Dynamical Systems Theory

Using a non-philosophical methodology, I propose to track the theological origins of the concept of trope through hermeneutics and into deconstruction, turning to biology for a recuperation of the root tropos as movement-towards. This movement-towards will be argued as a metaphor-in-the-last-instance of its unilateral relation with the real, extended into language and abstraction through the indication of movement and valuation in relation to environmental gradients. I argue that the focus on movement is a key non-philosophical strategy, as it necessitates a description that “requires its constant recasting  [and is] not determined but indefinitely rectified.” Movement’s perceptible necessity is change, a phenomenon that is describable, but never as a stable scene or calcified order.

Also inherent in this instance of movement is an allowance for the abstraction of that description of change in order to construct adaptive value environments through which to navigate and understand change. The evolutionary development of the organism requires this abstraction, which is determined by its environmentally devised tropisms, the most basic being geo-, photo-, and hydrotropism. The initiation of the trope through movement is a base form of conflationary description of the real with abstraction. This form will be used to reassess analogous gestures, in value and language. Its interrogation is necessary to think the dyadic philosophical relation of “the object and the thought of the object….the real and language in the mixture of the logos.” Its continued use and morphology throughout textual philosophical practices is further evidence of the trope’s “prohibition of knowledge in the name of interpretation.” Using dynamical systems theory to describe multiple field gradients, and the movements within them, I will attempt to apply this methodology to abstract metaphor environments. 

From a turn-towards, to a turn-towards-God, to a turn-of-phrase, the trope exemplifies the exploitation of the real through philosophy. This exploitation substitutes a constructed ideological value environment for that of the initial external biological stimulus, in a coercive attempt to manipulate various movements-towards. I argue that by reducing the trope to its biological constitutive element, that of the pivot, a perceivable reaction to the external stimulus of the real, we can both trace the monumental inflation of its extracted value from the real, and begin to attempt a methodology based on its determination in the last instance.

Gibson developed the affordance concept to complement his theory of direct perception that stands in sharp contrast with the prevalent inferential theories of perception. A comparison of the two approaches shows that the distinction between them also has an ontological aspect. We trace the history and newer formalizations of the notion of affordance and discuss some competing opinions on its scope. Next, empirical work on the affordance concept is reviewed in brief and the relevance of dynamical systems theory to affordance research is demonstrated. Finally, the striking but often neglected convergence of the ideas of Gibson and those of certain Continental philosophers is discussed.

Megaproject Planning and Management: Essential Readings contains the seminal articles from the growing body of research on megaproject planning and management along with an original introduction by the editor, Bent Flyvbjerg. The leading and most cited authority in the field, Flyvbjerg has used crowdsourcing and 25 years of experience to cherry-pick from several hundred articles and books the writings that define the field. This volume will be an indispensable source for those wishing to speak with authority about how megaprojects are prepared, delivered, and fought over. The target audience is students, academics, practitioners, and media pundits alike, as well as communities affected by megaprojects.

Different approaches towards cognition and creativity will be discussed and compared. Unified models of cognition are presented, such as SOAR, ACT*, neural network theories and hybrid models. Only a small part of cognition research is concerned with creativity, such as the creative cognition approach and information-processing models of design and creativity. Several Artificial Intelligence (AI) models concerning creativity will be presented. Most AI-models of creativity address only combinational and explorational creativity. Only some models, like Hofstadter’s COPYCAT and models based on genetic algorithms can produce transformational creativity. Dynamical system theory seems to offer a powerful new framework to describe cognition and creativity. Dynamical system theory provides a model that can capture the fluid complexity of human creativity, wherein the creative concept space evolves in time.

