PDF On Jan 1, 2003, Paul Glazier and others published Dynamical systems theory: A relevant framework for performance-oriented sports biomechanics research. 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.
The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.
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- Bifurcation theory 19 2.5. Saddle-node bifurcation 20 2.6. Transcritical bifurcation 21 2.7. Pitchfork bifurcation 21 2.8. The implicit function theorem 22 2.9. Buckling of a rod 26 2.10. Imperfect bifurcations 26 2.11. Dynamical systems on the circle 27 2.12. Discrete dynamical systems 28 2.13. Bifurcations of xed points 30 2.14.
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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 Φt 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 Φt 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 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 spaceTxM of the point x.) Given a smooth Φt, 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 Φt 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,
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 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:
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.
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.
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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
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, .., λν 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 λi – ∑ (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 Φt (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 Fμ can be characterized by the eigenvalues of the first derivative of the system DFμ(x0) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DFμ 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 Φt(A) and invariance of the phase space means that
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 Ut, the transfer operator,
By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φt. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φt 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?'
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 , with a manifold (locally a Banach space or Euclidean space), the domain for time (non-negative reals, the integers, ..) and f an evolution rule t → ft (with ) such that f t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain 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 .
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 . A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has . 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 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.
See also[edit]
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.
Further reading[edit]
Works providing a broad coverage:
- Ralph Abraham and Jerrold E. Marsden (1978). Foundations of mechanics. Benjamin–Cummings. ISBN978-0-8053-0102-1. (available as a reprint: ISBN0-201-40840-6)
- Encyclopaedia of Mathematical Sciences (ISSN0938-0396) has a sub-series on dynamical systems with reviews of current research.
- Christian Bonatti; Lorenzo J. Díaz; Marcelo Viana (2005). Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer. ISBN978-3-540-22066-4.
- Stephen Smale (1967). 'Differentiable dynamical systems'. Bulletin of the American Mathematical Society. 73 (6): 747–817. doi:10.1090/S0002-9904-1967-11798-1.
Introductory texts with a unique perspective:
- V. I. Arnold (1982). Mathematical methods of classical mechanics. Springer-Verlag. ISBN978-0-387-96890-2.
- Jacob Palis and Welington de Melo (1982). Geometric theory of dynamical systems: an introduction. Springer-Verlag. ISBN978-0-387-90668-3.
- David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press. ISBN978-0-12-601710-6.
- Tim Bedford, Michael Keane and Caroline Series, eds. (1991). Ergodic theory, symbolic dynamics and hyperbolic spaces. Oxford University Press. ISBN978-0-19-853390-0.CS1 maint: multiple names: authors list (link)
- Ralph H. Abraham and Christopher D. Shaw (1992). Dynamics—the geometry of behavior, 2nd edition. Addison-Wesley. ISBN978-0-201-56716-8.
Textbooks
- Kathleen T. Alligood, Tim D. Sauer and James A. Yorke (2000). Chaos. An introduction to dynamical systems. Springer Verlag. ISBN978-0-387-94677-1.
- Oded Galor (2011). Discrete Dynamical Systems. Springer. ISBN978-3-642-07185-0.
- Morris W. Hirsch, Stephen Smale and Robert L. Devaney (2003). Differential Equations, dynamical systems, and an introduction to chaos. Academic Press. ISBN978-0-12-349703-1.
- Anatole Katok; Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN978-0-521-57557-7.
- Stephen Lynch (2010). Dynamical Systems with Applications using Maple 2nd Ed. Springer. ISBN978-0-8176-4389-8.
- Stephen Lynch (2014). Dynamical Systems with Applications using MATLAB 2nd Edition. Springer International Publishing. ISBN978-3319068190.
- Stephen Lynch (2017). Dynamical Systems with Applications using Mathematica 2nd Ed. Springer. ISBN978-3-319-61485-4.
- Stephen Lynch (2018). Dynamical Systems with Applications using Python. Springer International Publishing. ISBN978-3-319-78145-7.
- James Meiss (2007). Differential Dynamical Systems. SIAM. ISBN978-0-89871-635-1.
- David D. Nolte (2015). Introduction to Modern Dynamics: Chaos, Networks, Space and Time. Oxford University Press. ISBN978-0199657032.
- Julien Clinton Sprott (2003). Chaos and time-series analysis. Oxford University Press. ISBN978-0-19-850839-7.
- Steven H. Strogatz (1994). Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering. Addison Wesley. ISBN978-0-201-54344-5.
- Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN978-0-8218-8328-0.
- Stephen Wiggins (2003). Introduction to Applied Dynamical Systems and Chaos. Springer. ISBN978-0-387-00177-7.
Popularizations:
- Florin Diacu and Philip Holmes (1996). Celestial Encounters. Princeton. ISBN978-0-691-02743-2.
- James Gleick (1988). Chaos: Making a New Science. Penguin. ISBN978-0-14-009250-9.
