as the evolution of the state for one time step to the next one, i.e. from t n → t n + 1 {\displaystyle t_{n}\to t_{n+1}} , or in the continuous case from t → t + δ t {\displaystyle t\to t+\delta t} . The maps Φ ^ n : x ∈ X , x n → x n + 1 {\displaystyle {\hat {\Phi }}_{n}:x\in X,x_{n}\to x_{n+1}} are the same map Φ {\displaystyle \Phi } for all time steps t n {\displaystyle t_{n}} , this can be interpreted also as an iteration over the map Φ ^ n = Φ n = Φ ∘ Φ ∘ ⋯ ∘ Φ {\displaystyle {\hat {\Phi }}_{n}=\Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi } . This convention is also related to the study of fixed points, and recursion. In the language of representation theory: time evolution in the state space X is the induced representation of time translation in the semigroup T, and the map Φ is the group action that induces the representation on X, i.e. there is a homomorphism beteeen the time group T and the state space X where T ( t 0 + t ) = T ( t ) ∘ T ( t 0 ) {\displaystyle T(t_{0}+t)=T(t)\circ T(t_{0})} . In the language of quantum mechanics the infinitesimal generator of time translation t → t + δ t {\displaystyle t\to t+\delta t} is the derivative in time ∂ t {\displaystyle {\partial }_{t}} , and also the time evolution operator.
Examples
Permutations Group actions Bernoulli scheme
Relation to geometric definition The measure theoretical definition assumes the existence of a measure-preserving transformation, in physical terminology the measure is a probability density function over the phase space. The intuition comes from Hamiltonian systems which are volume preserving. The fact that the energy is an integral of motion induces volume conservation on the phase space, or incompressibility of the flow, extra integrals of motions induce further foliations on the phase space.. Another intuition comes from permutations which are "incompressible". One can interpret the set of all possible permutations of a set as the automorphism group induced by the conservation of number of elements of the set. Many different invariant measures can be associated to any one evolution rule, in the same manner as multiple probability densities can induce the same macro-state. If the dynamical system is given by a system of differential equations the appropriate measure must be determined, the symmetries are not explicit and it may not even be possible to define a probability measure in certain patological cases. 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.
Informally and heuristically from an ergodic theorem perspective time averages are equal to space averages/ensemble averages. Ultimately time averages are constant of motions, they corresponds to symmetries due to the noether theorem, and ultimately symmetries corresponds to space averages
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. For dissipative systems typically a small region of phase space shrinks under time evolution, for chaotic systems a small region of phase space instead typically grows and expands at least to a full finite region of the phase space
Heuristically dissipation has an effect opposite to chaos it tends to dampen and limit motion in general but also it dampens turbulence and chaos, expanding to a finite region of phase space means to be sensitive to initial conditions, ergodic regimes tend to grow and cover in full a finite region of phase space, i.e to have dense periodic orbits, and chaotic regimes are also topologically transitive
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 dynamical system (T, X, Φ) on a locally compact and/or Hausdorff topological space X. Φ is a topological isomorphism and therefore a homeomorphism. Topological dynamics is typically done from a global perspective, i.e. considering all possible initial conditions and outcomes (e.g. equilibrium points, periodic orbits, attractors or chaos), the statistics and the groupings of the trajectories, the long term behaviour in time of the system, the homeomorphisms (i.e. continuous deformations) of the trajectories. Typically the trajectories are differentiable, and the systems are also flows.
These techniques are often topological in nature and complementary to ergodic dynamics, to symmetry and to integral of motions via the Noether theorem. For example one of the goals of topological dynamics is to classify topological conjugacy classes, groupings of type of motions with respect to an equivalence class of homeomorphism, one of the goals of ergodic theory instead is ergodic decomposition , one of the goals of symmetry is to classify all the group conjugacy classes of the automorphism group and an integrable system is when all integral of motions can be derived and are known.
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 A category X of mathematical objects has a semigroup G of homomorphisms acting on it (topological spaces have continuous maps, sets have arbitrary maps, groups, rings fields or algebras have homomorphisms, measure spaces have measurable maps). We can view each of these categories as a dynamical system. One can even include the category of dynamical systems with suitable homomorphisms. But this viewpoint is not a very useful in itself.
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:
Φ : U ⊆ ( T × X ) → X {\displaystyle \Phi :U\subseteq (T\times X)\to X}
with:
p r o j 2 ( U ) = X {\displaystyle \mathrm {proj} _{2}(U)=X}
(where p r o j 2 {\displaystyle \mathrm {proj} _{2}} is the 2nd projection map) and for any x in X:
Φ ( 0 , x ) = x {\displaystyle \Phi (0,x)=x}
Φ ( t 2 , Φ ( t 1 , x ) ) = Φ ( t 2 + t 1 , x ) , {\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),} for t 1 , t 2 + t 1 ∈ I ( x ) {\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)} and t 2 ∈ I ( Φ ( t 1 , x ) ) {\displaystyle \ t_{2}\