# Fractals arise in non-linear dynamics

## Chaos Fractal Structures or The Difficulty Bringing Nonlinear Dynamics to Students

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1 1 Chaos Fractal Structures or The Difficulty Bringing Nonlinear Dynamics to Students Friederike Korneck, Institute for Didactics of Physics, J.W. Goethe University Frankfurt Since the development of nonlinear dynamics, specialist didactics have been grappling with the elementaryization of this research area. For school and teacher training, a wide variety of approaches have emerged in recent years in order to bring the current research results closer to the students. Although non-linear dynamics have since been included in some current curricula and textbooks, it is rarely finding its way into schools. It is technically difficult to teach, very mathematical and therefore complex, and is viewed by teachers as a subject for specialists. In order to deal with this problem, some current curricula and textbooks are first analyzed and, on this basis, various suggestions are developed in order to establish the non-linear dynamics in schools after all. For me, chaos is mathematics - I have no desire for teacher training. What you never got to know in your studies, you don't acquire later in everyday teaching life. It was only through contact with you that I saw that non-linear dynamics can be elementized. I thought it was too hard for school. These are quotations from conversations with some upper school teachers based on the questions of whether they are of the opinion that nonlinear dynamics is a suitable subject for the upper school and whether they would teach nonlinear dynamics themselves if it were an optional subject in the curriculum would stand. The discussions showed that there are great concerns about teaching topics of non-linear dynamics oneself. They far outweigh the positive arguments, which are almost exclusively limited to the everyday reference of non-linear dynamics. Here are a few more quotes: You have other topics in it, you know the time frame. The subject is so complex and the mathematical background so complicated and I've never heard of it at university. In the Hessian plan that would fit in grade 11. On mechanics and thermodynamics. But we hardly manage the thermodynamics and certainly no extra topics. In 13/2 we have free choice of topics. But there the non-linear dynamics is certainly not mentioned. The following quotes come from the correspondence with a member of a curriculum committee: You are right to say that many teachers have a high inhibition threshold when it comes to this subject. We included the topic in the curriculum at the time, because we believe it has a paradigmatic character for a modern understanding of physics .... In the meantime, however, interest in the topic has declined noticeably. At that time, too, we carried out several teacher training courses on the subject. again 'gone out of style'. He describes the approach in the curriculum and handout, which provides access to chaos physics on the subject of vibrations, and continues: The question of access arose to me .... The systemic approach via self-organization and structure formation seems to me to be more sympathetic and educationally effective. But then the corresponding building block would have been an isolated block. 1. Inventory The following inventory of the current curricula and textbooks shows that most curriculum and textbook authors have followed this trend of integrating non-linear dynamics into the subject of vibrations. Unfortunately, to the disadvantage of the formation of structures, which is less taken into account in both curricula and textbooks. Many curricula are currently being revised: some have recently been adopted, others are in the consultation phase. Therefore, some of the current curricula that have incorporated nonlinear dynamics are presented first. The second part of the inventory is the analysis of teaching materials - current school books and teacher handouts - for materials

2 2 that most teachers refer to when preparing a new topic. In my dissertation [1], the various didactic approaches to non-linear dynamics were analyzed in detail. Therefore only one comment at this point: Although work has been added since then, the analysis is still largely up-to-date: In the last two years, research results on nonlinear dynamics have been processed almost exclusively at a very high level (e.g. article on the introduction of Turbulence, appeared in physics in school [2], [3]). In the publications, the last step from the elementary subject didactic content to the tried and tested teaching unit for school and teacher training is missing. The implementation is left to the teachers and most of the teachers complained about this in the conversation. Another issue will only be dealt with briefly: Not only in schools, but also in the lectures for student teachers, the non-linear dynamics are neglected. Even at the physics department of the Goethe University in Frankfurt, which had a special research area on nonlinear dynamics for years, the topic only appears in a block lecture. In the basic lectures it falls away with the same argument as in schools: It would be interesting and important, but unfortunately there is no time. Curricula Rhineland-Palatinate / major subject: Compulsory module mechanical vibrations I; Elective Modules Mechanical Vibrations II and Nonlinear Dynamic Systems Examples of nonlinear dynamic systems, chaos phenomena, characteristics and system conditions, description of chaotic phenomena, structures in chaos, sensitivity Methodological notes: In the case of internal physical access via vibrations, process the differential equations with a computer. An interdisciplinary approach leads to a discrete representation using difference equations. Fig.1: Excerpts from the physics curriculum, Rhineland-Palatinate [4] The current curriculum for Rhineland-Palatinate and a corresponding teacher manual recommend three modules for the major physics: The compulsory module Mechanical Vibrations I and the two optional modules Mechanical Vibrations II and Nonlinear Dynamic Systems . In the first two modules, expandable vibration examples are to be selected and thus the foundations for the non-linear dynamics are to be laid. The goal is a step-by-step procedure, at the end of which there is an appropriate and well-founded insight into chaos physics [5]. Figure 1 shows the content and methodological information of the elective module nonlinear dynamic systems. The latter only relate to the mathematical description of the non-linear systems. In this curriculum, an elective module for fluid dynamics is recommended for the major subject. Unfortunately, the opportunity to build on and couple the two subject areas of fluid dynamics and nonlinear dynamics was not used. The two building blocks stand next to each other completely independently. It would be a good idea to create teacher training courses and possibly a handout that shows the access to non-linear dynamics via phenomena from fluid dynamics, as worked out in my dissertation. Rhineland-Palatinate / basic subject: elective module chaos and fractals examples of chaos phenomena and fractal structures features and system conditions, description of chaotic phenomena structural similarity in different areas methodological notes: do without formalization that is inappropriate for the basic subject. Dealing with the logistic equation lends itself to making use of interdisciplinary references. Fig.2: Excerpts from the physics curriculum in Rhineland-Palatinate [4] For the basic subject (Fig. 2), an optional module Chaos and Fractals is provided with the aim of giving the student a contrastive insight into phenomena and the transdisciplinary character of the topic. [5]