- Ivar Ekeland (1990). Mathematics and the Unexpected (Paperback). University Of Chicago Press. ISBN978-0-226-19990-0.
- Ian Stewart (1997). Does God Play Dice? The New Mathematics of Chaos. Penguin. ISBN978-0-14-025602-4.
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.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Dynamical_system&oldid=914560182'
Published online 2012 Oct 4. doi: 10.3389/fpsyg.2012.00382
PMID: 23060844
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Dynamical Systems Theory (DST) has generated interest and excitement in psychological research, as demonstrated by the recent statement, “…the dynamical perspective has emerged as a primary paradigm for the investigation of psychological processes at different levels of personal and social reality” (Vallacher et al., , p. 263).
What is less clear to the authors is the degree to which this excitement is justified. Like many psychology researchers, we were initially unfamiliar with the concepts, terminology, and techniques used in DST modeling (an approach that was developed in physics), which made it difficult to judge applications of DST in the articles we encountered. After reading introductory DST material, we developed some opinions about how authors of DST-related articles could help psychologists who are not familiar with DST [hereafter referred to as “lay reader(s)”] to begin to make judgments about their work. In order for DST to be a useful methodology for psychology research, we believe, DST-based work must be reasonably accessible to other psychologists.
Although we are restricting our discussion to the application of DST methodology to clinical psychology research, we believe that the following three recommendations may be applied to the field of psychology more broadly.
Maintain a Distinction between Dynamical and Non-Dynamical Models
A defining feature of a dynamical model is that the values of the variables in a dynamical system at one time are modeled as functions of those same variables at earlier times. One characteristic, therefore, that distinguishes dynamical models from the statistical models commonly applied in clinical psychology research is that in dynamical models the same variables serve, in a sense, as both dependent and independent variables. Another way of saying this is that dynamical systems are, by definition, feedback models.
For example, X(t + 1) = aX(t) constitutes an extremely simple dynamical system with one variable (X) and a constant (the coefficient a) that, multiplied by X at time t, defines X at time t + 1.
In contrast, models in which dependent variables are distinct from independent variables, such as OLS regression and hierarchical linear modeling (HLM, which can also be used to perform non-linear growth curve analyses), are not feedback models, and thus are not dynamical systems.
In a 1994 review article, Barton seemed to blur this distinction, implying that all statistical models are dynamical and differ primarily in whether they involve linear or non-linear equations.
From a mathematical perspective, dynamics can be thought of as linear or non-linear… Linear equations… are… the cornerstone of statistics. When we perform an analysis of variance or enter data into a multiple regression equation, we are using linear equations to describe the relationships among variables (pp. 5–6).
In a response to Barton (), Mandel (1995, 107) clarified the distinction between “dynamical and static approaches (as well as linear and non-linear models),”such that OLS regression, for example, would be considered “linear static” and non-linear growth curve analysis via HLM would be considered “non-linear static.” However, this distinction is not always clearly maintained in the psychology literature.
For example, Hayes et al. () related their study, in which they examined non-linear trajectories of depression change during treatment, to DST, although there were no dynamical components to their model. That is, their dependent and independent variables were distinct (i.e., no feedback), and their data analyses were conducted via “static” approaches (non-linear growth curve analysis via HLM). Nonetheless, they described the focus of their study using DST terminology (e.g., “critical fluctuations,” p. 410). By doing so, they may have led lay readers to conclude, incorrectly, that the trajectories of depression change they reported fit into a DST framework, that their analyses constituted applications of DST theory and methods to clinical psychology, and that judgments of the presented research would be relevant to judging the usefulness of DST in clinical psychology research.
To maintain the distinction between dynamical and non-dynamical models, researchers reporting on non-dynamical models can simply omit any reference to DST. Researchers presenting non-dynamical models who choose to refer to DST terminology should explicitly state that their models are not dynamical, and, furthermore, should make clear what the relevance of DST is to the presented research. For example, are DST concepts being presented metaphorically? Do the researchers speculate that a dynamical process underlies their data, but refrain from examining a dynamical model? If so, what evidence supports the speculation, and why is a dynamical model not investigated? Maintaining a clear distinction between dynamical and non-dynamical models will assist the lay reader in making judgments about the usefulness of DST in clinical psychology research.
Maintain a Clear Distinction between Influences on the Variables from the Proposed Model and Other Influences.
DST, by its nature, involves the study of processes that unfold over time in a deterministic manner (absent any perturbations), from an initial state, based solely on the functional relationships among the variables in the system. In the context of clinical psychology, it may be difficult to identify variables that operate in such a deterministic manner or to construct models that adequately characterize their interactions. Difficulties may arise from a number of sources, including the intentional actions of participants and the difficulties in isolating psychological variables from the myriad environmental influences that affect human beings. Unless DST researchers explicitly state otherwise, lay readers may assume that any influences mentioned by the researchers originate from the proposed model, and thus be unable to accurately assess the usefulness of the model. Therefore, when researchers discuss the influences on variables they examine, we believe that it is incumbent on them to be particularly careful in distinguishing between those influences arising from the proposed model and other influences.
