Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolic
Complex Dynamics and Morphogenesis by Chaouqi Misbah is a recent book that introduces, first, concepts from nonlinear dynamics, and, second, pattern formation to students and interested researchers. The book is an extended revision of an earlier volume published in French in 2011. There are two main parts to the book: the first eight chapters cover various material from nonlinear dynamics such as bifurcation theory, chaos, and even catastrophe theory; the last four chapters concern spatio-temporal pattern formation and introduce the reader to some of the classic equations of this field like the Ginzburg-Landau equation and the Swift-Hohenberg equation. There are many exercises in the book and the entire last chapter of the book gives complete solutions. I have mixed feelings about solutions in the book since it makes it hard to assign problems unless you have very honorable students. Nevertheless, since some of the exercises are very extensive, it is nice to see them worked out.
The book is aimed at upper level undergraduate or graduate students and requires little more than calculus and perhaps a little bit of ordinary and partial differential equations. The many exercises make it useful as a textbook for a course in nonlinear dynamics; although, for undergraduates, the classic book by Strogatz [1] is superior.
The first two chapters describe static bifurcations (at a zero eigenvalue) through a series of mechanical examples. The problem I had with this book is that while there are many worked out examples, the author never presents any general theory for when there is a bifurcation. By focusing on scalar problems (which are, in a sense, already in “normal form”), the reader never gains an appreciation of (1) how general these methods are and (2) any recipes for computing the bifurcations. Scalar problems also obviate the need for any software or numerical methods which, to me is a ashame, since the problems that one encounters in the applied sciences are usually not possible to work out by hand. While it might be a bit much to ask of a textbook aimed at this level, it was surprising to me that no mention of the Liapunov-Schmidt or even the Fredholm alternative appears making it harder to apply the techniques to general problems. Chapter four covers the seven elementary catastrophes; I have to say that I was surprised to see catastrophe theory in a text book. Chapters five and six concern the Hopf bifurcation, demonstrated mainly through the examples of two-dimensional systems like the van der Pol equation. Like the chapters on bifurcation at a zero eigenvalue, there is no mention of the conditions that one looks for in the linearized system in order to detect the possibility of a Hopf bifurcation. In Chapter six, the author shows that the behavior near a Hopf bifurcation is universally described by the well-known amplitude equation, \(z'=z(a+b|z|^2)\). In addition, the fifth order approximation to illustrate systems that have both a stable fixed point and a limit cycle. One of the exercises asks the reader to apply the formula from Guckenheimer and Holmes [2] to determine the criticality of the Hopf, but, because the author never gives a general definition of the Hopf bifurcation, the formula has no context nor any sense of how it arises. Chapter seven covers parametric instabilities, a subject that is seldom covered in textbooks, but a welcome addition. The concept of phase-locking is introduced by focusing on nearly harmonic oscillators and thus depriving the students of an understanding of the generality of averaging methods. Chapter eight is the last chapter in the first part of the book and gives a very rapid tutorial in various aspects of chaos. This is the only chapter that focuses pretty exclusively on maps rather than flows since it is possible to concentrate on some simple scalar cases like the logistic equation, \(x_{n+1}=ax_n(1-x_n)\). Concepts like fractals and the Liapunov exponent are introduced but only through some elementary examples. The approach that Misbah takes in this first half of the book is definitely an engineering approach with perturbation techniques applied to simple one- and two-dimensional equations. It will not replace a more rigorous approach (such as [3]) or the clear exposition in Strogatz, but it combines some of the theory with hands-on perturbations that give students a first look at this type of useful, formal analysis.