Part I: Generalized Functions and Operator Theory An Introduction to Hyperfunctions and ?-expansions, G. Lumer Partial Inner Product Spaces of Analytic Functions, J.-P. Antoine Rigged Spectral States: A Proclivity for Eigenvalues, K. Gustafson Densities of Singular Measures and Generalized Spectral Decompositions, I. Antoniou and Z. Suchanecki Convolution Kernels and Generalized Functions, B. Baumer, G. Lumer, and F. Neubrander Spectral Theory of Closed Linear Operators on Banach Spaces from a Locally Convex Point of View, V. Wrobel Ultradistributions and the Levinson Condition, I. Cioranescu and L. Zsido Representation of the Derivatives and Products of the Delta Function in Hilbert Space, Yu. Melnikov Series Representations of the Complex Delta Function, I. Antoniou, Z. Suchanecki, and S. Tasaki Antieigenvalues: An Extended Spectral Theory, K. Gustafson Part II: Operator Theory and Dynamical Systems Laws of Nature, Probability, and Time Symmetry Breaking, I. Prigogine and T. Petrosky Extended Spectral Decompositions of Evolution Operators, I. Antoniou and S. Shkarin Some Little Things About Rigged Hilbert Spaces and Quantum Mechanics and All That, A. Bohm, M. Gadella, and S. Wickramasekara Axiomatics of Thermodynamics and Quantum Chaos, V. Maslov Stochastic Evolution on Product Manifolds. S. Albeverio, A. Daletskii, and Yu. Kondratiev Interaction Problems with Distributions and Hyperfunctions Data, G. Lumer Absolute Continuity of Convolutions of Singular Measures and New Branches of Spectrum of Liouvillians and Few-Body Hamiltonians, L Bos and B Pavlov On Scattering Theories Involving Moving Boundaries, G. F. Roach The Eigenvalue Problem for Networks of Beams, B Dekoninck and S Nicaise Generalized Perturbations and Operator Relations, P Kurasov and B Pavlov On Spectral Analysis of a Class of Integral-Difference Collision Operators, Yu. Melnikiv Dynamical Aspects of Processes with Long-Range Memory, M. Courbage

Ecological psychology has been criticized for ignoring the brain in its theory formation. In recent years, however, a number of researchers have started asking ecologically-inspired questions about the ways in which not only the embodied activity of the organism in its environment, but also the particulars of the organism's nervous system matter. This work has typically appeared in neuroscience journals, thereby potentially escaping the attention of ecological psychologists. The current article introduces a Special Issue of Ecological Psychology that aims to correct this problem. This Issue brings together one empirical and six theoretical articles that develop ideas about the contributions of the nervous system to perception-action. We briefly review each of the articles, identify common themes, and point out the surprising variety in theoretical positions. It is hoped that this Special Issue will help guide discussions amongst ecological psychologists and (ecological) neuroscientists as they confront the question “What should a ‘Gibsonian neuroscience’ look like?”

Invoking turn-taking to explain conversations has long blocked progress in the field. Not only are there no logical or empirical grounds for saying conversation is constructed out of 'turns', but acoustic records show talk to be inseparable from how, in a microtemporal dimension, individuals orchestrate their words. Conversations are dialogical activity irreducible to sequences of forms: phonetic substance matters. Talking of 'turns' obscures the significance of how people act. By subordinating action to word-based patterns, talk comes to be conceptualized independently of timing. In advocating the contrary view that timing is central to talk, the paper highlights pitch matching. Using acoustic measures, persons are shown to orchestrate the pitch of their voices so that interindividual patterns embody the interpersonal sense of events. Observations about timing thus clarify what listeners hear when utterances are spoken in a particular sense.

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit's unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Twobody parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.

