Dynamical systems

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

A dynamical system has a state determined by a collection of real numbers. 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&mdash;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.

Overview
The concept of dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation or difference equation.) To determine the state for all future times requires iterating the relation many times&mdash;each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. Once the system can be solved, given an initial point it is possible to determine all its future points, a collection known as a trajectory or orbit.

Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems. Numerical methods executed on computers 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: It was in the work of Poincaré that these dynamical systems themes developed.
 * The systems studied may only be known approximately&mdash;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.

Basic definitions
A dynamical system is a manifold M called the phase (or state) space and a smooth evolution function ft that for any element of t &isin; T, the time, maps a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.

The evolution function ft is often the solution of a differential equation of motion
 * $$ \dot{x} = v(x) \,.$$

The equation gives the time derivative, represented by the dot, of a trajectory x(t) on the phase space starting at some point x0. The vector field v(x) is a smooth function that at every point of the phase space M provides the velocity vector of the dynamical system at that point. (These vectors are not vectors in the phase space M, but in the tangent space TMx of the point x.)

There is no need for higher order derivatives in the equation, nor for time dependence in v(x) because these can be eliminated by considering systems of higher dimensions. For example, the system
 * $$ y = p(x), \dot{y} = q(x), \ddot{y} = r(x) $$

is equivalent to the
 * $$ y = p(x), \dot{y} = z, $$
 * $$ z = q(x), \dot{z} = r(x). $$

Other types of differential equations can be used to define the evolution rule:
 * $$ G(x, \dot{x}) = 0 $$

is an example of an equation that arises from the modeling of mechanical systems with complicated constraints.

The differential equations determining the evolution function ft are often ordinary differential equations: in this case the phase space M is a finite dimensional manifold. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds&mdash;those that are locally Banach spaces&mdash;in which case the differential equations are partial differential equations. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.

Linear dynamical systems
Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the &nu;-dimensional Euclidean space, so any point in phase space can be represented by a vector with &nu; numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).

Flows
For a flow, the vector field v(x) is a linear function of the position in the phase space, that is,
 * $$ v(x) = A x + b\,,$$

with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b &ne; 0 with A = 0 is just a straight line in the direction of b:
 * $$ f^t(x_1) = x_1 + b t \,. $$

When b is zero and A &ne; 0 the origin is an equilibrium (or singular) point of the flow, that is, if x0 = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x0,
 * $$ f^t(x_0) = e^{t A} x_0 \,.$$

When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.

The distance between two different initial conditions in the case A &ne; 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.



Maps
A discrete-time, linear dynamical system has the form
 * $$ x_{n+1} = A x_n + b \,,$$

with A a matrix and b a vector. As in the continuous case, the change of coordinates x &rarr; x + (1 - A)–1b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the form Anx0. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For example, if u1 is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along &alpha; u1, with &alpha; &isin; R, is an invariant curve of the map. Points in this straight line run into the fixed point.

Local dynamics
The qualitative properties of dynamical systems do not change under smooth change of coordinates (this is sometimes taken as a definition of qualitative):  a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that make the dynamical system as simple as possible.

Rectification
A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) &ne; 0, then there is a change of coordinate for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time around the orbit loops around phase space a different way, then it is impossible to rectify the vector field in the whole series of patches.

Near periodic orbits
In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x0 in the orbit &gamma; and consider the points in phase space in that neighborhood that are perpendicular to v(x0). These points are a Poincaré section S(&gamma;, x0), of the orbit. The flow now defines a map, the Poincaré map F : S &rarr; S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x0.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J &middot; x + O(x&sup2;), so a change of coordinates h can only be expected to simplify F to its linear part
 * $$ h^{-1} \circ F \circ h(x) = J \cdot x \,. $$

This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions analytic and in the process discovered the non-resonant condition. If &lambda;1,&hellip;,&lambda;&nu; are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form &lambda;i – &sum; (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.

Conjugation results
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.

In the hyperbolic case the theorem of Hartman and Grobman gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J &middot; x. The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.

The KAM theorem gives the behavior near an elliptic point.

