Introduction to Chaos!

Dr. Richard D. Neilson Department of Engineering, School of Engineering and Physical Sciences, University of Aberdeen, Aberdeen AB24 3UE E-Mail: [email protected]

Introduction z

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Introduction Review of Non-linear Dynamic Phenomena Examples of Systems Exhibiting Chaos Examples of Chaos in :– Logistic Map – Lorenz’s Equations – Two Potential Well Problem

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Conclusions Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

Definitions of Chaos (Kaos) z

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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. Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

Mathematical Definition of Chaos A 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! z

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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 z

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

•Linear Equation –Damped linear oscillator

•Nonlinear Equation –Damped Duffing oscillator –Damped pendulum

&x& + ζx& + x = 0 &x& + ζx& + x + εx 3 = 0 θ&& + ζθ& + sin θ = 0

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Non-linear Dynamic Phenomena Non-linear System

Linear System z

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Fixed natural frequency Responds at excitation frequency One solution or “attractor”

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Natural frequency depends on amplitude May respond at frequencies other than excitation frequency Possibility of multiple solutions or “attractors”

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Non-linear Phenomena – Variable Natural Frequency Linear - Small Amplitude Pendulum

Non-linear - Large Amplitude Pendulum

θ&& + θ = 0

θ&& + θ − εθ 3 = 0

θ (t) = Asin ωt ω =1

θ (t) ≈ Asin ωt ω = 1−

ε 3A2 4

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Non-linear Phenomena – Jump Phenomenon for Harmonic Excitation

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Non-linear Phenomena – Effects of Jump Phenomenon z

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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. Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

Non-linear Phenomena –Types of Response to Harmonic Excitation Harmonic Input

Linear System

Harmonic motion Harmonic motion

Harmonic Input

Nonlinear System

Sub-harmonic motion Super-harmonic motion Quasi-periodic motion Chaotic motion (random like)

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Examples of Systems Exhibiting Chaos

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Examples of Systems Exhibiting Chaos - Biological Systems z

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].

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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 z

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.

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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 z

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

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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] Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

Examples of Systems Exhibiting Chaos - Mechanical Systems z

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

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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 Map A Prey Predator Model z

Logistic Map - a prey-predator model for predicting the population of a species year on year xn +1 = rx n (1 − xn ) r - growth parameter - propensity for population to increase x - population at year n

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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

αi δ = → 4.66920... as i → ∞ βi

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β

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Period Doubling Bifurcation Feigenbaum Number z

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The Feigenbaum number is a universal constant irrespective of the dynamic system The 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 x& = −σ ( x − y ) y& = ρx − y − xz

z& = xy − β z

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 Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

Lorenz Equation - Conditions for Chaos z

Chaos occurs for σ = 10 ρ = 28 β = 8/3 – σ=10 corresponds to cold water

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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 z

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Divergence between close trajectories is measured by the Lyapunov exponent The 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 θ&& + ζθ& − θ + εθ 3 = f (t )

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Phase Plane Representation

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Two-Potential Well Divergence on Phase Plane

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Forced Two-Potential Well Chaos on Phase Plane

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Poincaré Map Representation Sampled Data

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Forced Two-Potential Well Poincaré Map - Strange Attractor

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Conclusions z

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

Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

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.

Physics Teachers CPD Day, University of Aberdeen, 26th May 2005

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.

Physics Teachers CPD Day, University of Aberdeen, 26th May 2005