The behavior of some systems is noncomputable in a precise new sense.  One infamous problem is that of the stability of the solar system:  Given the initial positions and velocities of several mutually gravitating bodies, will any eventually collide or be thrown off to infinity?  Many have made vague suggestions that this and similar problems are undecidable:  No finite procedure can reliably determine whether a given configuration will eventually prove unstable.  But taken in the most natural way, this is trivial.  The state of a system corresponds to a point in a continuous space, and virtually no set of points in space is strictly decidable.  A new, more pragmatic concept is therefore introduced:  A set is decidable up to measure zero (d.m.z.) if there is a procedure to decide whether a point is in that set and it only fails on some points that form a set of zero volume.  This is motivated by the intuitive correspondence between volume and probability:  We can ignore a zero-volume set of states because the state of an arbitrary system almost certainly will not fall in that set.  D.m.z. is also closer to the intuition of decidability than other notions in the literature, which are either less strict or apply only to special sets, like closed sets.  Certain complicated sets are not d.m.z., most remarkably including the set of know stable orbits for planetary systems (the KAM tori).  This suggests that the stability problem is indeed undecidable in the precise sense of d.m.z.  Carefully extending decidability concepts from idealized models to actual systems, we see that even deterministic aspects of physical behaviour can be undecidable in a clear and significant sense.

Reciprocal communication between couples is central to sustaining strong intimate relationships. Given that Asperger's Syndrome (AS) affects communication and social reciprocity, adults with this disorder are vulnerable to experiencing difficulties in relating to their "neurotypical" (NT) partner. As reported in a previous paper, prompt dependency was found to be a compensatory mechanism for some of the communication difficulties within AS-NT relationships. This paper draws on the same data-set to describe the impact of prompt dependency on AS-NT relationships. The research reported here is also used to derive a theoretical model that illustrates how a cycle of prompt dependency results in a communication "roundabout" for partners. Implications for practice and further research are discussed.

We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of the simulation involves a simpler simulation of the simulation, still further simulations thereof, and a universality conjecture. There are difficulties with that argument, but there are other, heuristic arguments supporting the qualitative results. It is urged, using this example, that absolute validation, while highly desirable, is overvalued. Simulations also provide valuable insights that we cannot yet (if ever) prove.

Reciprocal communication between couples is central to sustaining strong intimate relationships. Given that Asperger's Syndrome (AS) affects communication and social reciprocity, adults with this disorder are vulnerable to experiencing difficulties in relating to their " neurotypical " (NT) partner. As reported in a previous paper, prompt dependency was found to be a compensatory mechanism for some of the communication difficulties within AS-NT relationships. This paper draws on the same data-set to describe the impact of prompt dependency on AS-NT relationships. The research reported here is also used to derive a theoretical model that illustrates how a cycle of prompt dependency results in a communication " roundabout " for partners. Implications for practice and further research are discussed.

We investigate a spatial predator-prey system; predator under Allee effect. The predator-prey community dynamics is assumed to be of Holling-Tanner type. When community's local dynamics displays sustained periodic oscillations, the system's spatio-temporal dynamics supports periodic traveling waves. Dispersal of species is assumed to be a random process in one-dimension only. Two dimensional (2D) parameter scan studies reveal that the system dynamics is self-organized at Edges of Chaos (EOC). This is a vital result as existence of EOC has not been demonstrated in a reaction-diffusion system yet. As population dynamics of all animals are governed by two basic processes: a) growth and b) dispersal, existence of EOC puts a recent theory of ecological chaos on robust footing. Role of Allee effect in enhancing chaos presents an opportunity for its application in control of biological invasions and conservation of small populations. Extinction-persistence transitions were also explored.

This article reviews the major contributions of dynamic systems theory (DST) in advancing thinking about development, the empirical insights the theory has generated, and the key challenges for the theory on the horizon. The first section discusses the emergence of DST in developmental science, the core concepts of the theory, and the resonance it has with other approaches that adopt a systems metatheory. The second section reviews the work of Esther Thelen and colleagues, who revolutionized how researchers think about the field of motor development. It also reviews recent extensions of this work to the domain of cognitive development. Here, the focus is on dynamic field theory, a formal, neurally grounded approach that has yielded novel insights into the embodied nature of cognition. The final section proposes that the key challenge on the horizon is to formally specify how interactions among multiple levels of analysis and across multiple time scales create developmental change.

