The dynamical system concept is a mathematical Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions formalization In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) an expression from one or more other premises that are antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its for any fixed "rule" which describes the time Time has been defined as the continuum in which events occur in succession from the past to the present and on to the future. Time has also been defined as a one-dimensional quantity used to sequence events, to quantify the durations of events and the intervals between them, and to quantify and measure the motions of objects and other changes dependence of a point's position in its ambient space An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object. Examples include the mathematical models A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termed mathematical modelling . Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines, but also in the social sciences (such as economics, that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.
At any given time a dynamical system has a state In control theory, states are what characterize a system. In other words, state is the essence of the past that affects the future. With linear systems states are not unique, but can be transformed into equivalent states using certain linear transformations given by a set of real numbers In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an (a vector A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields) which can be represented by a point In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one in an appropriate state space (a geometrical manifold In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and sphere are two-dimensional manifolds, and so forth). Small changes in the state of the system correspond to small changes in the numbers. The evolution rule of the dynamical system is a fixed rule The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain that describes what future states follow from the current state. The rule is deterministic In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. Deterministic models thus produce the same output for a given starting condition: for a given time interval only one future state follows from the current state.
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Overview
The concept of a dynamical system has its origins in Newtonian mechanics In the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain boundary under the action of a system of forces. 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 A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines, difference equation In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms 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. 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 In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the phase space. Understanding the properties of orbits.
Before the advent of fast computing machines A computer is a programmable machine that receives input, stores and manipulates data//information, and provides output in a useful format, solving a dynamical system 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 In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by C1-small perturbations. Examples of such qualitative properties are numbers of fixed points and periodic orbits . Unlike Lyapunov stability, which considers perturbations of initial. 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 Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth 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 Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics 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 Statistical mechanics is the application of probability theory (which contains mathematical tools for dealing with large populations) to study the thermodynamic behavior of systems of a large number of particles. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties and of chaos Chaos theory is a field of study in mathematics, physics, economics and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions yield widely diverging outcomes for chaotic systems, rendering long-.
It was in the work of Poincaré Jules Henri Poincaré (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe]) was a French mathematician, theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime that these dynamical systems themes developed.
Basic definitions
Main article: Dynamical system (definition)A dynamical system is a manifold In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and sphere are two-dimensional manifolds, and so forth M called the phase (or state) space endowed with a family of smooth evolution functions Φ t that for any element of , the time, map 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.
Examples
The evolution function Φ t is often the solution of a differential equation of motion
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 In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other TMx of the point x.) Given a smooth Φ t, an autonomous vector field can be derived from it.
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. Other types of differential equations can be used to define the evolution rule:
is an example of an equation that arises from the modeling of mechanical systems with complicated constraints.
The differential equations determining the evolution function Φ t are often ordinary differential equations In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable: 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—those that are locally Banach spaces In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them. Many of the infinite-dimensional function spaces studied in analysis are examples of Banach spaces—in which case the differential equations are partial differential equations In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.
Further examples
- Logistic map The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first
- Dyadic transformation
- Tent map
- Double pendulum In mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior. The motion of a double pendulum is governed by a set of coupled ordinary differential equations. For certain energies its motion is chaotic
- Arnold's cat map
- Horseshoe map
- Baker's map is an example of a chaotic piecewise linear In mathematics, a piecewise linear function is a piecewise-defined function whose pieces are linear map
- Billiards and outer billiards
- Hénon map
- Lorenz system The Lorenz attractor, named for Edward N. Lorenz, is a fractal structure corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape. The map shows how the state of a dynamical system evolves over time in a complex, non-
- Circle map
- Rössler map
- List of chaotic maps
- Swinging Atwood's machine
- Quadratic map simulation system
- Bouncing ball dynamics
Linear dynamical systems
Main article: Linear dynamical systemLinear 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 In mathematics, a flow generalizes n-fold iteration of functions so that the iteration count n becomes a continuous parameter. It is used to formalize in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in engineering, physics and the study of ordinary differential equations. Informally,, the vector field Φ(x) is a linear function of the position in the phase space, that is,
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:
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,
When b = 0, the eigenvalues In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of of A determine the structure of the phase space. From the eigenvalues and the eigenvectors In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of 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 Chaos theory is a field of study in mathematics, physics, economics and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions yield widely diverging outcomes for chaotic systems, rendering long-.
Linear vector fields and a few trajectories.Maps
A discrete-time, affine In general, an affine transformation is composed of linear transformations and a translation (or "shift"). Several linear transformations can be combined into a single one, so that the general formula given above is still applicable dynamical system has the form
with A a matrix and b a vector. As in the continuous case, the change of coordinates x → 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 linear system A nx0. 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 α u1, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point.
There are also many other discrete dynamical systems.
Local dynamics
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.
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The . Dynamical Systems. group at Hasselt University has a strong background on singular perturbation problems. The candidate is expected to collaborate with different members of the group, as well as with the co-advisor at the University ...
