Abstract A course on nonlinear dynamics is taught for freshmen in Shanghai University. Focusing on the scientific concepts of chaos, fractal, and bifurcation, the course helps the students to understand diversity, uncertainty, and unpredictability in the real world. The paper surveys thetechnical contents of the course and highlights its pedagogical features. Student feedback is also reported.

Keywords  nonlinearity; chaos; fractal; bifurcation;

Introduction

Nonlinear dynamics or chaos theory emerged some four decades ago. Its impacts have extended beyond the natural sciences. Nonlinear dynamics has provided not only scientists and engineers with new views and new tools, but also social scientists and even people working in the humanists. Indeed, some of the associated terms, such as ‘chaos’ and ‘fractal’, are now part of common language.

 To help students to acquaint themselves with elementary nonlinear dynamics, a course titled “Chaos and Nonlinear Thinking” has been designed. Since 2011, it has been offered as an optional course for the freshmen in Shanghai Universityin the spring trimester;sophomores, juniors and seniors can also take the course in fall or winter trimesters. The main theme of the course is the scientific interpretations of chaos, fractal and bifurcation, as well as their cultural impacts. Nonlinearity is treated not only in mathematical models but is extended to thinking patterns. The objectives of the course are: to introduce the basic concepts of chaos, fractal, and bifurcation and to outline their historical evolution; to highlight the essential characteristics of nonlinear systems; and to encourage students to use nonlinear thinking to help them understand unpredictable and uncertain phenomena in the nature and society. Although nonlinear dynamics appears in some undergraduate curriculums, it has been rarely taught for freshmen.

 This paper summarizes the teaching of the course. It is organized as follows. The following section surveys technical contents of the course. Some of the pedagogical featuresof the course are highlighted, before some practical details of course delivery are set out (schedule, literature and grading). Some student feedback on the course is briefly reported before some concluding remarks are made.  

Technical Contents

The total course consists of 7 lectures.  

 Lecture 1 is an introductory description of the basic concept of nonlinearity: that the ratio of input to output is not constant. The mathematical definition of linearity is also presented, and the significances and the limits of linear models are discussed.

 Lecture 2 focuses on the logistic map. It begins with the derivation of the map as a model of biological populations with nonoverlapping generations. The period-1, period-2, and period-4 points are located, and their stabilities are analyzed. Chaos in the map is introduced through the properties of recurrence without periods, sensitiveness to initial states, as well as the bifurcation diagram with self-similarity and periodic windows.

 Lecture 3 elucidates the conceptual evolution of chaos: the culture background and the historical developments are outlined, and an illustrative example is given. The word “chaos” has been found in western classics such as The Theogony of Hesiod (translated by Huge G. Evelyn-White), The Metamorphoses of Ovid (translated by Horace Gregory), Holy Bible (King James Version), as well as Chinese classics such as Zhuang Zi. Some old sayings regarding the importance of initial steps have been collected. The rudiment of chaos theory, in the sense of nonlinear dynamics, can be traced back to James Clerk Maxwell’s Cambridge in 1873speech, in which he refers to instability, Jacques Hadamard’s 1898 paper on geodesic flow on a surface of negative curvature, and Pierre Duhem’s interpretation of Hadamard’s idea in 1906. Then Henri Poincaré founded chaos theory mathematically by discovering transverse homoclinic points in 1890 and conceptually by elucidating sensitivity to initial values. The contributions of other pioneers such as Steve Smale, Edward Norton Lorenz, Yoshisuke Ueda, Tien-Yien Li and James A. Yorke, are presented. The chapter ends with the demonstration of chaotic motion of a forced mass (nonlinear) spring oscillator, which was first studied by Ueda, via the time histories, the phase trajectories, and the Poincaré map. Chaos is represented by a trajectory that never closes or repeats, and that is located in a bounded region due to the recurrence of the motion.The Poincaré map of chaos is a set of infinite points that do not fill any loops or tori.

 Lecture 4 deals with fractal, in relation to the geometrical structure of chaos. Self-similarities in the nature are surveyed. The notion of fractal dimension is introduced with some classic examples such as the Cantor set, the Koch curve, the Koch snow, the Sierpinski triangle and carpet, and the Menger sponge. The fractal is introduced into dynamics. Strange attactors are illustrated via the Hénon map as the direct product of a one-dimensional manifold and a cantor set. Fractal attraction basin boundaries are discussed in relation to sensitive dependences. Lecture 4 ends with pictures of the Mandelbort set.

 Lecture 5 is concerned with bifurcation, with the emphasis on the routs to chaos. Static and dynamic bifurcations are explained with examples. Period-doubling cascade is demonstrated as an emerging process of chaos with the Feigenbaum constants; it is then investigated through variation of the system parameters. Intermittency, quasiperiodic torus breakdown, and crisis are briefly introduced.  

 Lecture 6 is devoted to the ubiquity of chaos. Examples are given from physics, chemistry, biology, astronomy, engineering, sociology, and economics. The influences of chaos on philosophy,aesthetics andliterature are briefly commented.

 Lecture 7 concludes the course, with some suggestions regarding nonlinear thinking. The linear view of the world is based on two “common sense” assumptions: that the whole is the sum of its parts; and the production is proportionate to the investment. Nonlinear thinking overthrows the linear view. It emphasizes diversity, uncertainty, and unpredictability.  

