dynamical symmetries

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. 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. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.

At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. 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 stochastic, in that random events also affect the evolution of the state variables.

In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.

The study of dynamical systems is the focus of dynamical systems theory, which 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.

Overview

The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as ''solving the system'' or ''integrating the system''. If the system can be solved, 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.

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.

Formal definition

In the most general sense,
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
:$\mathrm_(U) = X$ (where $\mathrm_$ 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) := \$ 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.
:$\Phi_x:I(x) \to X$
is called the flow through ''x'' and its graph trajectory through ''x''. The set
:$\gamma_x \equiv\$
is called the orbit through ''x''.
Note that the orbit through ''x'' is the image of the flow through ''x''.
A subset ''S'' of the state space ''X'' is called Φ-invariant if for all ''x'' in ''S'' and all ''t'' in ''T''
:$\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''.

More commonly there are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor.

Geometrical definition

In the geometrical definition, a dynamical system is the tuple $\langle \mathcal, \mathcal, f\rangle$. $\mathcal$ is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. $\mathcal$ 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 $t\in\mathcal$) such that ''f ^t'' is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain $\mathcal$ 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 $\mathcal$ .

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 locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is 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''. Note that 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''.

Discrete dynamical system

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 we call the system a ''semi-cascade''.

Cellular automaton

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.

Multidimensional generalization

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.

Compactification of a dynamical system

Given a global dynamical system (R, ''X'', Φ) on a locally compact and Hausdorff topological space ''X'', it is often useful to study the continuous extension Φ* of Φ to the one-point compactification ''X*'' of ''X''. Although we lose the differential structure of the original system we can now use 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.

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 $\Phi^\sigma \in \Sigma$. A map Φ is said to preserve the measure if and only if, for every ''σ'' in Σ, one has $\mu(\Phi^\sigma ) = \mu(\sigma)$. 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. 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.

Construction of dynamical systems

The concept of ''evolution in time'' is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems. But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example consider an initial value problem such as the following:

:$\dot=\boldsymbol(t,\boldsymbol)$
:$\boldsymbol, _=\boldsymbol_0$

where
*$\dot$ represents the velocity of the material point x
*''M'' is a finite dimensional manifold
*v: ''T'' × ''M'' → ''TM'' is a vector field in R^''n'' or C^''n'' and represents the change of velocity induced by the known forces acting on the given material point in the phase space ''M''. The change is not a vector in the phase space ''M'', but is instead in the tangent space ''TM''.

There is no need for higher order derivatives in the equation, nor for the parameter ''t'' in ''v''(''t'',''x''), because these can be eliminated by considering systems of higher dimensions.

Depending on the properties of this vector field, the mechanical system is called
*autonomous, when v(''t'', x) = v(x)
*homogeneous when v(''t'', 0) = 0 for all ''t''

The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above

:$\boldsymbol(t)=\Phi(t,\boldsymbol_0)$

The dynamical system is then (''T'', ''M'', Φ).

Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy

:$\dot-\boldsymbol(t,\boldsymbol)=0 \qquad\Leftrightarrow\qquad \mathfrak\left(t,\Phi(t,\boldsymbol_0)\right)=0$

where $\mathfrak:\to\mathbf$ is a functional from the set of evolution functions to the field of the complex numbers.

This equation is useful when modeling mechanical systems with complicated constraints.

Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations.

Examples

* Arnold's cat map
* Baker's map is an example of a chaotic piecewise linear map
* Billiards and outer billiards
* Bouncing ball dynamics
* Circle map
* Complex quadratic polynomial
* Double pendulum
* Dyadic transformation
* Hénon map">, 1)^\infty
: x \mapsto (x_0, x_1, x_2, ...