Introduction to Chaos!

Dr. Richard D. Neilson

Department of Engineering,School of Engineering and Physical Sciences,

University of Aberdeen,Aberdeen AB24 3UE

E-Mail: r.d.neilson@eng.abdn.ac.uk

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Introduction

IntroductionReview of Non-linear Dynamic PhenomenaExamples of Systems Exhibiting ChaosExamples of Chaos in :-– Logistic Map– Lorenz’s Equations– Two Potential Well Problem

Conclusions

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Definitions of Chaos (Kaos)

A condition or place of great disorder or confusion.A disorderly mass; a jumble: The desk was a chaos of papers and unopened letters.The disordered state of unformed matter and infinite space supposed in some views to have existed before the ordered universe.The Fifth Rider of the Apocalypse in Terry Pratchet’s“Thief of Time” alias Ronnie Soak the dairyman.

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Mathematical Definition of ChaosA dynamical system that has a sensitive

dependence on its initial conditions.

Note -Chaos can only occur in nonlinear systems!Chaos may look “random-like” but occurs in deterministic systems with no randomness in the input or the variables!

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Poincaré’s Note on Chaos

“If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had

no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the

same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions

produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible,

and we have the fortuitous phenomenon.”in a 1903 essay “Science and Method” by Poincaré

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Review of Non-linear Dynamic Phenomena

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Nonlinear systems

Nonlinear dynamic systems contain products or functions of the dependent variable.

•Linear Equation–Damped linear oscillator 0=++ xxx &&& ζ

•Nonlinear Equation–Damped Duffing oscillator–Damped pendulum

03 =+++ xxxx εζ &&&

0sin =++ θθζθ &&&

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Non-linear Dynamic Phenomena

Non-linear SystemNatural frequency depends on amplitudeMay respond at frequencies other than excitation frequencyPossibility of multiple solutions or “attractors”

Linear SystemFixed natural frequency

Responds at excitation frequency

One solution or “attractor”

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Non-linear Phenomena – Variable Natural FrequencyLinear - Small

Amplitude PendulumNon-linear - Large

Amplitude Pendulum

03 =−+ εθθθ&&0=+θθ&&

θ (t) ≈ Asinωtθ (t) = Asinωt

ω = 1−ε 3A2

4ω =1

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Non-linear Phenomena – Jump Phenomenon for Harmonic Excitation

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Non-linear Phenomena – Effects of Jump Phenomenon

Sudden changes in amplitude of vibration can occur for small changes in frequency.It is possible to have more than one stable solution at a particular frequency.The region E-B is unstable.Initial conditions determine which of the two solutions is attained, e.g. a large initial velocity may jump the system to the upper solution.

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Non-linear Phenomena –Types of Response to Harmonic Excitation

LinearSystem

NonlinearSystem

HarmonicInput

Harmonic motion

Sub-harmonic motion

Super-harmonic motion

Quasi-periodic motion

Chaotic motion (random like)

HarmonicInput

Harmonic motion

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Examples of Systems Exhibiting Chaos

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Examples of Systems Exhibiting Chaos - Biological Systems

Prey-predator models– Models describing the interaction between predators and

their prey to investigate species population year on year. Described initially by Robert May [1].

Human physiology– Brain - normal brain activity is thought to be chaotic. – Heart - normal heart activity is more or less periodic but has

variability thought to be chaotic. Fibrillation (loss of stability of the heart muscle) is thought to be chaotic.

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Examples of Systems Exhibiting Chaos - Fluid Systems

Weather systems– Models of the weather including convection, viscous effects

and temperature can produce chaotic results. First shown by Edward Lorenz in 1963 [2]. Long term prediction is impossible since the initial state is not known exactly.

Turbulence– Experiments and modelling show that turbulence in fluid

systems is a chaotic phenomenon.