For example, Peluso et al. (, 51), presented non-linear dynamical models of the changes over time of psychotherapist and client emotional valences, in which each participant’s emotional valence was modeled as deterministic functions of both the other participant’s emotional valence and their own emotional valence at the immediately preceding time.
The authors also suggested that psychotherapists be “mindful of” and “monitor” these valences and how strongly they impact one another, presumably with the idea that the psychotherapists would adjust their own values in order to improve therapy outcomes. Implicit, then, was an assumption that there were two sources of influence on psychotherapist emotional valence – one from a dynamical system, in which the emotional valence changes deterministically, and the other from outside of the dynamical system, involving direct volitional changes. However, because the authors did not make this distinction explicit, the lay reader may assume, incorrectly, that volitional changes in variables are consistent with their model, when in fact they are inconsistent both with the specific model and also with the deterministic framework of DST more generally.
In a different type of example, Chow et al. () used a linear dynamical model to describe periodic fluctuations in hedonic level. Empirical results showed a weekly periodicity in hedonic level; specifically, the undergraduate subjects in the study, who were studied in their natural environment, enjoyed themselves more on weekends than on weekdays.
Dynamical System Theory
In this case, the hypothesized dynamical model posited was a simple model in which hedonic level at one time varied only in relation to hedonic level at a previous time. However, the fact that the hedonic cycle appeared to be entrained to a weekly calendar cycle suggests that other influences may have been at work (i.e., behavioral demands are different on weekdays versus weekends for students). Because Chow et al. () did not explicitly state that the entrainment of periodic hedonic fluctuations to an external environmental (calendar) cycle was not part of their model, the lay reader may assume, incorrectly, that it was.
When researchers present DST-related work, it might be helpful for them to include two separate sections in the discussion: one for influences on the variables that arise from the proposed model; and another for other influences. This would put lay readers in a better position to judge the usefulness of the model, and thus to better evaluate the role of DST in psychology research.
Include a Time Series Plot from the Empirical Data or the Model, and if Both are Available, Show Them Together
Dynamical systems involve changes in variables that unfold over time. Although the graphical techniques specific to DST are important to include because they show specific DST-related properties of models, we think that it is also useful to include time series plots to help the reader conceptualize the model, to make judgments about the plausibility of the model, and, where both modeled and empirical data are available and plotted together, to make judgments about how well the model fits the data.
When results are presented from simulations in the absence of empirical data (e.g., Peluso et al., ), showing a few representative time series plots of the simulated data could help readers gain a better sense of how the patterns described in other ways (e.g., other plots, text descriptions) would look in empirical data. Readers might be able to get some sense of the plausibility of the model based on the look of these simulated time series plots. When empirical data have been collected (e.g., Cook et al., 1995; Gottman et al., 1999; Chow et al., ; Boker and Laurenceau, 2006; Fisher et al., ), presenting time series plots of these data along with time series plots generated from the models (on the same scale), would allow the reader to get a visual sense of how well the models fit. If different models are fit to the same data (e.g., Hufford et al., ; Witkiewitz et al., ), a time series for each model should be included. The time series plots should be generated at the same level (e.g., individual, group) for which the dynamical system variables are described in the model.
In our opinion, because time series plots do not require technical expertise for interpretation, showing empirical and modeled time series plots together is a way of presenting results that is particularly accessible to lay readers. While we did not encounter this kind of presentation in any of the articles relevant to clinical psychology that we looked at, an example from the biological sciences can be found in Figures 4B and 4D of Shiferaw et al. (). In these figures, an empirical time series plot of calcium transients in a stimulated rat cardiac muscle cell is shown alongside a corresponding time series plot generated from a non-linear dynamical model. Despite slight differences between the empirical and model plots, we believe that the overwhelming resemblance of the two plots provides a compelling illustration, even to readers with no knowledge of DST or cell biology, of the excellent match of the dynamical model to the empirical data. We believe that similar presentations in psychology articles would provide much clearer evidence than model fit statistics, or other statistical measures, of the value of dynamical models.
Conclusion
Dynamical Systems Theory Pdf File
Is DST a useful approach for clinical psychology research? Has it already made contributions to the field? We are not sure, and believe that it is impossible, at this point, for non-expert readers to determine.
We hope that by providing clearer information about the role of DST models in their work, and about the fit of their models to data, researchers applying DST to psychological variables will better enable the psychology research community to answer these questions.
Acknowledgments
We wish to thank Nicholas Forand, Steven Hollon, Louis Littman, and Robert Rusling for their comments and suggestions, and Martin Gelfand and Jeff Gerecht for verifications and clarifications regarding dynamical systems theory.
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