Dynamical Systems Theory Pdf

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Systems theory also enables us to understand the components and dynamics of client systems in order to interpret problems and develop balanced inter- vention strategies, with the goal of enhancing the “goodness of fit” between individuals and their environments. Basic Theory of Dynamical Systems A Simple Example. Let us start oﬀby examining a simple system that is mechanical in nature. We will have much more to say about examples of this sort later on. Basic mechanical examples are often grounded in New-ton’s law, F = ma. For now, we can think of a as simply the acceleration.

Dynamical systems theory is an area of mathematics used to describe the behavior of the complexdynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

This theory deals with the long-term qualitative behavior of dynamical systems,^[1] and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.

The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.

3Concepts
4Related fields
5Applications

Overview[edit]

Dynamical systems theory and chaos theory deal 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 steady states?', or 'Does the long-term behavior of the system depend on its initial condition?'

An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.

Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos.^[2] The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.

History[edit]

The concept of dynamical systems theory 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.

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.

Some excellent presentations of mathematical dynamic system theory include (Beltrami 1990), (Luenberger 1979), (Padulo & Arbib 1974), and (Strogatz 1994).^[3]

Concepts[edit]

Dynamical systems[edit]

The dynamical system concept is a mathematical formalization for any fixed 'rule' that describes the time dependence of a point's position in its ambient space. Examples include the mathematical models 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.

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. 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—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 may be deterministic (for a given time interval only one future state follows from the current state) or stochastic (the evolution of the state is subject to random shocks).

Dynamicism[edit]

Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic cognition, is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues that differential equations are more suited to modelling cognition than more traditional computer models.

Nonlinear system[edit]

In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle.^[1] Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.

Related fields[edit]

Arithmetic dynamics[edit]

Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function.

Chaos theory[edit]

Chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.

Complex systems[edit]

Complex systems is a scientific field that studies the common properties of systems considered complex in nature, society, and science. It is also called complex systems theory, complexity science, study of complex systems and/or sciences of complexity. The key problems of such systems are difficulties with their formal modeling and simulation. From such perspective, in different research contexts complex systems are defined on the base of their different attributes.

The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neurosciences, social sciences, meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation, economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves.

Control theory[edit]

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with influencing the behavior of dynamical systems.

Ergodic theory[edit]

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.

Functional analysis[edit]

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.

Graph dynamical systems[edit]

The concept of graph dynamical systems (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.

Projected dynamical systems[edit]

Projected dynamical systems it is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.

Symbolic dynamics[edit]

Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.

System dynamics[edit]

System dynamics is an approach to understanding the behaviour of systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system.^[4] What makes using system dynamics different from other approaches to studying systems is the use of feedback loops and stocks and flows. These elements help describe how even seemingly simple systems display baffling nonlinearity.

Topological dynamics[edit]

Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.

Applications[edit]

In human development[edit]

In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence.^[5] Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).^[5]

According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from adolescence to young adulthood, and after reaching the critical point of 18 years of age (young adulthood), a peak in testosterone is observed in males^[6] and the period of optimal fertility begins in females.^[7] Similarly, at age 30 optimal fertility begins to decline in females,^[8] and at the stage transition from middle adulthood to older adulthood at 48 years, the average age of onset of menopause occurs.^[8]

These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA^[9] and self-organizing properties of the Fibonacci numbers that converge on the golden ratio.

Dynamical Systems Theory Pdf Examples

In biomechanics[edit]

In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems.^[10] There is no research validation of any of the claims associated to the conceptual application of this framework.

In cognitive science[edit]

Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the neo-Piagetian theories of cognitive development. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and AI. It also believed that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.

In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. Self-organization (the spontaneous creation of coherent forms) sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process. These links form the structure of a new state of order in the mind through a process called scalloping (the repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, idiosyncratic and unpredictable.^[11]

Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error.^[12]

In second language development[edit]

The application of Dynamic Systems Theory to study second language acquisition is attributed to Diane Larsen-Freeman who published an article in 1997 in which she claimed that second language acquisition should be viewed as a developmental process which includes language attrition as well as language acquisition.^[13] In her article she claimed that language should be viewed as a dynamic system which is dynamic, complex, nonlinear, chaotic, unpredictable, sensitive to initial conditions, open, self-organizing, feedback sensitive, and adaptive.

