Modelling emergent dynamics in complex systems

Overview

In January 2015, SIAM published my book titled Model Emergent Dynamics in Complex Systems on rational mathematical modelling of dynamics.

Via the web page https://tuck.adelaide.edu.au/modelling.php you may create a PDF book of draft supplementary material tailored to the interests you specify.

Maurício Kritz (2019) commented in reference to the book: develops methods, techniques, and algorithms for studying the behaviour of systems ... necessity of having simple models, equations and mathematics as brilliantly explained ... the huge and beautiful knowledge accumulated recently about dynamical systems.

Errata of printed first edition

• p.8, Fig.1.1: \(x=1-\frac12\epsilon+\cdots\), not \(+\frac12\epsilon\).
• p.52, l.-6: \(\lim_{x\to0}\frac{\cos x}2\) not the negative.
• p.60, Ex. 2.11: the ODE \(x^2y''+xy'+xy^2=0\), not \(x^2y''+x^2y'+xy^3=0\).
• p.61, Ex. 2.14: change the hint to writing \(x^2y''-xy'+y\) as \(x^2[x(y/x)']'\).
• p.75, l.7: \(\vartheta\) denotes the phase of the left-travelling wave (not the right-travelling).
• p.158, Ex. 4.6 (a slow subspace): use \(\dot x=-ay\), not \(\dot x=-ax\).
• p.160, Ex. 4.12 (instability at higher order): construct the slow manifold to higher order, such as \(O(x^3+y^3)\), and find that the conclusion is not quite correct!
• p.206, Ex 5.4 and 5.5: in the answers there is no need for the error component ``\(,\alpha^2\)'', omit.
• p.246, Eqn (7.5): omit the \(\theta\) derivatives (in the last term).
• p.265--6, Eqn (7.20): the right-hand side should be "\(=-k(\cdots)\)", not "\(=k(\cdots)\)"; consequently, the second iteration answer should be "\(\partial s/\partial t \approx (k/s)\left[1-\frac13k+\frac13s_x^2+\frac13ss_{xx} -\frac23ks_x^2\right]\)".
• p.253: last line of the first displayed maths should read \(=(uU-U^2)w\frac{\partial^2C}{\partial x^2}\).
• p.299, just before section 9.1.4: "\([-2\gamma +\frac23 gamma^2+\)" should be "\([-2\gamma -\frac23gamma^2+\)".
• p.300: "decay rate \(\pi/4\)" should be "decay rate \(\pi^2/4\)".
• p.304, lateral momentum equation: the RHS of the update PDE needs a factor of \(1/Re\); the residual of the tangential stress equation needs to be negated; and "change to the pressure field is due to viscous stresses at the free surface and the need to accelerate the flow vertically" should read "change to the horizontal velocity field is due to viscous drag at the bed, flagged by factor~\(\gamma\), and the acceleration due to spatial variations".
• p.305, last line before ``Iterative construction'': boundary condition "\(\hat u=0\)" should read "\(\partial\hat u/\partial Z=0\)".
• p.353, (11.17): "\(-\alpha_\ell X_\ell\)" should be "\(-\alpha_\ell\hat x_\ell\)"; and the same three lines after in the text.
• p.474, line 13: capital \(U\) in the PDE should be little \(u\).
• p.490, in the first paragraph the \(f_{2j}\) should be in the numerators of the fractions, not the denominators. Similarly in the second paragraph, \(f_{2,\min}\) should also be in the numerator.
• p.708--712, Algorithm 21.4--8 does not work when there is large noise in the slow variables. It does appear to be OK when the slow variable noise is small or zero.