Dynamical systems can be defined in a fairly abstract way, but we prefer to start with a few examples of historical importance before giving general definitions. This will allow us to specify the class of systems that we want to study, and to explain the differences between the.
Published online 2012 Oct 4. doi: 10.3389/fpsyg.2012.00382
PMID: 23060844
This article has been cited by other articles in PMC.
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.
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
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.
References
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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]
Dynamical Systems Theory Pdf Example
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]
3d max 2010 free download. 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.
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.
See also[edit]
Related subjects
Related scientists
Notes[edit]
^ abBoeing, 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
^ abSacco, 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.
^ abBlanchflower, 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.
Dynamical Systems Theory Pdf
Further reading[edit]
Abraham, Frederick D.; Abraham, Ralph; Shaw, Christopher D. (1990). A Visual Introduction to Dynamical Systems Theory for Psychology. Aerial Press. ISBN978-0-942344-09-7. OCLC24345312.
Beltrami, Edward J. (1998). Mathematics for Dynamic Modeling (2nd ed.). Academic Press. ISBN978-0-12-085566-7. OCLC36713294.
Hájek, Otomar (1968). Dynamical systems in the plane. Academic Press. OCLC343328.
Luenberger, David G. (1979). Introduction to dynamic systems: theory, models, and applications. Wiley. ISBN978-0-471-02594-8. OCLC4195122.
Michel, Anthony; Kaining Wang; Bo Hu (2001). Qualitative Theory of Dynamical Systems. Taylor & Francis. ISBN978-0-8247-0526-8. OCLC45873628.
Padulo, Louis; Arbib, Michael A. (1974). System theory: a unified state-space approach to continuous and discrete systems. Saunders. ISBN9780721670355. OCLC947600.
Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Addison Wesley. ISBN978-0-7382-0453-6. OCLC49839504.
Nonlinear Dynamics Pdf
External links[edit]
Dynamical Systems Theory Pdf Online
Dynamic Systems Encyclopedia of Cognitive Science entry.
Definition of dynamical system in MathWorld.
DSWeb Dynamical Systems Magazine
Dynamical Systems Theory Pdf Converter
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