The purpose of this paper is to analyse in how far the new field of embodied cognitive science is compatible with the traditional cognitive science program, with a side view on possible differences in the explanatory power between the two different programs.
Two conclusions are drawn. First, the various approaches in embodied cognitive science should be classified into two versions ‐ one of which is compatible with traditional approaches, the second of which is not. In particular, weak embodied theories are still compatible with the core of the traditional
program because they still share the latterʹs computational strategies. By contrast, radical embodied theories are not compatible with the traditional program since they completely abandon the computational approach in favour of the complex interactions between the system, its body and the world.
Second, weak and radical embodiment have different status. Weak embodied cognitive science seems likely to exceed the traditional program in explanatory power, because it can extend the pool of traditional concepts in interesting ways. By contrast, radical embodiment at current seems to be a theoretical position rather than a mature experimental research program, because it is in a process of conceptual and methodological clarification that predates substantial empirical research.

Reciprocal communication between couples is central to sustaining strong intimate relationships. Given that Asperger's Syndrome (AS) affects communication and social reciprocity, adults with this disorder are vulnerable to experiencing difficulties in relating to their " neurotypical " (NT) partner. As reported in a previous paper, prompt dependency was found to be a compensatory mechanism for some of the communication difficulties within AS-NT relationships. This paper draws on the same data-set to describe the impact of prompt dependency on AS-NT relationships. The research reported here is also used to derive a theoretical model that illustrates how a cycle of prompt dependency results in a communication " roundabout " for partners. Implications for practice and further research are discussed.

Many social relationships are a locus of struggle and suffering, either at the individual or interactional level. In this paper we explore why this is the case and suggest a modeling approach for dyadic interactions and the well-being of the participants. To this end we bring together an enactive approach to self with dynamical systems theory. Our basic assumption is that the quality of any social interaction or relationship fundamentally depends on the nature and constitution of the individuals engaged in these interactions. From an enactive perspective the self is conceived as an embodied and socially enacted autonomous system striving to maintain an identity. This striving involves a basic two-fold goal: the ability to exist as an individual in one's own right, while also being open to and affected by others. In terms of dynamical systems theory one can thus consider the individual self as a self-other organized system represented by a phase space spanned by the dimensions of...

has recently spearheaded a movement to challenge the dominance of connectionist and classicist models in cognitive science. The dynamical conception of cognition is van Gelder's replacement for the computation bound paradigms provided by connectionism and classicism. He relies on the Watt governor to fulfill the role of a dynamicist Turing machine and claims that the Motivational Oscillatory Theory (MOT) provides a sound empirical basis for dynamicism. In other words, the Watt governor is to be the theoretical exemplar of the class of systems necessary for cognition and MOT is an empirical instantiation of that class. However, I shall argue that neither the Watt governor nor MOT successfully fulfill these prescribed roles. This failure, along with van Gelder's peculiar use of the concept of computation and his struggle with representationalism, prevent him from providing a convincing alternative to current cognitive theories.

This authoritative treatment covers theory, optimal estimation and a range of practical applications. The first book on the subject, and written by leading researchers, this clear and rigorous work presents a comprehensive theory for both the stability boundary and the stability regions of a range of nonlinear dynamical systems including continuous, discrete, complex, two-timescale and non-hyperbolic systems, illustrated with numerical examples. The authors also propose new concepts of quasi-stability region and of relevant stability regions and their complete characterisations. Optimal schemes for estimating stability regions of general nonlinear dynamical systems are also covered, and finally the authors describe and explain how the theory is applied in applications including direct methods for power system transient stability analysis, nonlinear optimisation for finding a set of high-quality optimal solutions, stabilisation of nonlinear systems, ecosystem dynamics, and immunisation problems.

