Tobias Galla
Introduction to Non-linear physics
Lecture 1
Anne JuelNonlinear Dynamics
room [email protected]://www.maths.manchester.ac.uk/~ajuel/
Introduction to Nonlinear PhysicsLecture I
Adapted from T. Galla’s slides
Why did you choose this course ?
Nonlinear physics
• interesting and timely topic
• combination of mathematical methods, numerical studiesand 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 Physics education.
Reasons why you should choose this course
Reasons why you shouldnot choose this course
Reasons why you shouldnot choose this course
None
Why did I choose to give this course ?
Nonlinear physics
• Research in complexity and soft matter: focus on instabilities in fluid dynamics and solid mechanics, 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 systems.
• Lots of interest in the public
e.g. game theory and evolutionary dynamics, biological systems (gene
regulation, epidemics, metabolic systems), models in economics,
models of socio-dynamics (traffic, opinion spreading, decision making).
http://en.wikipedia.org/wiki/Fractal
Nonlinear dynamics in a nutshell
1.
Bifurcation diagram of the logistic map:Universality in chaos
http://en.wikipedia.org/wiki/Logistic_map
2.
[www.metoffice.gov.uk] [wikipedia]
Lorenz model for weather prediction
3.
"Predictability: Does the Flap of aButterfly's Wings in Brazil Set Offa Tornado in Texas?"
Edward Norton Lorenz (May 23 1917 - April 16 2008)
[wikipedia]
Historical perspective
Nonlinear dynamics and chaos theory is
contemporary science!
Archimedes - Eureka
What year ?
[wikipedia]
Archimedes - Eureka
What year ?
287-212 BC
Answer:
[wikipedia]
Galileo objects fall at rate independent of
mass
?
[wikipedia]
Galileo objects fall at rate independent of
mass
1589
[wikipedia]
?
[wikipedia]
Isaac Newton "Mathematical Principles of Natural Philosophy"
Unified the three laws of motion
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
?
[wikipedia]
Andre Ampere force on electric current in a
magnetic field
1820
[wikipedia]
James Clerk Maxwellequations
?
[wikipedia]
James Clerk Maxwellequations
1864
[wikipedia]
Schroedingerequation
?
[www.nobelprice.org]
Schroedingerequation
1926
[www.nobelprice.org]
Murray Gell-Mann (+Zweig)“Quarks”
?
[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 chaos1961 Lorenz chaos theory1962 Mandelbrot Mandelbrot set
[wikipedia]
Nonlinear dynamics and chaos
1961 Lorenz chaos theory1962 Mandelbrot Mandelbrot set1975 Feigenbaum universality in
nonlinear systems
[wikipedia]
[http://www.rockefeller.edu/research/labmembers.php?id=38&memberId=6
2]
Nonlinear dynamics and chaos1961 Lorenz chaos in weather prediction model 1962 Mandelbrot fractals1975 Li & Yorke coin the term “Chaos”1975 Feigenbaum universality1976 May chaos in logistic map
[wikipedia][http://www.zoo.ox.ac.uk/staff/
academics/may_r.htm]
1987 Chaos becomes trendy, “Chaos” is published by James Gleick.
Two views of classical physics
Deterministic
• Laplace: We ought to regard the present state of the universe as the effect of the preceding state and the cause of the succeeding state.
• Newtonian dynamics: Present state Predict future.
• Examples of deterministic processes are: -Planetary motions, -Fluid motion, -Weather prediction, -Containment of plasmas.
Probabilistic
• Maxwell: The true logic of the world is the calculus of probabilities.
• Examples of probabilistic processes are: -Many body problems, -Coin toss, -Throw of dice.
Two views of classical physics
Deterministic
• Write down equations. Cannot solve them!
• Is it just the case of waiting until a large enough computer is available?
• Three body problem -restricted versions have exact solutions but general form… Looks like a random process!
Probabilistic
• In principle, each of the above is deterministic, but the probabilistic approach is a useful means of representing complicated problems.
• Success of probabilistic approach in statistical mechanics, for instance.
Nonlinearity
• The source of the ‘difficulty’ is nonlinearity.
• Newton's equations of motion for a ‘simple’ problem such as the three body problem are nonlinear due to coupling between the three bodies.
• Nonlinearity means that the output of a system is not proportional to its input.
• Often no analytical solution to nonlinear equations!
Poincaré1903
Deterministic dynamics: linear system
Deterministic dynamics: nonlinear system
‘Chaos’ in common parlance
• The scientific meaning of chaos is much more subtle. • It refers to a well-characterised state of a dynamical system.• An important signature of chaos is the `Butterfly effect, i.e. extreme
sensitivity to initial conditions.
Football crowd.
Extreme sensitivity to initial conditions
• Two velocity time series records obtained from numerical integrations of the equations of motion of the parametrically excited pendulum in the chaotic regime.
• Angular position q versus time.
• Starting conditions differ by 1 part in 106.
• 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 dimensionstability analysis, bifurcations, numerical methods
Nonlinear systems in two and three dimensionsstability analysis, impossibility of chaos in 2D
Spatio-temporal dynamics and pattern formation.chaos in fluid dynamics.
Chaos in discrete mapslogistic map, Lyapunov exponent, Feigenbaum diagram
Fractalsfractal (non-integer) dimensions, Mandelbrot set.
Strange attractorsstretching and folding, evolution of volumes in phase space
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, population dynamics• plasma physics• chemical kinetics• non-linear electronics• 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 feedback
• complain if too fast or too slow
• tell me if something is unclear
• tell me how the course can be improved
Some general points:
• during the lectures
• immediately after the lectures
• come to my office (G.12 Schuster)
• email: [email protected]
Never be afraid to ask questions:
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)
J. Gleick, “Chaos: making of a new science” (Heinemann 1988)
Further reading
