Tuesday, January 19, 2010

NONLINEAR DYNAMICS

NON-LINEAR DYNAMICS and APPLICATIONS for PHYSICS
(work in progress, stay tuned)

Much of this blog article is my interpretation of things learned from Cornell University Mathematics professor Dr. Stephen Strogatz's 1994 book:  Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering

Outline

I. Dynamics - HISTORY
II. Dynamics - Mathematics - forthcoming, see Calculus IV in the meantime 
III. Linear Dynamics - Applications in Physics
IV. Nonlinear Dynamics - Applications in Physics
V. Nonlinear Dynamics - Applications in Other Fields
VI. Personal History

I. Dynamics - HISTORY

A. Issac Newton - Differential Equations and the Two-Body Problem
B. Henri Poincaré - Qualitative Geometric Approach; Chaos
C. Early-Mid 20th Century Mathematicians (Birkhoff, Kolmogorov, Arnold, Moser)- Nonlinear oscillators; Complex behavior in Hamiltonian mechanics
D. Edward Lorenz - 1963 Discovery of Chaotic Motion on a Strange Attractor
E. 1970's Mathematicians (Ruelle & Takens, May, Mandelbrot) - Turbulence in Fluids; Population Biology; Fractals
F. 1970's Physics - Mitchell Feigenbaum: Universality and renormalization; Chaos and Phase transitions linked
G. 1970's Biology - Arthur Winfree: Nonlinear oscillators in biology
H. 1980's to present - Widespread interest in chaos, fractals, oscillators and their applications

II. Dynamics - Mathematics - forthcoming

III. Linear Dynamics - Applications in Physics
      
     A. N= 1 variable (Growth, Decay, or Equilibrium)
          
          1. Exponential Growth
          2. RC Circuit
          3. Radioactive decay 

      B. N=2 variables (Oscillations)

          1. Linear oscillator
          2. Mass and spring
          3. RLC Circuit
          4. 2-body problem (Kepler, Newton)

     C. N>=3 variables

          1. Civil engineering, structures
          2. Electrical Engineering
     
     D. N >>> 1 variables (Collective phenomena)

          1. Coupled harmonic oscillators
          2. Solid-state physics
          3. Molecular dynamics
          4. Equilibrium statistical dynamics

     E. Continuum (Waves and patterns)

          1. Elasticity
          2. Wave equations
          3. Electromagnetism (Maxwell)
          4. Quantum mechanics (Schrodinger, Heisenberg, Dirac)
          5. Heat and diffusion
          6. Acoustics
          7. Viscous fluids
 


IV. Nonlinear Dynamics - Applications in Physics

       A. N=1 variable

          1. Fixed points
          2. Bifurcations
          3. Overdamped systems, relaxation mechanics
          4. Logistic equation for single species

       B. N=2 variables
          
          1. Pendulum
          2. Anharmonic oscillators
          3. Limit cycles
          4. Biological oscillators (neurons, heart cells)
     
      C. N>=3 variables  (Chaos)

          1. Strange attractors  (Lorenz)
          2. 3-body problem (Poincare)
          3. Chemical kinetics
          4. Iterated maps (Feigenbaum)
          5. Fractals (Mandelbrot)
          6. Forced nonlinear oscillators (Levinson, Smale)
          7. Practical uses of chaos
          8. Quantum chaos ?

       D. N >>> 1 variables

          1.  Coupled nonlinear oscillators
          2. Lasers, nonlinear optics
          3. Nonequilibrium statistical mechanics
          4. Nonlinear solid-state physics (semiconductors)
          5. Josephson arrays
          6. Heart cell synchronization
          7. Neural networks
          8. Immune systems
          9. Ecosystems
          10. Economics

       E. Continuum  (Spatio-temporal complexity)

          1. Nonlinear waves (shocks, solitons)
          2. Plasmas
          3. Earthquakes
          4. General Relativity (Einstein)
          5. Quantum field theory
          6. Reaction-diffusion, biological and chemical waves
          7. Fibrillation
          8. Epilepsy
          9. Turbulent fluids (Navier-Stokes)
          42. Life
 


V. Nonlinear Dynamics - Applications in Other Fields - forthcoming


Click here  to see a highly interactive version of Brian Castellani's Complexity Map, as shown below:




VI. Personal History

      I'm as happy as a Philosophy Grad Student at the beginning of  his first lecture of his first teaching job teaching "Introduction to Plato". *

Why?

Because, I've finally found my specialty, thanks to United Parcel Service delivering the following book from Amazon to my doorstep yesterday:



By the way, I really hate, loathe and despise the term "Chaos Theory." The proper description would be "Structure-in-Chaos" Theory. It's young, it's happening, it's taking off in many different fields, and the high-speed computers of Computer Scientists and their wonderful Algorithms are its very best friend. We have miles to go before we sleep. Time to get cracking! :-)

From Wikipedia, at which an input of "Nonlinear Dynamics" directs to "Nonlinear differential equations" under "Nonlinear systems." Under that section it states:

A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics, the Lotka–Volterra equations in biology, and the Black–Scholes PDE in finance.
One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.

Going to the top of the page:
 
In mathematics, a nonlinear system is a system which is not linear, that is, a system which does not satisfy the superposition principle, or whose output is not proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables.
Nonlinear problems are of interest to physicists and mathematicians because most physical systems are inherently nonlinear in nature. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos. The weather is famously nonlinear, where simple changes in one part of the system produce complex effects throughout.