This paper challenges arguments that systematic patterns of intelligent behavior license the claim that representations must play a role in the cognitive system analogous to that played by syntactical structures in a computer program. In place of traditional computational models, I argue that research inspired by Dynamical Systems theory can support an alternative view of representations. My suggestion is that we treat linguistic and representational structures as providing complex multi-dimensional targets for the development of individual brains. This approach acknowledges the indispensability of the intentional or representational idiom in psychological explanation without locating representations in the brains of intelligent agents.

This paper proposes how the theoretical framework of ecological dynamics can provide an influential model of the learner and the learning process to pre-empt effective behaviour changes. Here we argue that ecological dynamics supports a well-established model of the learner ideally suited to the environmental education context because of its emphasis on the learner–environment relationship. The model stems from perspectives

Several theoretical studies deal with the stability transition curves of coupled and damped Mathieu equations utilizing numerical and asymptotic methods. In this contribution, we exploit the fact that symplectic maps describe the dynamics of Hamiltonian systems. Starting with a Hamiltonian system, a particular dissipation is introduced, which allows the extension of Hamiltonian and symplectic matrices to more general γ-Hamiltonian and μ-symplectic matrices. A proof is given that the state transition matrix of any γ-Hamiltonian system is μ-symplectic. Combined with Floquet theory, the symmetry of the Floquet multipliers with respect to a μ-circle, which is different from the unit circle, is highlighted. An attempt is made for generalizing the particular dissipation to a more general form. The methodology is applied for calculation of the stability transition curves of an example system of two coupled and damped Mathieu equations.

In 2016, a peace agreement was reached between the government of Colombia and the Revolutionary Armed Forces (FARC), a Marxist insurgency group, ending a 52-year old civil war that left around 220,000 people dead and nearly 6 million people displaced. The peace deal, one of the first of its kind to account for a gender perspective, was highly praised for its innovation and inclusive nature. Not only were women involved in the negotiation and drafting of the agreement, making up 20% of the government team and 43% of FARC delegates, but a gender sub-commission was also formed to review all peace documents from a gender focus. Accordingly, the final agreement dedicates an entire chapter to gender. Much of the literature points to the involvement of women as an indicator of success, focusing solely on the impact of conflict on women as well as the role of women as “peacebuilders.” While this paper does not attempt to disprove this claim, it widens the scope of this view, seeking to understand the overall cultural context that reinforces the gender norms that contribute to the nature of conflict. Such an analysis is necessary in order to properly assess the impact of gender-based intervention strategies on peace-building processes. While the research presented is confined to the bounded-system that is the country of Colombia, the findings may provide implications that could be applied to the peace-building strategies of other nations in the promotion of sustainable peace where the importance of gender in conflict analysis has not been fully realized.

This paper studies the concept of genericity for two dimensional dynamic systems. The basic model is the differential model, proposed by S. Smale, of constructing generic properties for diffeomorphisms defined on compact varieties . The generic property concept for dynamic systems defined on an open set G, is proposed. The aim of the construction is to test the conditions for which, in two-dimensional case, the structural stability of dynamic systems can be a generic property. 2000 Mathematical Subject Classification: 37C20 1 The differential model of S. Smale S. Smale [4] defined the dynamic systems by a diffeomorphism , f : M → M , with M compact differential bundle. Most of results are also available for dynamic systems defined by a first order ordinary differential equation.

During the last few decades the interest in Complexity Theory (CT) methods in sociology has increased, partly as a result of the increased complexity in sociological theory and partly of that these methods have become more powerful and accessible. However, despite this increased interest, the application of CT-methods still holds a peripheral existence within mainstream sociology. Based on a review of the contemporary discussion regarding computer-based modeling in sociology, it is argued that the reasons for this include a strong focus on the distinction between social and natural systems as well as a lack of connections from CT-models to existing theories. As an alternative to this, the application of more abstract models with the explicit purpose of theory development is suggested to complement traditional sociological methods. Ways of connecting models to theories to enable such theory development are proposed. This discussion is exemplified in a case study with the purpose of f...

