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.
50-631: Hinke Maria Osinga (born 25 December 1969) is a Dutch mathematician and an expert in dynamical systems . She works as a professor of applied mathematics at the University of Auckland in New Zealand. As well as for her research, she is known as a creator of mathematical art . Osinga earned a master's degree in 1991 and a Ph.D. in 1996 from the University of Groningen . Her doctoral dissertation, jointly supervised by dynamical systems theorist Henk Broer and computational geometer Gert Vegter,
100-420: 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 . 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
150-467: A flow , the vector field v( x ) is an affine 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 : Cliodynamics Cliodynamics ( / ˌ k l iː oʊ d aɪ ˈ n æ m ɪ k s / )
200-401: 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 if we take one of
250-421: 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
300-463: 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,
350-469: 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
400-522: 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. 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 . Given
450-463: 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
500-632: 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
550-420: 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
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#1732790117955600-445: 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: Many people regard French mathematician Henri Poincaré as
650-626: A stainless steel sculpture that provides another interpretation of the same mathematical system. Osinga was an invited speaker at the International Congress of Mathematicians in 2014, speaking on "Mathematics in Science and Technology". In 2015 she was elected as a fellow of the Society for Industrial and Applied Mathematics "for contributions to theory and computational methods for dynamical systems." In October 2016 she became
700-498: A wide range of historical processes. This typically involves building massive stores of evidence. The rise of digital history and various research technologies have allowed huge databases to be constructed in recent years. Some prominent databases utilized by cliodynamics practitioners include: As of 2016, the main directions of academic study in cliodynamics are: There are several established venues of peer-reviewed cliodynamics research: Critics of cliodynamics often argue that
750-404: Is infinite-dimensional . 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 . A discrete dynamical system , discrete-time dynamical system is a tuple ( T , M , Φ), where M is a manifold locally diffeomorphic to
800-413: Is non-empty , compact and simply connected . 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
850-478: Is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain T {\displaystyle {\mathcal {T}}} 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 T {\displaystyle {\mathcal {T}}} . A real dynamical system , real-time dynamical system , continuous time dynamical system , or flow
900-454: Is a transdisciplinary area of research that integrates cultural evolution , economic history / cliometrics , macrosociology , the mathematical modeling of historical processes during the longue durée , and the construction and analysis of historical databases. Cliodynamics treats history as science. Its practitioners develop theories that explain such dynamical processes as the rise and fall of empires, population booms and busts , and
950-399: 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 , the dynamical system is finite-dimensional ; if not, the dynamical system
1000-420: 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
1050-507: Is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ − 1 σ ∈ Σ {\displaystyle \Phi ^{-1}\sigma \in \Sigma } . A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has μ ( Φ − 1 σ ) = μ ( σ ) {\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )} . Combining
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#17327901179551100-487: Is so called because the whole phenomenon is represented as a system consisting of several elements (or subsystems ) that interact and change dynamically (i.e., over time). More simply, it consists of taking a holistic phenomenon and splitting it up into separate parts that are assumed to interact with each other. In the dynamical systems approach, one sets out explicitly with mathematical formulae how different subsystems interact with each other. This mathematical description
1150-446: Is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. M {\displaystyle {\mathcal {M}}} 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 (with t ∈ T {\displaystyle t\in {\mathcal {T}}} ) such that f
1200-561: Is the model of the system, and one can use a variety of methods to study the dynamics predicted by the model, as well as attempt to test the model by comparing its predictions with observed empirical , dynamic evidence. Although the focus is usually on the dynamics of large conglomerates of people, the approach of cliodynamics does not preclude the inclusion of human agency in its explanatory theories. Such questions can be explored with agent-based computer simulations. Cliodynamics relies on large bodies of evidence to test competing theories on
1250-594: Is the study of how and why phenomena change with time. The term was originally coined by Peter Turchin in 2003, and can be traced to the work of such figures as Ibn Khaldun , Alexandre Deulofeu , Jack Goldstone , Sergey Kapitsa , Randall Collins , John Komlos , and Andrey Korotayev . Many historical processes are dynamic , in that they change with time: populations increase and decline, economies expand and contract, states grow and collapse, and so on. As such, practitioners of cliodynamics apply mathematical models to explain macrohistorical patterns—things like