Chapters 9-14 form the second part of the book and introduce the “morphogenesis” part of the book’s title. What these chapters represent is a greatly condensed and simplified version of the two books [4,5]. Chapter 9 begins quite naturally with the Turing instability using a two component generic reaction-diffusion equation as an example and a nice description of the instability. Misbah follows this with several examples two-variable systems such as the Schnackenberg model and the more realistic Lengyel-Epstein model. He mentions that “as a general rule” numerical investigation is a good way to determine what happens in these systems beyond the linear stability analysis yet he offers no help in this regard. For me, the strongest part of this chapter was the section on Rayleigh-Benard convection which I had not seen before. Chapter 10 introduces the general amplitude equation near a Turing instability. Rather than continuing with a reaction-diffusion example, he instead introduces a scalar model to avoid any tedious calculations. However, in doing so, Misbah obscures the generality of the methods, which to my view, are not so difficult to explain and derive. Using multiscale analysis, he derives the classic Ginzburg-Landau equation for the amplitude of a pattern in an infinite one-dimensional domain:
\(A_T = c_0 A_{XX} + c_1 A - c_3 A|A|^2. \quad\quad \hbox{(GL)}\)
He does not mention that the diffusive term is there only because the domain is infinite; if the domain is finite with boundary conditions, then all that is left are the terms involving powers of \(A\). Because the coefficients \(c_j\) are real, (GL) admits a variational approach, which the author describes. When \(c_j=1\) (WLOG), equation (GL) admits a family of solutions of the form \(A_0(X)=\sqrt{1-q^2}e^{iqX}\) and a band of these is stable. The loss of stability for \(q\) large enough is called the Eckhaus instability. Unfortunately, here the typos in the book catch up to the author and a reader must be careful of the ensuing calculations. Nevertheless, Chapter 10 was an otherwise clear exposition on this famous equation. Chapter 11 describes wavefronts between the stable states of equation (GL) and is quite short and is burdened by too many mechanical analogies which may make it a bit less appealing to the general applied mathematician. Chapter 12 introduces a very important equation in the analysis of patterns in infinite domains: the complex Ginzburg-Landau (CGL) equation which is exactly equation (GL), but the coefficients, \(c_j\) are generally complex. This equation arises when there is a Hopf bifurcation in a spatially extended infinite domain and is fundamental to the understanding of complex spatio-temporal dynamics. Misbah derives the equations from basic symmetry argument but never provides an example where such patterns exist. The CGL admits a family of plane waves, \(A(X,T)=r(q) e^{i(qX+\Omega(q) T)}\) as well as many other exotic solutions (such as “target” and “spiral” patterns analyzed in [6], but never cited here). The author shows that the plane waves lose stability (Benjamin-Feir instability) and that in some cases, the dynamics can be very complex leading to several types of spatio-temporal chaos. Indeed, near some of these complex bifurcations does the famed Kuramoto-Sivashinsky equation arise. Chapter 13 of the book delves into stationary two-dimensional patterns and provides the nonlinear selection mechanisms for hexagons, squares, and rolls. In the last chapter on space-time dynamics, Misbah provides a list of mechanisms for wavelength selection. This “problem” is an artifact of looking at pattern formation on an infinite domain. In finite domains, the wavelengths are determined by the discrete eigenvalues of the linearized operator. The last chapter of the book is a short conclusion listing some questions that cannot be answered by amplitude equations.
In sum, the book provides a survey of some of the classic methods used to study nonlinear dynamics near the onset of instability. I’d say that the approach is a little bit dusty and the problems attacked are also a bit dated. Nevertheless, the book covers a lot of ground and is quite accessible; if you haven’t seen this type of analysis and don’t have time to read more comprehensive texts, then it will make a nice addition to your library.
BIBLIOGRAPHY
- S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview press, 2014.
- J. Guckenheimer and Ph. J. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Vol. 42, Springer Science & Business Media, 2013.
- Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 112, Springer Science & Business Media, 2013.
- M. Cross and H. Greenside, Pattern Formation And Dynamics In Nonequilibrium Systems, Cambridge University Press, 2009.
- R. B. Hoyle, Pattern Formation: an Introduction To Methods, Cambridge University Press, 2006.
- N. Kopell and L. N. Howard, Target pattern and spiral solutions to reaction-diffusion equations with more than one space dimension, Advances in Applied Mathematics, 2.4 (1981), pp. 417-449.