This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

References

References
1.
Strogatz
,
S.
,
2001
,
Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering
,
1st ed.
,
Westview Press
Boulder, CO.
2.
Schneider
,
T.
,
Marinc
,
D.
, and
Eckhardt
,
B.
,
2010
, “
Localized Edge States Nucleate Turbulence in Extended Plane Couette Cells
,”
J. Fluid Mech.
,
646
, pp.
441
451
.10.1017/S0022112009993144
3.
Willis
,
A.
, and
Kerswell
,
R.
,
2009
, “
Turbulent Dynamics of Pipe Flow Captured in a Reduced Model: pu? Relaminarization and Localized Edge States
,”
J. Fluid Mech.
,
619
, pp.
213
233
.10.1017/S0022112008004618
4.
Duguet
,
Y.
,
Schlatter
,
P.
, and
Henningson
,
D. S.
,
2009
, “
Localised Edge States in Plane Couette Flow
,”
Phys. Fluids
,
21
, p.
111701
.10.1063/1.3265962
5.
Manneville
,
P.
, and
Pomeau
,
Y.
,
2009
, “
Transition to Turbulence
,”
Scholarpedia
,
4
(
3
), p.
2072
.10.4249/scholarpedia.2072
6.
Subramanian
,
P.
,
Mariappan
,
S.
,
Sujith
,
R. I.
, and
Wahi
,
P.
,
2010
, “
Bifurcation Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube
,”
Int. J. Spray Combus. Dyn.
,
2
, pp.
325
355
.10.1260/1756-8277.2.4.325
7.
Gaster
,
M.
,
1969
, “
Vortex Shedding From Slender Cones at Low Reynolds Numbers
,”
J. Fluid Mech.
,
38
(
3
), pp.
565
576
.10.1017/S0022112069000346
8.
Venkatraman
,
D.
,
2013
, “
Computational Models of Dorsal Coverts on Birds' Wings
,” Ph.D. thesis, University of Genova, Italy.
9.
Rand
,
R.
,
Lecture notes on Nonlinear Vibrations. Version 52
. Available at: http://www.tam.cornell.edu/randdocs/ed
10.
Hinch
,
E. J.
,
1991
,
Perturbation Methods
,
Cambridge University Press
,
Cambridge, UK
.
11.
Johnson
,
R. S.
,
2005
,
Singular Perturbation Theory
,
Springer ebooks
.
12.
Holmes
,
M. H.
,
1991
, In
Introduction to Perturbation Methods
,
Springer-Verlag
,
New York
.
13.
Kevorkian
,
J.
, and
Cole
,
J. D.
,
1996
,
Multiple Scale and Singular Perturbation Methods
,
Springer-Verlag
,
New York
.
14.
Mickens
,
R. E.
,
1981
,
An Introduction to Nonlinear Oscillations
,
1st ed.
,
Cambridge University Press
Cambridge, UK.
15.
Mickens
,
R. E.
,
1996
,
Oscillations in Planar Dynamic Systems
,
World Scientific Publishing Co.
,
River Edge, New Jersey
.
16.
Chatterjee
,
A.
,
2002
,
An Elementary Continuation Technique
, http://home.iitk.ac.in/anindya/continuation.pdf
17.
Morgan
,
A.
,
2009
,
Solving Polynomial Systems using Continuation for Engineering and Scientific Problems
, Classics in Applied Mathematics, SIAM.
18.
Stoker
,
J. J.
,
1950
,
Nonlinear Vibrations in Mechanical and Electrical Systems
,
John Wiley & Sons
,
New York
.
19.
Verhulst
,
F.
,
1996
,
Nonlinear Differential Equations and Dynamical Systems
,
2nd ed.
,
Springer
,
New York
.
20.
Lanczos
,
C.
,
1986
,
The Variational Principles of Mechanics
,
Dover Publications
,
New York
.
21.
Hill
,
G. W.
,
1886
, “
On the Part of the Motion of the Lunar Perigee is a Function of the Mean Motions of the Sun and the Moon
,”
Acta Math.
,
8
, pp.
1
36
.10.1007/BF02417081
22.
Magnus
,
W.
, and
Winkler
,
S.
,
2004
,
Hill's equation
,
Dover Publications
, Mineola, NY.
23.
Rana
,
N.
, and
Joag
,
P.
,
2011
,
Classical Mechanics
,
30th ed.
,
Tata-McGraw Hill
.
24.
Sparrow
,
C.
,
1982
,
The Lorenz Equations: Bifurcations, Chaos and Strange Attractors
,
Applied Mathematical Sciences 41, Springer-Verlag
,
New York
.
25.
Rugonyi
,
S.
, and
Bathe
,
K.
,
2003
, “
An Evaluation of the Lyapunov Characteristic Exponent of Chaotic Continuous Systems
,”
Int. J. Numer. Methods Eng.
,
56
, pp.
145
163
.10.1002/nme.560
26.
Wiggins
,
S.
,
2003
,
Introduction to Applied Nonlinear Dynamics and Chaos
,
2nd ed.
,
Springer
,
New York
.
27.
Hale
,
J. K.
,
1963
,
Oscillations in Nonlinear Systems
,
1st ed.
,
Dover Publications Mineola, NY
.
28.
Drazin
,
P. G.
,
1992
,
Nonlinear Systems
,
1st ed.
,
Cambridge University Press,
Cambridge, UK.
29.
Grimshaw
,
R.
,
1991
,
Nonlinear Ordinary Differential Equations
,
Taylor & Francis
,
London
.
30.
Glendinning
,
P.
,
1994
,
Stability, Instability and Chaos
,
Cambridge University Press
, Cambridge, UK.
You do not currently have access to this content.