Session 6A

Dynamical systems theory (DST) is a mathematical theory. In that form DST has been applied to many fields in the natural sciences and engineering. As several authors have shown (see Feldman 2019, Mitchell 2009, Strogatz 2015), DST can in principle be applied to fields that are amenable to mathematical modelling and this kind of fully operationalised application of DST to such fields yields testable predictions for dynamic processes specific to that field. 
Various authors (e.g., Larsen-Freeman 2012) decided that DST should be applied to SLA without its mathematical core and argue for an application at a metaphorical level. In this colloquium we explore an alternative. We ask 'Can a mathematically-based DST be applied to SLA as it was in science?' 
We do not intend to compare this alternative with the perspective established by Larsen-Freeman and colleagues. Such an exercise would be interesting only once the mathematically-based alternative has been sketched out in sufficient detail to be testable. 
In this colloquium we will focus on some of the fundamental issues involved in implementing a mathematically-based DST approach to SLA, i.e., to a field outside the science arena: Which aspects of SLA can be operationalised in such a way that they can be implemented using DST-based mathematical models? What is the status of predictions in DST and in its application to SLA? When are predictions possible? How can we capture the constraints that operate on 'chaos'? How can developing linguistic systems be modelled according to the mathematical requirements of DST and with non-specialist training in mathematics? What are the implications of our alternative approach for L2 pedagogy? 
All papers result from collaboration between the three presenters, but each author will take responsibility for the presentation of an individual paper in the colloquium. 
As the discussant Professor Dr Kristin Kersten Professor of SLA and English Language Teaching, University of Hildesheim, Germany will open up key threads between the papers and critical issues for wider discussion with the audience.
References:
Feldman, D. P. (2019). Chaos and dynamical systems. Princeton: Princeton University Press.
Larsen-Freeman, D. (2012). Complex, dynamic systems: A new transdisciplinary theme for applied linguistics? Language Teaching 45(2), 202-214.
Mitchell, M. (2009). Complexity. A guided tour. New York: Oxford University Press.
Strogatz, S. H. (2015). Nonlinear dynamics and chaos. With applications to physics, biology, chemistry and engineering (2nd edition). Boulder: Westview.

This paper addresses some of the challenges that arise when applying dynamical systems theory (DST) to second language acquisition (SLA). We argue that any attempt to pursue this endeavour beyond a mere metaphorical level needs careful consideration of some of the core mathematical concepts underlying DST. In this paper, we explain key concepts of DST and discuss the lessons of dynamical systems that are potentially relevant to the study of SLA. The concepts include the nature of dynamical systems, the notion of nonlinearity, the butterfly effect as well as the mathematical understanding of 'chaos' (Feldman 2012, 2019; Strogatz 2015). A key focus will be on the issue of predictability and the constraints on chaos. 
We introduce the logistic map, which is considered to be a prime example of a dynamical system, as "it illustrates many of the fundamental notions of nonlinear dynamics" (Saha & Strogatz 1995, 42). Using the logistic map, we demonstrate the essentially deterministic nature of dynamical systems. This has crucial implications for the scope of and relationship between order and chaos together with their predictability. This mathematical approach to chaos contrasts with the common understanding of the latter term that equates chaos with unpredictability and/or randomness. We demonstrate that the emergence and the scope of chaos in dynamical systems can be predicted with great precision. Building on these insights, we argue that taking a mathematically-based DST perspective on SLA can allow for predictions related to selected aspects of both development and variation. 
This dynamical perspective on SLA is compatible with testable predictions about L2 developmental trajectories that are shared between learners. At the same time, this view does not conflict with the view of language as a complex system and the fact that there is substantial individual learner variation in L2 acquisition. We argue that prediction and explanation are key tenets in SLA research and that applying DST to SLA by taking the core mathematical basis into account has the potential to enhance our understanding of the complex processes involved in SLA. 
References:
Feldman, D. P. (2012). Chaos and fractals. An elementary introduction. Oxford: Oxford University Press. 
Feldman, D. P. (2019). Chaos and dynamical systems. Princeton: Princeton University Press.
Saha, P. & S. H. Strogatz (1995). The birth of period three. Mathematics Magazine, 68(1), 42-47.
Strogatz , S. H. (2015). Nonlinear dynamics and chaos. With applications to physics, biology, chemistry and engineering (2nd edition). Boulder: Westview.

