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Tutorial

# Nonlinear Dynamical Systems, Their Stability, and ChaosLecture 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,
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

## Abstract

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

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## Figures

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.

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.

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.

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).

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.

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.

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.

Fig. 8

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

Fig. 9

Subcritical and supercritical bifurcations; this figure is taken from Wikipedia

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.

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.

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.

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).

Fig. 14

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

Fig. 15

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

Fig. 16

Harmonic response of Duffing equation for different forcing amplitudes γ

Fig. 17

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

Fig. 18

Inverted pendulum, harmonically excited in y-direction

Fig. 19

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

Fig. 20

Trajectories of the logistic map for different values of r

Fig. 21

Bifurcation diagram of a logistic map

Fig. 22

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

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

Fig. 24

Numerical calculation of the largest Lyapunov exponent

Fig. 25

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

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