Notes[edit]

^ ^a^bBoeing, G. (2016). 'Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction'. Systems. 4 (4): 37. arXiv:1608.04416. doi:10.3390/systems4040037. Retrieved 2016-12-02.
^Grebogi, C.; Ott, E.; Yorke, J. (1987). 'Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics'. Science. 238 (4827): 632–638. doi:10.1126/science.238.4827.632. JSTOR1700479.
^Jerome R. Busemeyer (2008), 'Dynamic Systems'. To Appear in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008. Archived June 13, 2008, at the Wayback Machine
^MIT System Dynamics in Education Project (SDEP)Archived 2008-05-09 at the Wayback Machine
^ ^a^bSacco, R.G. (2013). 'Re-envisaging the eight developmental stages of Erik Erikson: The Fibonacci Life-Chart Method (FLCM)'. Journal of Educational and Developmental Psychology. 3 (1): 140–146. doi:10.5539/jedp.v3n1p140.
^Kelsey, T. W. (2014). 'A validated age-related normative model for male total testosterone shows increasing variance but no decline after age 40 years'. PLoS One. 9 (10): e109346. doi:10.1371/journal.pone.0109346. PMC4190174.
^Tulandi, T. (2004). Preservation of fertility. Taylor & Francis. pp. 1–20.
^ ^a^bBlanchflower, D. G. (2008). 'Is well-being U-shaped over the life cycle?'. Social Science & Medicine. 66 (8): 1733–1749. CiteSeerX10.1.1.63.5221. doi:10.1016/j.socscimed.2008.01.030. PMID18316146.
^Perez, J. C. (2010). (2010). 'Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618'. Interdisciplinary Sciences: Computational Life Sciences. 2 (3): 228–240. doi:10.1007/s12539-010-0022-0. PMID20658335.
^Paul S Glazier, Keith Davids, Roger M Bartlett (2003). 'DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research'. in: Sportscience 7. Accessed 2008-05-08.
^Lewis, Mark D. (2000-02-25). 'The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development'(PDF). Child Development. 71 (1): 36–43. CiteSeerX10.1.1.72.3668. doi:10.1111/1467-8624.00116. PMID10836556. Retrieved 2008-04-04.
^Smith, Linda B.; Esther Thelen (2003-07-30). 'Development as a dynamic system'(PDF). Trends in Cognitive Sciences. 7 (8): 343–8. CiteSeerX10.1.1.294.2037. doi:10.1016/S1364-6613(03)00156-6. PMID12907229. Retrieved 2008-04-04.
^'Chaos/Complexity Science and Second Language Acquisition'. Applied Linguistics. 1997.

External links[edit]

Dynamic Systems Encyclopedia of Cognitive Science entry.
Definition of dynamical system in MathWorld.
DSWeb Dynamical Systems Magazine

Retrieved from 'https://en.wikipedia.org/w/index.php?title=Dynamical_systems_theory&oldid=917647092'

The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (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.^[1]^[2] 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.'^[3] 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 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,^[4]^[5] biology,^[6] chemistry, engineering,^[7] economics,^[8] 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.

3Basic definitions
4Linear dynamical systems
5Local dynamics
7Ergodic systems

Overview[edit]

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, 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.

History[edit]

Many people regard 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 dynamic 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 SystemsBirkhoff'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 is 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.

Basic definitions[edit]

A dynamical system is a manifoldM called the phase (or state) space endowed with a family of smooth evolution functions Φ^t that for any element of t ∈ T, 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[edit]

The evolution function Φ^t is often the solution of a differential equation of motion

{displaystyle {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 fieldv(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 spaceT_xM 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:

{displaystyle G(x,{dot {x}})=0}

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

Dynamical Systems Theory Pdf Free

The differential equations determining the evolution function Φ^t 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—those that are locally Banach spaces—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.