Pedagogical Features

The course is accessible for freshmen in diverse fields such as science, engineering, economics, and humanities. Little ability in either mathematics or physical sciences is assumed. Actually, most of the students are studying calculus and university physics, which is helpful but not necessary.

 As the prerequisites of the course are rather limited, it is taught in an intuitive way. Videos and pictures are used to visualize the scientific concepts and to rouse student interests. For example, to demonstrate the unpredictability of chaos, a shout video of toy (see Fig. 1) is played in the class; to demonstrate the sensitivity to initial states, a video about double pendulums produced by Howard Stone at Harvard University is played in the class; to illustrate the self-similarity, pictures of a cauliflower,  broccoli, dendrites, a wadi, Moon craters, and so on are presented. Students are encouraged to do some simple experiments. There are three pendulums on a rotating wheel (shown in Fig. 2) in the lobby of a teaching building. The students can observe chaotic motion by spinning the wheel.

Fig. 1 A chaotic toy.

Fig. 2 An experimental device.

 Some vivid sayings are cited to explain scientific concepts. For instance, to exemplify the significance of the initial state, a famous rhyme is presented

“For want of a nail the shoe was lost.

For want of a shoe the horse was lost.

for want of a horse the rider was lost.

For want of a rider the message was lost.

For want of a message the battle was lost.

For want of a battle the kingdom was lost.

And all for the want of a horseshoe nail.”

In addition, Lorenz’s vivid version of chaos as the butterfly effect is discussed -- “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”

 Reference is made to works of literature to highlight the scientific concepts. For example, to descript self-similarity, the first 4 lines of William Blake’s poem ‘Auguries of Innocence’ (1863) are cited

“To see a world in a grain of sand

And a heaven in a wild flower,

Hold infinity in the palm of your hand

And eternity in an hour.”

 Mathematics is kept to a minimum, although necessary equations are used to conduct mathematical arguments and to clarify the exactness and the soundness of the knowledge. For example, fixed point, period-2 points and period-4 points in the logistic map are detailed via pre-calculus mathematics, while the Lagrange mean value theorem has to be employed to determine their stabilities. Pre-calculus mathematics plus the concept of limit is also applied to calculate the dimension ofthe Cantor set, the Koch curve, the Sierpinski triangle and carpet, and the Menger sponge. 

 The course is essentially interdisciplinary, as it includes elements of the sciences, engineering, and humanists. It provides the students with a broad version, beyond mathematics and science. In fact, the course has applications not only in engineering but also in history, sociology and methodology of science. Some impacts of chaos theory on philosophy,aesthetics andliterature are also discussed.

Time schedule, References and Grading

The course lasts ten weeks, with two class periods each week. Each lecture needs two class periods except that lectures 3 and 4 need six and four class periods, respectively.

 The students in the class are required to read at least in part, one of six references [1-6] as they choose. They are also recommended to read some historically significant but not too technically difficult papers, such as [7-12].

 The grading depends on two reports, each contributing 50%. In the eighth week, all students should submit either a reading report or a project report, or both if they like. The reading report surveys some materials related to nonlinear dynamics, for example, a reference book (or a chapter or even a section of it), or a paper. Students select the reading materials themselves. The project report discusses the possible applications of nonlinear dynamics to the fields in which the students are going to major. At the end of the course, all students should submit a summary report (of less than 1000 Chinese characters) to outline what they have gained from the course and to give some suggestions to the instructor, if applicable.

Students’ Feedback

Most students have found the course novel and stimulating and have enjoyed pursing new knowledge. Even so, some of them, especially those majoring in humanities, felt it too abstruse: they found the historical aspects and the images are attractive, but the mathematical terms and operations hard to understand. Some engineering-oriented students felt the course is too abstruse or at least too academically oriented, as few practical applications are mentioned. Anyway, some students are very interested in the subject.

Concluding Remarks

The course informally introducesthe basic of nonlinear dynamicsthrough its historical developments, and highlights the essential characteristics of nonlinear systems. It makes students observe, analyze, understand unpredictable and uncertain phenomena in natural and in society via nonlinear thinking.

References

[1]      L.A.Smith,Chaos: a Very Short Introduction (Oxford University Press, Oxford, 2007).

[2]      J. Gleick, Chaos: Making a New Science (Viking Press, New York, 1987).

[3]      E.N. Lorenz, The Essence of Chaos (University of Washington Press, Seattle, 1993).

[4]      D. Ruelle, Chance and Chaos (Princeton University Press, Princeton, 1991).

[5]      I. Stewart, Does God Play Dice? The New Mathematics of Chaos (Blackwell Publishing, Oxford, 1989).

[6]      P.Smith,Explaining Chaos (Cambridge University Press, Cambridge, 1998).

[7]      E. N. Lorenz,Deterministic Nonperiodic Flow, J. Atmos.Sci.,20(1963), 130-141.

[8]      R.M. May, Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos, Science186(1974), 645-647.

[9]      T.-Y. Li and J. Yorke, Period There Implies Chaos, Amer. Math. Monthly82(1975), 985-992.

[10]  M. Hénon, A Two-dimensional Mapping with a Strange Attractor, Commun. Math. Phys.50(1976), 69-79.

[11]  R. M. May, Simple Mathematical Models with Very Complicated Dynamics, Nature261(1976), 459-467.

[12]  S. Smale, Finding a Horseshoe on the Beaches of Bio, Math. Intel.20(1998), 39-44.

Published in: International Journal of Mechanical Engineering Education, 2013, 41(2): 93-98