Dynamical system

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, 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.

Many people regard French mathematician Henri Poincaré as the founder of dynamical systems.[9] 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.[10] 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.[11]

In the most general sense,[12][13] 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

${\displaystyle \Phi :U\subseteq (T\times X)\to X}$

with

${\displaystyle \mathrm {proj} _{2}(U)=X}$ (where ${\displaystyle \mathrm {proj} _{2}}$ is the 2nd projection map)

and for any x in X:

${\displaystyle \Phi (0,x)=x}$

${\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}$

for ${\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)}$ and ${\displaystyle \ t_{2}\in I(\Phi (t_{1},x))}$, where we have defined the set ${\displaystyle I(x):=\{t\in T:(t,x)\in U\}}$ for any x in X.

In particular, in the case that ${\displaystyle U=T\times X}$ we have for every x in X that ${\displaystyle I(x)=T}$ 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

${\displaystyle \Phi _{x}(t)\equiv \Phi (t,x)}$

${\displaystyle \Phi ^{t}(x)\equiv \Phi (t,x)}$

if we take one of the variables as constant. The function

${\displaystyle \Phi _{x}:I(x)\to X}$

is called the flow through x and its graph is called the trajectory through x. The set

${\displaystyle \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}$

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

${\displaystyle \Phi (t,x)\in S.}$

Thus, in particular, if S is Φ-invariant, ${\displaystyle I(x)=T}$ 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.

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:

${\displaystyle {\dot {\boldsymbol {x}}}={\boldsymbol {v}}(t,{\boldsymbol {x}})}$

${\displaystyle {\boldsymbol {x}}|_{t=0}={\boldsymbol {x}}_{0}}$

where

• ${\displaystyle {\dot {\boldsymbol {x}}}}$ represents the velocity of the material point x
• M is a finite dimensional manifold
• v: T × M → TM is a vector field in Rⁿ or Cⁿ 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

${\displaystyle {\boldsymbol {x}}(t)=\Phi (t,{\boldsymbol {x}}_{0})}$

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

${\displaystyle {\dot {\boldsymbol {x}}}-{\boldsymbol {v}}(t,{\boldsymbol {x}})=0\qquad \Leftrightarrow \qquad {\mathfrak {G}}\left(t,\Phi (t,{\boldsymbol {x}}_{0})\right)=0}$

where ${\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} }$ 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.

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 N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N 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 an affine function of the position in the phase space, that is,

${\displaystyle {\dot {x}}=v(x)=Ax+b,}$

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 ≠ 0 with A = 0 is just a straight line in the direction of b:

${\displaystyle \Phi ^{t}(x_{1})=x_{1}+bt.}$

When b is zero and A ≠ 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,

${\displaystyle \Phi ^{t}(x_{0})=e^{tA}x_{0}.}$

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 ≠ 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.

Linear vector fields and a few trajectories.

The qualitative properties of dynamical systems do not change under a 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 makes the dynamical system as simple as possible.

When the evolution map Φ^t (or the vector field it is derived from) depends on a parameter μ, 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 μ₀ 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 μ. 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 a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point x₀ of a system family F_μ can be characterized by the eigenvalues of the first derivative of the system DF_μ(x₀) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DF_μ on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.

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

In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-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) × (momentum). The flow takes points of a subset A into the points Φ ^t(A) and invariance of the phase space means that

${\displaystyle \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}$

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 the same energy form an energy shell Ω, 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(Ω).

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 φ ^t. This introduces an operator U ^t, the transfer operator,

${\displaystyle (U^{t}a)(x)=a(\Phi ^{-t}(x)).}$

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

The Liouville measure restricted to the energy surface Ω 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(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) 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.

• Arnold, Vladimir I. (2006). "Fundamental concepts". Ordinary Differential Equations. Berlin: Springer Verlag. ISBN 3-540-34563-9.
• Chueshov, I. D. Introduction to the Theory of Infinite-Dimensional Dissipative Systems. online version of first edition on the EMIS site .
• Temam, Roger (1997) [1988]. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer Verlag.

Wikimedia Commons has media related to Dynamical systems.

• Arxiv preprint server has daily submissions of (non-refereed) manuscripts in dynamical systems.
• Encyclopedia of dynamical systems A part of Scholarpedia — peer reviewed and written by invited experts.
• Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
• Sci.Nonlinear FAQ 2.0 (Sept 2003) provides definitions, explanations and resources related to nonlinear science

Online books or lecture notes

• Geometrical theory of dynamical systems. Nils Berglund's lecture notes for a course at ETH at the advanced undergraduate level.
• Dynamical systems. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.
• Chaos: classical and quantum. An introduction to dynamical systems from the periodic orbit point of view.
• Learning Dynamical Systems. Tutorial on learning dynamical systems.
• Ordinary Differential Equations and Dynamical Systems. Lecture notes by Gerald Teschl

Research groups

• Dynamical Systems Group Groningen, IWI, University of Groningen.
• Chaos @ UMD. Concentrates on the applications of dynamical systems.
• , SUNY Stony Brook. Lists of conferences, researchers, and some open problems.
• Center for Dynamics and Geometry, Penn State.
• Control and Dynamical Systems, Caltech.
• Laboratory of Nonlinear Systems, Ecole Polytechnique Fédérale de Lausanne (EPFL).
• Center for Dynamical Systems, University of Bremen
• Systems Analysis, Modelling and Prediction Group, University of Oxford
• Non-Linear Dynamics Group, Instituto Superior Técnico, Technical University of Lisbon
• Dynamical Systems, IMPA, Instituto Nacional de Matemática Pura e Applicada.
• Nonlinear Dynamics Workgroup, Institute of Computer Science, Czech Academy of Sciences.
• UPC Dynamical Systems Group Barcelona, Polytechnical University of Catalonia.
• Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara.