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Home , Bifurcation diagram, James Gleick, Nonlinear system

Introduction to Nonlinear Physics IntroductionLecture I to Non-linear physics

Lecture 1

Tobias Galla

Anne Juel Nonlinear Dynamics room G.12 [email protected] http://www.maths.manchester.ac.uk/~ajuel / Adapted from T. Galla’s slides Nonlinear physics

Why did you choose this course ? Reasons why you should choose this course

• interesting and timely topic

• combination of mathematical methods, numerical studies and experimental applications.

• most phenomena in physics due to nonlinear effects

• applications in a variety of different fields (in physics, but also biology, epidemiology, economics ... you name it)

• essential for a broad . Reasons why you should not choose this course Reasons why you should not choose this course

None Nonlinear physics

Why did I choose to give this course ? • Research in and soft matter : focus on instabilities in and , which are inherently nonlinear phenomena.

• Manchester Centre for Nonlinear Dynamics : multi- pronged approach of quantitative experiments and mathematical analysis/numerical simulations.

• Concepts of nonlinear dynamics fundamental to understanding complex . e.g. and evolutionary dynamics, biological systems (gene regulation, epidemics, metabolic systems), models in economics, models of socio-dynamics (traffic, opinion spreading, decision making). • Lots of interest in the public Nonlinear dynamics in a nutshell

1.

http://en.wikipedia.org/wiki/Fractal 2.

Bifurcation diagram of the logistic : Universality in chaos

http://en.wikipedia.org/wiki/Logistic_map 3.

Lorenz model for weather prediction

[www.metoffice.gov.uk ] [wikipedia ] Historical perspective

Nonlinear dynamics and is contemporary !

Edward Norton Lorenz (May 23 1917 - April 16 2008)

": Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?"

[wikipedia ] Archimedes - Eureka

What year ?

[wikipedia ] Archimedes - Eureka

What year ?

Answer: 287-212 BC

[wikipedia ] Galileo objects fall at rate independent of

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[wikipedia ] Galileo objects fall at rate independent of mass

1589

[wikipedia ] "Mathematical Principles of Natural Philosophy " Unified the three laws of motion

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[wikipedia ] Isaac Newton "Mathematical Principles of Natural Philosophy " Unified the three laws of motion

1687

[wikipedia ] Andre Ampere force on electric current in a magnetic field

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[wikipedia ] Andre Ampere force on electric current in a magnetic field

1820

[wikipedia ] James Clerk Maxwell

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[wikipedia ] James Clerk Maxwell equations

1864

[wikipedia ] Schroedinger

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[www.nobelprice.org ] Schroedinger equation

1926

[www.nobelprice.org ] Murray Gell-Mann (+Zweig) “Quarks”

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[http://asymptotia.com/2009/09/16/happy-birthday-murray-gell-mann/] Murray Gell-Mann (+Zweig) “Quarks”

1964

[http://asymptotia.com/2009/09/16/happy-birthday-murray-gell-mann/] Nonlinear dynamics and chaos

1961 Lorenz chaos theory

[wikipedia ] Nonlinear dynamics and chaos 1961 Lorenz chaos theory 1962 Mandelbrot

[wikipedia ] Nonlinear dynamics and chaos 1961 Lorenz chaos theory 1962 Mandelbrot Mandelbrot set 1975 Feigenbaum universality in nonlinear systems

[http://www.rockefeller.edu/research/l abmembers.php?id=38&memberId=6 2] [wikipedia ] Nonlinear dynamics and chaos 1961 Lorenz chaos in weather prediction model 1962 Mandelbrot 1975 Li & Yorke coin the term “Chaos” 1975 Feigenbaum universality 1976 May chaos in

[http://www.zoo.ox.ac.uk/staff/ academics/may_r.htm] [wikipedia ] 1987 Chaos becomes trendy, “Chaos” is published by James Gleick. Two views of

Deterministic Probabilistic • Laplace: We ought to regard • Maxwell: The true logic of the present state of the the world is the calculus of universe as the effect of the probabilities. preceding state and the cause of the succeeding state. • Newtonian dynamics: Present state Predict future.