And that's it for today. For all my regular readers (all 4 of you ... it would be 5 but Mom passed away in 2008) I'm afraid I will spend less time on-line and at this blog, as I have much to read. I won't go away completely, but for the most part I'll be incognito. Cheers and farewell, and here's hoping I do Mom proud, wherever she is, when I publish my first paper in 6 months to 3 years, or so.

* - most of whom start off with: "I am SO ENVIOUS of you people! You are about to hear about Plato for the FIRST time!" They have their point.

Sincerely,
S'Colyer

P.S. For your viewing and listening pleasure (subjective), a VERY non-linear song:



Here is the Number One most popular song in America today, Empire State of Mind by Jay-Z and Alicia Keys. Keys' bits are beautifully linear, Jay-Z's nonlinear. Somehow, they blend well:

8 comments:

Jérôme CHAUVET said...

Ya-hoooo!!! Dear Steven, you seem to be quite much very highly motivated by your project !!!... Nonlinearity is some fascinating area of mathematics, good choice (a passion of mine too...). In fact, when you understand it, you will surely think linearity is the definition of boredom in the Nature. All interesting things are non linear,that is the rule.

When you feel you are ready to, we can think of collaborating for some project, at least discuss about it.

Keep going!

Best,

Jérôme CHAUVET said...

Probably due to your particular way of thinking here and there, anyway and everywhere, it is to me no big surprise that chaos theory and you both have this reciprocal feeling of kinship (see what I mean?)

Best,

Steven Colyer said...

Thanks for your lovely and kind thoughts, but actually Jerome, the whole "Romeo-and-Juliet" LINEAR Dynamics in Strogatz' book interests me greatly. As in: FUNNY! And funny in a wonderful way.

It's totally worthy of it's own blarticle, "The Mathematics of Love Affairs," but it also reminds me that a website such as blogspot may not be the way to go, as Tex/LaTex is definitely needed to explain it, so perhaps I shall start a second, and hopefully last blog, at ScientificBlogging.com, unless you or another can show me how to introduce dot notation, here.

Jérôme CHAUVET said...

For putting formulas into your column:

(1) Write your formulas in Word Equation Editor. Save the corresponding ".doc" file.

(2) Load your ".doc" from Openoffice. Doing so, you automatically are in the word processor.

(3) Open Openoffice drawing program. Copy-paste formulas from the text file to the drawing file.

(3) Export the whole page as ".bmp" file.

(4)Load the ".bmp" from GIMP or Photoshop. As the formulas may appear in the middle of a large white page, you should cut the white margins all around so as to frame your formulas.

(5)Save as ".jpg"

(6) Load image into your post.

Another solution is to blog with wordpress, which seems to allow shoving LateX code into the posts.

Best,

Steven Colyer said...

Thanks, Jérôme, I like that last solution the best!

Currently, I'll be outlining the field a bit more here based on Strogatz's book and other reading in the field. Next up, an outline based on Strogatz's table of the many fields within ND based on the Degree of linearity vs Number of variables ... then after a week decide which of them to zone in on.

I've got some screwy thoughts on QM + ND in the interior of a 3-particle baryon, such as the proton and neutron.

Quantum chromodynamics is another of my (far too many) favorite topics.

Steven Colyer said...

I have completed the outline on parts III and IV. - Applications.

Review and comment pls, thanks.

Steven Colyer said...

by Strogatz, page 8 Sec. 1.2

Why Are Nonlinear Problems So Hard?

As we've mentioned earlier, most nonlinear systems are impossible to solve analytically. Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts.

But many things in nature don't act this way. Whenever parts of a system interfere, or cooperate, or compete, there are nonlinear interactions going on. Most of everyday life is nonlinear, and the principle of superposition fails spectacularly. If you listen to your two favorite songs at the same time, you won't get double the pleasure! Within the realm of physics, nonlinearity is vital to the operation of a laser, the formation of turbulence in a fluid, and the superconductivity of Josephson junctions.

Jérôme CHAUVET said...

Solving nonlinear problems, especially in the case of differential equations, will result in trying to solve high degree polynomials, of which we know they require new functions or some complicated hypergeometric series to be solved.

There are however ways to solve pure nonlinear systems (i.e. nonlinear systems for which dimensions cannot be uncoupled thanks to some inner symmetries) using the W Lambert function, a forgotten though important function (it is the inverse function of f(x)=x.exp x)

What is fascinating in nonlinearity is that the modification of a single paramater may lead to a wide variety of behaviors. This is what happens with the celebrated logistic equation, which is still extensively studied for some particular values of its lambda parameter nowadays, although the basic map is drastically simple :

x = L*x*(1-x) with 0 < L < 4

The issue is then to determine a measure probability fitting with the dynamics, which is not automatically obtained from one value of L to the other. Each value of the control parameter is an outstanding challenge.

Implementing nonlinearity is crucial because it models an intensivity relation between variables, which is current in the Nature and cannot be bypassed. In Volterra's ecosystem model, two species encounters according to an intensive relation, which one models by multiplying their number at each step of time.

Addressing nonlinearity promises much, the reason why it is irresistible.

Best,