3 3 In contrast to the major, an inappropriate formalization should be avoided here. Nevertheless, it is recommended to study the logistic equation. The hearing version of the Baden-Württemberg education plan [6] for the four-hour profile and specialty subject corresponds to the Rhineland-Palatinate curriculum in the first three points of the elective module Nonlinear Dynamic Systems. As a regional delicacy, the synergetics has also been included here, thus taking into account the structure formation in addition to the chaos physics (Figure 3). BaWü (hearing version 12/2000): Elective modules Nonlinear dynamic systems and flow physics Chaos phenomena Features and system conditions Structures in deterministic chaos Synergetics Methodological notes on NLD: Experimental mathematics; e.g. Opportunities for independent development, e.g. with appropriate computer software From preliminary remarks: Philosophical and methodological aspects of physics: - e.g .: What is meant by causality and determinism? Fig.3: Excerpts from the educational plan for the course level of the grammar school, Baden-Württemberg [6] Put the general preliminary remarks for this curriculum, with the question of causality and determinism, including the philosophical aspect, in the center of the teaching. The discussion of this question can be handled excellently in the elective module Nonlinear, Dynamic Systems [6]. The methodological notes on the elective modules themselves, however, point in a different direction (see Figure 3). In this plan, the two elective modules flow physics and nonlinear dynamic systems are directly below each other, so that a thematic connection is obvious. But this is not provided for in the curriculum. As in Rhineland-Palatinate, cooperation would also be here, e.g. conceivable for the conception of teacher's handouts. In the framework plan for Bremen, the non-linear dynamics only appears as an additional topic in the field of mechanics, sub-topic mechanical vibrations. Apart from the reference to the thread pendulum, this plan does not contain any suggestions. Bremen 1998: subject area mechanics, extension topic non-linear vibrations (e.g. thread pendulum) Hesse 1994: content area vibrations and waves, extension topic chaotic systems Possibility: additional content area with free choice of topic Fig. 4: Extracts from the specialist framework plan Bremen [7] and course structure plan Hessen [8] Also in Hessian course structure plan, the non-linear dynamics is only an extension of the content area vibrations and waves. However, the course structure plan in Hessen offers the possibility of freely choosing a content area. These free topics are mostly taught in 13/2. The 30-hour teaching unit of my work on fluid dynamics and nonlinear dynamics was originally designed for this period. A reform of the upper level is being sought in Hesse. Parallel to this reform, new curricula are also to be drawn up so that greater consideration of the non-linear dynamics can possibly be enforced. Finally, the North Rhine-Westphalian curriculum, which takes the non-linear dynamics into account the most. First of all, in this plan, non-linear vibrations are provided as a compulsory subject for basic and advanced courses within the subject area mechanics (see Figure 5). NRW (1999): Subject area mechanics, topic mechanical vibrations Nonlinear vibrations Predictability of the vibration behavior Fig. 5: Excerpts from the guidelines and curricula for the secondary level II, NRW [9] Another very detailed subject area on nonlinearity and chaos can be found in the subject area thermodynamics than more general

