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Tutorial

Nonlinear Dynamical Systems, Their Stability, and Chaos
Lecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013

[+] Author and Article Information
Amol Marathe

Birla Institute of Technology and Science,
Pilani 333031, India
e-mail: marathe.amol@gmail.com

Rama Govindarajan

TIFR Centre for Interdisciplinary Sciences,
Tata Institute of Fundamental Research,
Narsingi, Hyderabad 500075, India
e-mail: rama@tifrh.res.in

We assume that X˜(t) is diagonalizable.

Manuscript received July 8, 2013; final manuscript received December 6, 2013; published online March 24, 2014. Assoc. Editor: Ardeshir Hanifi.

Appl. Mech. Rev 66(2), 024802 (Mar 24, 2014) (16 pages) Paper No: AMR-13-1048; doi: 10.1115/1.4026864 History: Received July 08, 2013; Revised December 06, 2013

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.

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References

Strogatz, S., 2001, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, 1st ed., Westview Press Boulder, CO.
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. [CrossRef]
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. [CrossRef]
Duguet, Y., Schlatter, P., and Henningson, D. S., 2009, “Localised Edge States in Plane Couette Flow,” Phys. Fluids, 21, p. 111701. [CrossRef]
Manneville, P., and Pomeau, Y., 2009, “Transition to Turbulence,” Scholarpedia, 4(3), p. 2072. [CrossRef]
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. [CrossRef]
Gaster, M., 1969, “Vortex Shedding From Slender Cones at Low Reynolds Numbers,” J. Fluid Mech., 38(3), pp. 565–576. [CrossRef]
Venkatraman, D., 2013, “Computational Models of Dorsal Coverts on Birds' Wings,” Ph.D. thesis, University of Genova, Italy.
Rand, R., Lecture notes on Nonlinear Vibrations. Version 52. Available at: http://www.tam.cornell.edu/randdocs/ed
Hinch, E. J., 1991, Perturbation Methods, Cambridge University Press, Cambridge, UK.
Johnson, R. S., 2005, Singular Perturbation Theory, Springer ebooks.
Holmes, M. H., 1991, In Introduction to Perturbation Methods, Springer-Verlag, New York.
Kevorkian, J., and Cole, J. D., 1996, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York.
Mickens, R. E., 1981, An Introduction to Nonlinear Oscillations, 1st ed., Cambridge University Press Cambridge, UK.
Mickens, R. E., 1996, Oscillations in Planar Dynamic Systems, World Scientific Publishing Co., River Edge, New Jersey.
Chatterjee, A., 2002, An Elementary Continuation Technique, http://home.iitk.ac.in/anindya/continuation.pdf
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Stoker, J. J., 1950, Nonlinear Vibrations in Mechanical and Electrical Systems, John Wiley & Sons, New York.
Verhulst, F., 1996, Nonlinear Differential Equations and Dynamical Systems, 2nd ed., Springer, New York.
Lanczos, C., 1986, The Variational Principles of Mechanics, Dover Publications, New York.
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. [CrossRef]
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Sparrow, C., 1982, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Applied Mathematical Sciences 41, Springer-Verlag, New York.
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. [CrossRef]
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Figures

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Fig. 1

A sample solution of the velocity profiles in a boundary layer, as described by the Falkner–Skan equation. Both solutions satisfy all three boundary conditions. Here β = -0.1. The solid line corresponds to f"(0) = 0.1644, whereas the dashed line corresponds to a separated velocity profile, with f"(0) = -0.0545.

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Fig. 2

Standard fixed points and their phase portraits, along with their canonical linear equations. Made using1. The tiny arrows indicate the direction in which the solutions move as time progresses.

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Fig. 3

A cylindrical blob of fluid displaced slightly from its original location in a stratified fluid will display simple harmonic motion in the absence of diffusion. The z-axis is upwards and density increases as z decreases.

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Fig. 4

A phase plane of a nonlinear system with multiple fixed points. There are two saddles, at (0,0) and (1.1, −1.59), a stable node, at (-0.59,-0.77), and an unstable node at (0.82,1.17).

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Fig. 5

The flow past a cylinder of square cross section. At a Reynolds number Re = 30, a steady bubble is formed behind the cylinder. A Hopf bifurcation occurs at a Reynolds number of about 46, and oscillatory states with Karman vortex streets are evident at Re = 50 and 100. Note that the length scale in different at Re = 30. Figure courtesy: Srikanth Toppaladoddi.

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Fig. 6

Phase portrait in a transcritical bifurcation. The exchange of stabilities between the two fixed points as r crosses zero is evident in this figure. In this and subsequent figures, filled circles correspond to stable fixed points, and open circles indicate unstable fixed points.

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Fig. 7

The nature of fixed points in a transcritical bifurcation. As r crosses zero, the two fixed points exchange stabilities. In such diagrams solid lines correspond to stable fixed points, whereas dashed lines show how unstable fixed points move as r varies.

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Fig. 8

Phase portrait for a prototypical saddle-node bifurcation. The two fixed points annihilate each other.

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Fig. 9

Subcritical and supercritical bifurcations; this figure is taken from Wikipedia

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Fig. 10

Top view of the boundary layer on a flat plate, sketch of Roddam Narasimha. The thick vertical line marks the transition from periodic to chaotic flow.

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Fig. 11

A schematic of the Craik mode of secondary instability: an example of spatial period doubling. The lines show maxima in the disturbance amplitude, which move downstream as well as in the spanwise direction in a sinusoidal manner.

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Fig. 12

Phase plane of van der Pol oscillator with ϵ = 0.1. The dash-dot trajectory emerges out of the origin and approaches the limit cycle, while the pink line begins from infinity and moves inwards towards the limit cycle. The limit cycle is where the pink and blue trajectories meet.

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Fig. 13

Multiple periodic solutions for the forced Duffing oscillator with α = 1, δ = 0.1, β = 1, γ = 1, ω = 2.5 with initial conditions IC1 = (1,0) (small amplitude) and IC2 = (7,0) (large amplitude).

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Fig. 14

Harmonic response of the Duffing equation for different strengths of nonlinearity α

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Fig. 15

Nonlinear harmonic response of the Duffing equation for different damping coefficients δ

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Fig. 16

Harmonic response of Duffing equation for different forcing amplitudes γ

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Fig. 17

Poincaré section for the forced Duffing oscillator for δ=0.1,β=1,α=0.25,γ=1.38,ω=2

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Fig. 18

Inverted pendulum, harmonically excited in y-direction

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Fig. 19

Movement of Floquet multipliers with respect to parameters of f(t)

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Fig. 20

Trajectories of the logistic map for different values of r

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Fig. 21

Bifurcation diagram of a logistic map

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Fig. 22

Phase portrait for σ = 10.0,b = (8/3),r = 28.0 with IC (0.01,0,0)

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Fig. 23

Two trajectories for the Lorenz system with ICs (1.5,0,0) and (1.5 + 0.001,0,0) diverge appreciably after t = 25

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Fig. 24

Numerical calculation of the largest Lyapunov exponent

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Fig. 25

Three Lyapunov exponents of the Lorenz system for σ = 10.0,b = (8/3),r = 28.0

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