We investigate a spatial predator-prey system; predator under Allee effect. The predator-prey community dynamics is assumed to be of Holling-Tanner type. When community's local dynamics displays sustained periodic oscillations, the system's spatio-temporal dynamics supports periodic traveling waves. Dispersal of species is assumed to be a random process in one-dimension only. Two dimensional (2D) parameter scan studies reveal that the system dynamics is self-organized at Edges of Chaos (EOC). This is a vital result as existence of EOC has not been demonstrated in a reaction-diffusion system yet. As population dynamics of all animals are governed by two basic processes: a) growth and b) dispersal, existence of EOC puts a recent theory of ecological chaos (Upadhyay, 2009; Rai and Upadhyay, 2006) on robust footing. Role of Allee effect in enhancing chaos presents an opportunity for its application in control of biological invasions and conservation of small populations. Extinction-persistence transitions were also explored.

The coordination between horizontal pelvic and thoracic rotations during treadmill walking was studied in ten subjects. Previous studies have considered the relative phase using the mean to characterize coordination mode, and the S.D. as an index of its stability. However, to use S.D. of relative phase as a measure for coordinative stability, the underlying oscillations have to be phase-locked at a certain value. Random fluctuations around this value can then be viewed as stochastic perturbations of a stable fixed point, resulting in a unimodal distribution of relative phase. Using methods of circular statistics this study shows that these conditions were not met in pelvis-thorax coordination. Spectral analyses revealed that, as walking velocity increased, a triphasic component emerged in the pelvic rotations, while the thoracic rotations remained harmonic across all walking velocities. These findings refute the use of standard relative phase measures to capture pelvis-thorax coordination. An alternative measure is introduced, namely the difference between the continuous Fourier phases of the component oscillations as determined for the main frequency of the thorax oscillation. With this measure, pelvis-thorax coordination was found to evolve from in-phase coordination towards antiphase coordination as walking velocity increased. This method may be used to assess reliably the properties of pelvis-thorax coordination in both healthy and pathological gait patterns in the future.

Gibson developed the affordance concept to complement his theory of direct perception that stands in sharp contrast with the prevalent inferential theories of perception. A comparison of the two approaches shows that the distinction between them also has an ontological aspect. We trace the history and newer formalizations of the notion of affordance and discuss some competing opinions on its scope. Next, empirical work on the affordance concept is reviewed in brief and the relevance of dynamical systems theory to affordance research is demonstrated. Finally, the striking but often neglected convergence of the ideas of Gibson and those of certain Continental philosophers is discussed.

To appear in A. Ericsson, N. Charness, P. Feltovich & R. Hoffman (Eds.), Cambridge Handbook on Expertise and Expert Performance. New York: Cambridge University Press, pp. 127-145. ... Observation of Work Practices in Natural Settings ... The Florida Institute for Human ...

The purpose of this study was to investigate the different coordination strategies used following obstacle clearance during running. Ten subjects ran over a level surface and over obstacles of six different heights (10, 12.5, 15, 17.5, 20 and 22.5% of their standing height). Analysis based upon the dynamical systems theory (DST) was used and the phasing relationships between lower extremity segments were examined. The results demonstrated that the increasing obstacle height elicited behavioral changes. The foot and the leg became more independent in their actions, while the leg and the thigh strengthened their already stable relationship. The 15% obstacle height seems to be a critical height for the observed changes.

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals.  We introduce such a notion, decidability in a measure μ (or d-μ), which is particularly appropriate for physics and in some ways more intuitive than Ko’s (1991) recursive approximability (r.a.). For Lebesgue measure λ, d-λ implies r.a.  Sets with positive λ- measure that are sufficiently “riddled” with holes are never d-λ but are often r.a.  This explicates Sommerer and Ott’s (1996) claim of uncomputable behavior in a system with riddled basins of attraction.  Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-λ.