4 4 Structure Theory. It also offers irreversible phenomena, real systems with high complexity and non-equilibrium systems the necessary conceptual and conceptual basis [9, p. 119]. Figure 6 shows content and concepts from the nonlinear dynamics that this plan recommends for use in class. NRW (1999): Subject area mechanics, subject mechanical vibrations Subject area thermodynamics, subject nonlinearity and chaos Self-organization and dissipative structures Symmetry and symmetry breakage Sensitivity, causality and strong causality principle Phase diagrams and attractors Fig tree diagrams, bifurcations and self-similarity A corresponding treatment of fractals and fractal dimensions can be are not based on long-term experience and must ... be tried out and assessed independently. Fig. 5: Excerpts from the guidelines and curricula for the upper secondary level, North Rhine-Westphalia [9] This is the only curriculum that also takes the structure into account to a sufficient extent. This master plan refers - perhaps as the only one - to the didactic research work of the last few years. However, he also points to a problem that affects didactic work in recent years: the investigation of non-linear systems has only led to important findings in the last thirty years, precisely because of the extensive computer simulations that have become possible during this time. A classroom treatment of such issues cannot therefore be based on long-term experience and must be tried out and assessed by the teachers independently. [9, p. 127] It is precisely this prospect of untested teaching proposals that makes teachers shy away from the non-linear dynamics. This becomes clear again and again in conversations. We must not leave the teachers alone with the testing and evaluation! This is originally the task of the didactic specialists. The non-linear dynamics in school books and teachers' handouts Time and again, teachers complain that the non-linear dynamics in school books are not sufficiently taken into account. It is not so. This is shown by the analysis of some current physics books for upper secondary school [10] [15]. Figure 7 lists textbooks that deal with nonlinear dynamics. A teacher's handout [5] on nonlinear dynamics, which was written by teachers for teachers, is particularly interesting. That is why it was included in the analysis. However, since it has a different target group, the content that affects it has been marked in green. From this handout, only the teaching units - not the factual analysis - were evaluated. Main topics Nonlinear Concepts Contributions to nonlinear dynamics in textbooks Phenomena Methods Impulse Physics 2, Klett 1996 Kuhn Physik, Volume 2, Westermann 2000 and Textbook of Physics, 1989 Oberstufe Physik Volume 12/13, Cornelsen 1999 Dorn Bader, Physik 12/13, Schroedel 2000 Metzler Physik, Schroedel 1998 Non-linear dynamic systems and chaos, handout for the physics curriculum, R.-P Fig.7: Analysis of teaching materials on nonlinear dynamics The evaluation of the school books and the handout is based on the following categories: According to main topics to which the nonlinear dynamics are assigned has been; according to the processed phenomena; According to the non-linear concepts and the presentation methods Main topics In four out of five textbooks and in the teacher manual, chaos physics is dealt with under the topic of vibrations. In two books it appears as a separate topic.

5 5 The structure can be found - analogous to the curricula - in only two out of five school books. In both, it is located under thermodynamics. Phenomena The phenomena presented in the chapters on nonlinear dynamics have been assigned to chaos physics and structure formation for the following list. Phenomena chaos physics (6) magnetic pendulum, Pohl's wheel with imbalance (6 each) electrical oscillating circuit, double pendulum, population dynamics (4 each) fractals (3) three-body problem, dripping tap, ecosystems (2 each) stock exchange, Lorenz system, frictional oscillation, transition from laminar to turbulent , Butterfly effect, yo-yo wagon, two-spring cross pendulum, double inclined plane, Duffingoszi. (1 each) Everyday phenomena (weather, balloons, games of chance, billiards ...) (5 each) medical applications, heart rhythm, processes in the brain, (3 each) structure formation (2) Bénard system (2) Biological applications, living beings as dissipative structures ( 2 each) sand corrugation, bulk solids, mixing and unmixing, transition to turbulence, chemical oscillations, various self-excited oscillators (1 each) Fig.8: Non-linear phenomena in school books and teacher's handout The listing is hierarchical according to the popularity of the phenomena. The number of publications in which the respective phenomena are dealt with is given in brackets. In all publications, the magnetic pendulum and the Pohl wheel are regarded as suitable systems for introducing the concepts of non-linear dynamics. This is followed in the ranking list by the electrical oscillating circuit, double pendulum and the population dynamics and the fractals. In all publications apart from the teacher's handout, everyday phenomena are also discussed, with medical applications being the most popular alongside weather forecasting. As already mentioned, structure formation is addressed in two textbooks. The Bénard system has a central position in both books. In Cornelsen's upper school belt, the formation of structures with everyday phenomena such as sand reefs, bulk materials, mixing and unmixing takes up a lot of space. Fortunately, one recognizes the didactic work of Nordmeier and Schlichting again. Nonlinear Concepts Figure 9 shows the physical concepts of nonlinear dynamics, which should be worked out in the classroom according to the analyzed textbooks and the manual.Non-linear concepts Chaos physics (6) Non-linearity (6) weak and strong causality, bifurcation of strange attractor, Laplace demon, (5 each) sensitivity to disturbances, stable and unstable equilibrium, forecast horizon, linear models, deterministic chaos, universal path into chaos, order in Chaos, phase space, different attractor types (4 each) fractals, exponential error growth, superposition principle in linear systems (3 each) period doubling, scale invariance (2 each) degree of freedom, control parameters, synchronization (1 each) structure formation (2) open systems / non-Gg systems (2 ) Dissipation, energy devaluation (2) Nature is non-linear (2) Spontaneous structure formation above a critical value (2) Symmetry breaking (2) Chance as a creative element (2) Feedback (2) Synergetics: self-organization, collective behavior of the particles, order parameters, selection, Enslavement, circular causality (1 each) Fig.9: Nonlinear concepts in school books and Teacher's Guide Compared to classic teaching topics such as the fundamentals of mechanics or optics, where the physical concepts are relatively uniform regardless of the textbook author, there is a wide range of different physical concepts for non-linear dynamics. Non-linearity plays a central role as a prerequisite for chaotic behavior. Followed by the concepts of violation of the principle of weak causality, bifurcation and the strange attractor. The

