In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. 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.
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, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.
Dynamical systems can be categorized according to their characteristics: discrete, continuous differentiable, smooth, deterministic, ergodic, stochastic or chaotic. Properties of dynamical systems that are investigated include: the existence and uniqueness of solutions; integrability (i.e. the existence of conserved quantities).
Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems.One of the first to get the intuition of numerical computations for weather forecasting is Richardson, he imagined a set of human people doing computations 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:
Hénon map Irrational rotation Kaplan–Yorke map Lorenz system Quadratic map simulation system Rössler map Swinging Atwood's machine Tent map
Any mathematical map can be treated as the definition of a dynamical system for example:
List of chaotic maps
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.Dynamical systems, Oliver Knill, Harvard University, Spring semester, 2005 https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf Classes of Dynamical systems pp 6 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.Dynamical systems, Oliver Knill, Harvard University, Spring semester, 2005 https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf A FANCY DEFINITION pp 6
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).Dynamical Systems in Number Theory https://www.math.uzh.ch/gorodnik/dyn_num/index.html
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.Here a few examples where time is generalized into real, complex or discrete n-dimensional vector spaces https://mathoverflow.net/questions/105071/dynamical-systems-with-multidimensional-complex-and-other-exotic-kinds-of-time
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.