Dynamical systems theory applied to the simulation of L2 simplification: Paper 2 in the proposed colloquium, 'An alternative proposal for the application of dynamical systems theory to SLA'
L2 simplification is one of the key areas in which learner variation manifests itself. In this paper we examine the role of the internal dynamics of the developmental pathways of L2 simplification and its differential effect on inter-learner variation. More specifically, we will demonstrate that the specific variational pathways chosen by individual learners determine whether the learner stabilises as a result of the developmental dynamics inherent in her or his interlanguage system.
In our approach L2 developmental dynamics are treated as a dynamical system as defined in dynamical systems theory (DST). As several authors have shown (see Feldman 2019; Mitchell 2009, Strogatz 2015), DST can in principle be applied to fields that are amenable to mathematical modelling and this kind of fully operationalised application of DST to such fields yields testable predictions for dynamic processes specific to that field. In our approach we have operationalised such a mathematical application of DST to SLA by using an established agent-based modelling environment called NetLogo (Wilensky 1999). Using NetLogo we have developed an agent-based model of L2 simplification.
We demonstrate that our AI simulation model is capable of simulating the time course of L2 simplification processes focussing on equational structures. We validate these simulations by comparing them with the time course of the simplification of equational structures produced by five SLA learners who were studied longitudinally over periods lasting from 12 to 18 months. This also includes a precise match of the initial state and the time intervals between the simulations and the natural L2 data. Each of the five longitudinal studies is based on spontaneously produced L2 data that were fully transcribed. The empirical validation of the simulations is based on a full quantitative analysis of the natural L2 data. The validation process reveals that the simulations closely match the naturally occurring L2 simplification processes based on measures such as the Normalized Root-Mean-Squared Error. Our results suggest that in future studies it may be viable to use L2 simulations in situations where natural data are hard or impossible to be obtained.
Using the NetLogo platform has a strategic advantage for the feasibility of applying the mathematical formalisms of DST to SLA: researchers can make their ABMs available to the research community. This strategy reduces the formal requirements for new researchers to enter the field of maths-based DST research in SLA.


Feldman, D. P. (2019). Chaos and dynamical systems. Princeton: Princeton University Press. Mitchell, M. (2009). Complexity. A guided tour. New York: Oxford University Press. 
Strogatz, S. H. (2015). Nonlinear dynamics and chaos. With applications to physics, biology, chemistry and engineering (2nd edition). Boulder: Westview. 
Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

This paper engages with the pedagogic consequences of claims made that all aspects of L2 learning are equivalent in their complexity at all times and that patterns of individual variation cannot be mapped onto patterns identified for groups (Lowie & Verspoor 2019). The consequences of claims such as this are seen in Fogal (2020), who, in support of an argument that there is extensive and meaningful variability in the course of L2 development claims that traditional SLA perspectives are linear and unidimensional. We argue for an alternative understanding that positions variation in a continuing and precisely-specified relationship to predictable sequences of development (for some L2 features) such that these sequences co-exist with variation within and between learners. Our argument is not that there is no variability. Instead, we argue that the significant, observable and potentially predictable variation is PART of a larger picture that includes shared, predictable sequences of development. Using data derived from our investigation of L2 stabilisation, we show how the pedagogic implications of the two co-existing but different developmental and variational dimensions have different pedagogic correlates and consequences. We show how this position overcomes the dichotomy between development and variation that is foregrounded in metaphoric approaches to dynamical systems.
We discuss the nature of the evidence for predictable sequences in L2 learning and the linguistic and methodological constraints that are associated with those claims. We show how this view of predictable sequences can be reconciled with a view of language use as complex and dynamic and highlight the analytic techniques that are required to support the claim.
In addressing the above relationships, we point to the necessity of nuancing the claimed relationships between language, language use and language learning. We provide evidence in support of a less simplistic view than the claim that all aspects of all three of these phenomena are similarly dynamic in similarly complex ways.
Using the outlines of the mathematically-informed approach presented in the two other papers in this colloquium, we provide examples to show how L2 pedagogy requires a view that coherently and consistently relates shared learning sequences, learner variation and professional sensitivity to individual needs and creativity. Using some fo the insights from our work on stabilisation we point to examples of how regularity and variation intersect in both teacher decision-making processes and learning challenges.


Fogal, G. (2020). Investigating variability in L2 development: Extending a complexity theory perspective on L2 writing studies and authorial voice. Applied Linguistics 41(4), 575-600.
Lowie, W. & Verspoor, M. (2019). Individual differences and the ergodicity problem. Language Learning 69(S1), 184-206.