• Examples of deterministic • Examples of probabilistic processes are: processes are: -Planetary motions, -Many body problems, -Fluid motion, -Coin toss, -Weather prediction, -Throw of dice. -Containment of plasmas. Two views of classical physics

Deterministic Probabilistic • Write down equations. • In principle, each of the Cannot solve them! above is deterministic, but the probabilistic approach is • Is it just the case of waiting a useful means of until a large enough representing complicated computer is available? problems. • Three body problem - • Success of probabilistic restricted versions have approach in statistical exact solutions but general mechanics, for instance. form… Looks like a random process! Nonlinearity • The source of the ‘difficulty’ is nonlinearity . • Newton's for a ‘simple’ problem such as the three body problem are nonlinear due to coupling between the three bodies.

Poincaré 1903

• Nonlinearity means that the output of a is not proportional to its input. • Often no analytical solution to nonlinear equations! Deterministic dynamics: Deterministic dynamics: ‘Chaos’ in common parlance

Football crowd.

• The scientific meaning of chaos is much more subtle. • It refers to a well-characterised state of a . • An important signature of chaos is the `, i.e. extreme sensitivity to initial conditions. Extreme sensitivity to initial conditions • Two velocity records obtained from numerical integrations of the equations of motion of the parametrically excited in the chaotic regime. • Angular position q versus time. • Starting conditions differ by 1 part in 10 6. • Butterfly effect Unpredictable dynamics Poincaré (1903): pioneer of chaos • After tackling the 3-body problem, Poincaré identified the phenomenon of sensitive dependence on initial conditions (SDIC), this provided a definition of “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 is 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.” [H. Poincaré, 1903] Some highlights

Nonlinear systems in one dimension stability analysis, bifurcations, numerical methods Nonlinear systems in two and three dimensions stability analysis, impossibility of chaos in 2D Spatio-temporal dynamics and . chaos in fluid dynamics. Chaos in discrete maps logistic map, Lyapunov exponent, Feigenbaum diagram Fractals (non-integer) dimensions, Mandelbrot set. Strange stretching and folding, of volumes in At the end of this course you should • be able to analyse (simple) non-linear systems

• have a good control over the fundamental mathematical and numerical techniques used to study nonlinear systems

• understand what constitutes chaotic behaviour

• know the basic concept of a fractal and to explain the idea of a non-integer dimension

• know how these ideas can be used to progress our understanding of complicated behaviour in practice Some applications of non-linear dynamics

basically everything in physics/science • pattern formation, fluid dynamics, turbulence • lasers, non-linear optics • heart cell synchronisation • eco-systems, predator-prey, • plasma physics • chemical kinetics • non-linear • brain, neural networks, cortex • learning dynamics, game theory • ... Some general points: • Two lectures a week: •Monday 9-10am (Chaplaincy) •Wednesday 12-1pm (Moseley, Schuster)

• Course is going to be fairly mathematical, so you have to study and revise between lectures.

• There will be 4-5 example sheets, you should do these problems either by yourself or in groups

• Material will be posted on Blackboard 9

• Exam (January): 1hr30min Finally

• give me

• complain if too fast or too slow

• tell me if something is unclear

• tell me how the course can be improved Some general points:

Never be afraid to ask questions:

• during the lectures

• immediately after the lectures

• come to my office (G.12 Schuster)

• email: [email protected] Recommended textbook

S. H. Strogatz, “Nonlinear dynamics and chaos”, Perseus Publishing 2000

Useful references G.L. Baker and J.P. Gollub “Chaotic dynamics: an introduction” 2nd ed. (CUP 1996)

D.W. Jordan and P. Smith, “Nonlinear ordinary differential equations” 3rd ed. (OUP 1999)

T. Mullin “The nature of chaos” (OUP 1993) Further reading J. Gleick, “Chaos: making of a new science” (Heinemann 1988)