Further examples[edit]

Baker's map is an example of a chaotic piecewise linear map
Billiards and outer billiards

Linear dynamical systems[edit]

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[edit]

For a flow, the vector field Φ(x) is an affine function of the position in the phase space, that is,

{displaystyle {dot {x}}=phi (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.

Maps[edit]

A discrete-time, affine dynamical system has the form of a matrix difference equation:

{displaystyle x_{n+1}=Ax_{n}+b,}

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 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.

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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 αu₁, 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[edit]

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.

Rectification[edit]

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) ≠ 0, then there is a change of coordinates 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(x) = 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 the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

Near periodic orbits[edit]

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 γ and consider the points in phase space in that neighborhood that are perpendicular to v(x₀). These points are a Poincaré sectionS(γ, x₀), of the orbit. The flow now defines a map, the Poincaré mapF : S → 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 · x + O(x²), so a change of coordinates h can only be expected to simplify F to its linear part

{displaystyle h^{-1}circ Fcirc h(x)=Jcdot 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 to be analytic and in the process discovered the non-resonant condition. If λ₁, .., λ_ν 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 λ_i – ∑ (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[edit]

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 Hartman–Grobman theorem 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 · 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 Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.

Bifurcation theory[edit]

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.

Ergodic systems[edit]

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.

Nonlinear dynamical systems and chaos[edit]

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly 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?'

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Note that the chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—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.

Geometrical definition[edit]

A dynamical system is the tuple ${displaystyle langle {mathcal {M}},f,{mathcal {T}}rangle }$ , with ${displaystyle {mathcal {M}}}$ a manifold (locally a Banach space or Euclidean space), ${displaystyle {mathcal {T}}}$ the domain for time (non-negative reals, the integers, ..) and f an evolution rule t → f^t (with ${displaystyle tin {mathcal {T}}}$ ) such that f^t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain ${displaystyle {mathcal {T}}}$ 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 ${displaystyle {mathcal {T}}}$ .

Measure theoretical definition[edit]

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X, Σ, μ, τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X, Σ, μ) is a probability space. A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has ${displaystyle tau ^{-1}sigma in Sigma }$ . A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has ${displaystyle mu (tau ^{-1}sigma )=mu (sigma )}$ . Combining the above, a map τ 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 quadruple (X, Σ, μ, τ), for such a τ, is then defined to be a dynamical system.

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates ${displaystyle tau ^{n}=tau circ tau circ cdots circ tau }$ for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

Examples of dynamical systems[edit]

Baker's map is an example of a chaotic piecewise linear map
Billiards and Outer billiards

Multidimensional generalization[edit]

Dynamical systems are defined over a single independent variable, usually 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.

References[edit]

^Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.
^Katok, A.; Hasselblatt, B. (1995). Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press. ISBN978-0-521-34187-5.
^'Nature'. Springer Nature. Retrieved 17 February 2017.
^Melby, P.; et al. (2005). 'Dynamics of Self-Adjusting Systems With Noise'. Chaos: An Interdisciplinary Journal of Nonlinear Science. 15 (3): 033902. Bibcode:2005Chaos.15c3902M. doi:10.1063/1.1953147. PMID16252993.
^Gintautas, V.; et al. (2008). 'Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics'. J. Stat. Phys. 130. arXiv:0705.0311. Bibcode:2008JSP..130.617G. doi:10.1007/s10955-007-9444-4.
^Jackson, T.; Radunskaya, A. (2015). Applications of Dynamical Systems in Biology and Medicine. Springer.
^Kreyszig, Erwin (2011). Advanced Engineering Mathematics. Hoboken: Wiley. ISBN978-0-470-64613-7.
^Gandolfo, Giancarlo (2009) [1971]. Economic Dynamics: Methods and Models (Fourth ed.). Berlin: Springer. ISBN978-3-642-13503-3.
^Holmes, Philip. 'Poincaré, celestial mechanics, dynamical-systems theory and 'chaos'.' Physics Reports 193.3 (1990): 137-163.

External links[edit]

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.
[1], 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.

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Dynamical systems[edit]

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Control theory[edit]

Ergodic theory[edit]

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Nonlinear dynamical systems and chaos[edit]

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Multidimensional generalization[edit]

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