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Examples of Systems Exhibiting Chaos - Mechanical Systems

Systems with clearance– Gear systems - gears can “rattle” against each other in a

chaotic manner– Rotor systems - clearance in bearings can induce chaos

which can be used to diagnose bearing faults

Two potential well system– If a pendulum or the tip of a cantilever beam is set up

between two strong magnets the pendulum or cantilever will be attract to one or other magnet. The final solution of which attractor is achieved is chaotic. The beam problem was reported by Frank Moon [3]

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Examples of Systems Exhibiting Chaos - Mechanical Systems

Systems with clearance– Gear systems - gears can “rattle” against each other in a

chaotic manner– Rotor systems - clearance in bearings can induce chaos

which can be used to diagnose bearing faults

Double Pendulum– A double pendulum can exhibit regular motion or highly

irregular chaotic motion. Starting the pendulum from similar positions results in different motion in the long term.

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Chaos in the Logistic Map

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Example 1 - Logistic MapA Prey Predator Model

Logistic Map - a prey-predator model for predicting the population of a species year on year

r - growth parameter - propensity for population to increasex - population at year n

xn +1 = rxn(1 − xn )

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Route to Chaos - Period Doubling Bifurcation

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Routes to Chaos

r =2.5

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Routes to Chaos

r =3.3

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Routes to Chaos

r =3.5

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Routes to Chaos

r =3.55

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Routes to Chaos

r =3.8

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Period Doubling Bifurcation -Feigenbaum Number

α βδ =

αi

βi

→ 4.66920... as i → ∞

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Period Doubling Bifurcation -Feigenbaum Number

The Feigenbaum number is a universal constant irrespective of the dynamic systemThe value of 4.66920… is achieved as the bifurcations tend to infinity i.e. the ratio of the range of growth parameter between consecutive bifurcations tends to the Feigenbaum number as more bifurcations are occur.

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Divergence of Close Solutions of Logistic Map

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Chaos in the Lorenz’s Equations

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Example 2 -Lorenz’s Equations An Atmospheric Model

zxyzxzyxy

yxx

βρσ

−=−−=−−=

&

&

& )(

x - Rotation of the eddy- clockwise +ve, anticlockwise -ve

y - Horizontal temperature distribution - if x and y have the same sign then then warmer fluid is on the side of the eddy which is rising

z - Vertical temperature profile

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Lorenz Equation - Conditions for Chaos

Chaos occurs for

– σ=10 corresponds to cold water

σ = 10ρ = 28β = 8 / 3

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Routes to Chaos

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Divergence of Close Solutions of Lorenz Equation

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Divergence of Close Solutions Lyapunov Exponent

Divergence between close trajectories is measured by the Lyapunov exponentThe Lyapunov exponent is calculated by propagating two initially close trajectories and measuring the divergence in each dimension with time.One positive Lyapunov exponent for a system implies chaotic motion.

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Chaos in the Two Potential Well Problem

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Example 3 - Two potential Well A Mechanical Model

Pendulum attracted by two magnets

Governing equation

)(3 tf=+−+ εθθθζθ &&&

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Phase Plane Representation

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Two-Potential WellDivergence on Phase Plane

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Forced Two-Potential Well Chaos on Phase Plane

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Poincaré Map RepresentationSampled Data

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Forced Two-Potential Well Poincaré Map - Strange Attractor

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Conclusions

Chaos occurs only in non-linear systemsChaos occurs in a wide variety of systemsChaotic systems have a high sensitivity to initial conditionsChaotic motion may appear “random” but has underlying structure - the chaotic attractor

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Any questions?

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References

1. May, R. M., (1976), Simple mathematical models with very complex dynamics, Nature, 261, 459-467.

2. Lorenz, E. N., (1963), Deterministic nonperiodic flow, Journal of Atmospheric Science, 20, 130-141.

3. Moon, F.C. and Holmes, P. J.,(1979), A magnetoelastic strange attractor, Journal of Sound and Vibration, 65, 285-296.

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Other Reading

1. Gleick, J., (1988), Chaos : making a new science. New York, U.S.A. : Penguin, 1988.

2. Moon F. C. (1992), Chaotic and fractal dynamics : an introduction for applied scientists and engineers , New York, U.S.A. : Wiley, 1992.