6 6 Laplace's demon as the epitome of a deterministic worldview thematized. Although the chapters on nonlinear dynamics in the school books are partly are designed in a very appealing way, the wide range makes it difficult to select the relevant teaching content on chaos physics. The selection of the concepts for structure formation in both textbooks, which deal with this topic in addition to chaos physics, is clearer. They both deal with the conditions that a system must meet in order to be able to form self-organized structures - it must be open and dissipative and the particles must obey non-linear equations of motion. Under these conditions, structures can develop spontaneously above a critical value of the control parameter. Whether this clarity of the relevant concepts is due to the fact that the structure formation is only included by two authors, or whether it is due to the matter, cannot be clearly determined. Methods Methods Chaos physics (6) Mathematical representation methods: Schematic representations (e.g. for open systems), Bifurcation diagram, potential representation, fig tree diagram, phase diagram (4 each), iterative calculations with predefined equations, differential equations, Poincaré representation (2 each) non-linear characteristics, potential equations, estimates (1 each), experiment instructions (6) for magnetic pendulum, oscillating circuit, double pendulum (4 each) ; Rotary pendulum (3); dripping tap (2); ... practical applications of chaos theory (4) mnemonics, anecdotes, analogies, (3 each) tasks: calculate / program (5); discuss (4), home experiments (1) structure formation (2) experiment instructions (2) for the Bénard experiment (2), Bärlapp on a vibrating plate, mixing and unmixing ... Practical applications of structure formation / natural phenomena (2 each) schematic representations (2) memorandum , Anecdotes, analogies, freehand experiments, (1 each) Tasks: calculate / program (-); discuss (2), Heimversuche (1) Figure 10: Representation methods for non-linear dynamics in school books and teacher handouts It is striking that the handout, which was created by teachers for teachers, deals exclusively with non-linear systems that can be mathematically recorded and modeled. This seems to be the main criterion for whether or not a system has been included in the manual: Without modeling, use of computers and programming effort, the treatment does not make sense [5, p. 10] So most of the teaching time, if you follow this manual, inevitably goes through the modeling and programming required. The non-linear concepts are by-products, some of which can only be mentioned, hardly interpreted. In comparison, the textbooks place more emphasis on discussion and interpretation of non-linear concepts. Here, the mathematization of the theory is of secondary importance. All five textbooks and the manual deal with chaos physics, while only two textbooks deal with structure formation. The chapters on structure formation proceed purely phenomenologically and the introduction of the synergetic concepts does not include a mathematical representation of the theory. 3. Summary Some current school books contain the nonlinear dynamics - partly very appealingly presented with student experiments, freehand experiments and interesting tasks. The non-linear dynamics can also be found as an elective topic in more and more curricula. Unfortunately, existing didactic approaches and approaches are hardly taken into account. Thus the teachers benefit little from our work. One example is the mentioned teacher's handout: The most recent of the cited subject-related didactic works dates from 1994, so it is seven years old! Although textbooks and curricula incorporate the nonlinear dynamics, many teachers reject them as impractical. The inhibition threshold to deal with this topic is too high. At the same time, teacher training events on the subject are poorly attended (e.g. the training event on chaos in HELP last year had to be merged with another event so that it was not canceled due to a lack of participation). It is possible that in the meantime all those teachers who found the nonlinear dynamics interesting in their own right have received further training. But apparently these teachers do not act as multipliers. So now we need to reach those teachers whose concerns have been too great to deal with the subject. Otherwise the non-linear dynamics will definitely not reach the students!