1300-424: 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 where G : ( T × M ) M → C {\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} } is a functional from the set of evolution functions to
1350-468: 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
1400-607: 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
1450-478: The edge of chaos concept. 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
1500-653: 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 Φ n = Φ ∘ Φ ∘ ⋯ ∘ Φ {\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi } for every integer n are studied. For continuous dynamical systems,
1550-460: 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 ). For
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1600-532: The complex social formations of the past cannot and should not be reduced to quantifiable, analyzable 'data points', for doing so overlooks each historical society's particular circumstances and dynamics. Many historians and social scientists contend that there are no generalisable causal factors that can explain large numbers of cases, but that historical investigation should focus on the unique trajectories of each case, highlighting commonalities in outcomes where they exist. As Zhao notes, "most historians believe that
1650-908: 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 . 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 with and for any x in X : for t 1 , t 2 + t 1 ∈ I ( x ) {\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)} and t 2 ∈ I ( Φ ( t 1 , x ) ) {\displaystyle \ t_{2}\in I(\Phi (t_{1},x))} , where we have defined
1700-488: 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
1750-408: 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
1800-670: The first female mathematician elected to the Royal Society of New Zealand. She was awarded the Aitken Lectureship in 2017. In 2017 Osinga was selected as one of the Royal Society Te Apārangi's " 150 women in 150 words ", celebrating the contributions of women to knowledge in New Zealand. The same year she received the Moyal Medal from Macquarie University. Dynamical system At any given time,
1850-602: 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. In the geometrical definition, a dynamical system is the tuple ⟨ T , M , f ⟩ {\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle } . T {\displaystyle {\mathcal {T}}}
1900-440: The following: where 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 The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above The dynamical system
1950-408: 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
2000-412: The importance of any mechanism in history changes, and more importantly, that there is no time-invariant structure that can organise all historical mechanisms into a system." Starting in the 1940s, Isaac Asimov invented the fictional precursor to this discipline, in what he called psychohistory , as a major plot device in his Foundation series of science fiction novels Robert Heinlein wrote
2050-513: The map Φ is understood to be a finite time evolution map and the construction is more complicated. 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
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2100-464: The observed statistics of hyperbolic 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
2150-564: 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
2200-417: The rise of empires , social discontent , civil wars , and state collapse . Cliodynamics is the application of a dynamical systems approach to the social sciences in general and to the study of historical dynamics in particular. More broadly, this approach is quite common and has proved its worth in innumerable applications (particularly in the natural sciences ). The dynamical systems approach
2250-582: The second in New Zealand. In 2004 Osinga created a crocheted visualization of the Lorenz manifold, an invariant manifold for the Lorenz system , and published the crochet pattern for her work with her husband Bernd Krauskopf ; the resulting mathematical textile artwork involved over 25,000 crochet stitches, and measured nearly a meter across. Osinga and Krauskopf later collaborated with artist Benjamin Storch on
2300-551: The set I ( x ) := { t ∈ T : ( t , x ) ∈ U } {\displaystyle I(x):=\{t\in T:(t,x)\in U\}} for any x in X . In particular, in the case that U = T × X {\displaystyle U=T\times X} we have for every x in X that I ( x ) = T {\displaystyle I(x)=T} and thus that Φ defines
2350-442: The spread and disappearance of religions. These theories are translated into mathematical models. Finally, model predictions are tested against data. Thus, building and analyzing massive databases of historical and archaeological information is one of the most important goals of cliodynamics. The word cliodynamics is composed of clio- and -dynamics . In Greek mythology, Clio is the muse of history. Dynamics , most broadly,
2400-511: 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
2450-496: The variables as constant. The function is called the flow through x and its graph is called the trajectory through x . The set is called the orbit through x . 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 Thus, in particular, if S is Φ- invariant , I ( x ) = T {\displaystyle I(x)=T} for all x in S . That is,
2500-517: Was on the computation of invariant manifolds . After postdoctoral studies at The Geometry Center and the California Institute of Technology , and a short-term lecturership at the University of Exeter , she became a lecturer at the University of Bristol in 2001, and was promoted to reader and professor there in 2005 and 2011, respectively. She moved to Auckland in 2011, becoming the first female mathematics professor at Auckland and
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