Differential Equations


by Cédric Villani

Differential equations are, without any doubt, one of the most important and fruitful branches of mathematics. Modern analysis started with them in the middle-1600’s, and they have been a key to the development of industry since then. They found applications in all natural sciences, and human sciences as well; they were also associated with the development of other fields of mathematics such as Fourier analysis, differential topology or functional analysis.

Still, to the beginner the qualitative theory of differential equations poses major conceptual challenges; no wonder that it is only at the end of the 19th century that it started to be systematically developed. And over the years, teaching the subject again and again across the world, I have often found it so difficult to initiate the students with the spirit of this theory. They think that solving differential equations is about computing and finding out formulas; they discover that we don’t care about formulas but want them to draw a phase portrait, a qualitative sketch of the flow, little arrows and wiggling curves. Actually, these notions can prove fatal!

No wonder that differential equations pose a pedagogical challenge then. Among hundreds of textbooks, only a handful have stood the test of time, among which the most prominent is Arnold’s celebrated course, a true heir of the tradition by Poincaré. Its informal explanations, well-chosen examples, sketches and drawings pay tribute to the author’s legendary sense of pedagogy. Many courses on differential equations, including my own, are paraphrases of Arnold’s. Of course, decade after decade the quality of illustrations has improved, and students have also been encouraged to write their own code and recreate for themselves the properties of differential equations, so that the book experiment is enriched by complementary activities and personal fight.

But this book is of a different nature: truly a book, with all the work done by the authors to make it easy for the reader; but also truly interactive. Thanks to more than 60 experiments embedded in the text, the magic of differential equations comes to life in an streamlined, effortless way. Every concept is illustrated and lets you play with it by tweaking parameters. Especially in the tablet version, it truly becomes a child’s game. You wish to see the pendulum in action: just tap it gently to make it swing in one direction or another. You want to see the qualitative difference between strong friction and weak friction: just drag the button to change the value and recompute the trajectories. How about seeing the systematic convergence to the Van der Pol attractor: just tap again and again directly in the phase space of the Van der Pol equation, and you will see the fateful cycle being drawn by the countless trajectories. And to get a feeling of the sensitivity to initial conditions, what will be better than contemplating the dance of two points in the phase space of the Lorenz equation, initially so close that they are hardly distinguishable, but soon playing hide and seek amidst the corridors of the labyrinthic attractor.

In short, playing and admiring the beauty: these two notions are so proudly heralded by Chazottes and Monticelli, that at times they do not hesitate to present experiments in an impressionistic way, and even to present beautiful phase portraits just for the beauty of them!

Apart from the magic of the digital visual experiments, the authors have opted for a very pedagogical plan, starting with a gallery of examples, then going for equations in dimension one, then two, then three and higher. Along the way, qualitative reasonings are highlighted, with more technical formulas being deferred to special notes that can be popped up; and the more advanced concepts such as bifurcation are touched several times on various examples, before general theorems are stated. Some of the famous modern developments of the field, dating from the seventies or eighties, are incorporated with the rest; and some directions of further developments are sketched, such as partial differential equations and the Turing instability, or the theory of fractals. Curious readers will have a lot more to dive in!

Going through all these jewels, tapping and stroking the pad as I was testing this ebook, I felt some jealousy for the students who would enjoy this experience and see differential equations coming into life; a little bit as I felt jealous for my kids when they were discovering classical geometry with GeoGebra. Indeed, this is exactly the same kind of experience that Chazottes and Monticelli are offering you: no magical recipe to make you an expert, but a way to feel and grasp the concepts in a beautifully interactive, flexible and playful way.

© Marie-Lan Nguyen / Wikimedia Commons

Cédric Villani
Professor at University of Lyon Claude Bernard
Director of Institut Henri Poincaré (CNRS/UPMC)
Member of the French Academy of Sciences
Fields Medal 2010