Bifurcations
When the evolution map ft (or the vector field it is derived from) depends on a parameter &mu;, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value &mu;0 is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter &mu;. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point x0 of a map family F&mu; can be characterized by the eigenvalues of the first derivative DF(x0) of the map computed at the bifurcation point. The bifurcation will occur when there are eigenvalues of DF on the unit circle. If there is an isolated eigenvalue of value 1 on the unit circle, then the bifurcation is a saddle-node bifurcation. If there is an isolated eigenvalue –1 on the unit circle, then it is a flip bifurcation. And if there is a pair of complex conjugate eigenvalues on the unit circle, then it is a Hopf bifurcation.

Some bifurcations can lead to very complicated structures in phase space. The Ruelle-Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. The Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of doublings of its period.

Ergodic systems

 * See main article ergodic theory.

In many dynamical systems it is possible to choose the coordinates of the system so that volume (really a &nu;-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) &times; (momentum). The flow takes points of a subset A into the points f t (A) and invariance of the phase space means that
 * $$ \mathrm{vol} (A) = \mathrm{vol} ( f^t(A) ) \,. $$

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with same energy form an energy shell &Omega;, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's  derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(&Omega;).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function ft. This introduces an operator Ut, the transfer operator,


 * $$ (U^t a)(x) = a(f^{-t}(x)) \,. $$

By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of ft. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving ft gets mapped into an infinite-dimensional linear problem involving U.

The Liouville measure restricted to the energy surface &Omega; is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(&minus;&beta;H). This idea has been generalized by Sinai, Bowen, and Ruelle to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

Chaos theory
Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random. (Remember that we are speaking of completely deterministic systems!). This unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

Note that the chaotic behavior of complicated systems is not the issue. Meteorology has been known for years to involve complicated&mdash;even chaotic&mdash;behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

Formal definition
There are two formal 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. The measure theoretical definitions assumes the existence of a measure-preserving transformation. This appears to exclude dissipative systems, as in a dissipative system a small region of phase space shrinks under time evolution. A simple construction (sometimes called the Krylov-Bogoliubov theorem) shows that 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.

Many different invariant measures can be associated to any one evolution rule. In ergodic theory the choice is assumed made, but if the dynamical system is given by a system of differential equations the appropriate measure must be determined. 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 many dissipative chaotic 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 hyperbolic dynamical systems, the SRB 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 difficulty in constructing the natural measure for a dynamical system 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.

Geometrical definition
A dynamical system is the tuple $$ \langle \mathcal{M}, f, \mathcal{T}\rangle $$, with $$\mathcal{M}$$ a manifold (locally a Banach space or Euclidean space), $$\mathcal{T}$$ the domain for time (non-negative reals, the integers, ...) and an evolution rule ft (with $$t\in\mathcal{T}$$) a diffeomorphism of the manifold to itself.

Measure theoretical definition

 * See main article measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet $$(X,\Sigma,\mu,\tau)$$. Here, X is a set, and &Sigma; is a topology on X, so that $$(X, \Sigma)$$ is a sigma-algebra. For every element $$\sigma \in \Sigma$$, &mu; is its finite measure, so that the triplet $$(X,\Sigma,\mu)$$ is a probability space. A map $$\tau:X\to X$$ is said to be &Sigma;-measurable if and only if, for every $$\sigma \in \Sigma$$, one has $$\tau^{-1}\sigma \in \Sigma$$. A map &tau; is said to preserve the measure if and only if, for every $$\sigma \in \Sigma$$, one has $$\mu(\tau^{-1}\sigma ) = \mu(\sigma)$$. Combining the above, a map &tau; is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is &Sigma;-measurable, and is measure-preserving. The quadruple $$(X,\Sigma,\mu,\tau)$$, for such a &tau;, is then defined to be a dynamical system.

The map &tau; embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates $$\tau^n=\tau \circ \tau \circ \ldots\circ\tau$$ for integer n are studied. For continuous dynamical systems, the map &tau; is understood to be finite time evolution map and the construction is more complicated.

Examples of dynamical systems

 * Logistic map
 * Double pendulum
 * Arnold's cat map
 * Horseshoe map
 * Baker's map is an example of a chaotic piecewise linear map
 * Billiards and Outer Billiards
 * Henon map
 * Lorenz system
 * Circle map
 * Rossler map
 * List of chaotic maps
 * Swinging Atwood's Machine (SAM)
 * Bouncing Ball
 * Mechanical Strings