Transitions to Chaos
Phase transitions... Liquid state is where complex behavior can arise
Solids - Total order
Gases - Total Randomness
Liquids - Chaos
Equilibrium --> Periodicity --> Nonperiodic/Chaos
How does a system Turn Chaotic?
Up until now we have been describing the properties of chaos.
Different things can go Chaotic in the same way - Interdisciplinary
Three Big Ideas
1) Iterated Maps
2) Bifurcations and the transition to chaos
3) Linearity vs Nonlinearity
Solids - Total order
Gases - Total Randomness
Liquids - Chaos
Equilibrium --> Periodicity --> Nonperiodic/Chaos
How does a system Turn Chaotic?
Up until now we have been describing the properties of chaos.
Different things can go Chaotic in the same way - Interdisciplinary
Three Big Ideas
1) Iterated Maps
2) Bifurcations and the transition to chaos
3) Linearity vs Nonlinearity
Iterated Maps
LosiSquare root button, type a number and keep pressing squareroot.
You are iterating a map which is the square root function.
Rule that takes a number to another number.
Iterated maps are repetitive processes where you feed an output back into the map/function as an input.
Two other iterated maps, one with a rounded top.
Other is a straight line.
Iterated maps one of the two main types of dynamics along with differential equations.
Stroboscopic view.
Iterated maps provide simplest model from transitioning from order to chaos.
Bank Account
10% annual interest... 1.1^n
Biology and Insect populations.
Non-overlapping generations. Insects lay some eggs. Parents die. A year later
Parents never live same time as the kids.
Population now (x) vs Population next year (y)
y=rx(1-x)
r= growth rate.
Bent hump shaped nonlinear relationship.
If the population small, then lots of food for next year.
If population too big, than hard to find mates.
Large populations followed by small... Booms and Busts...
Try to forecast far into the future.
Bank account - No bifurcations. You money does not start to oscillate of go chaotic.
I fork or splitting.
Bifurcation - A dynamical system branching off into a new type of behavior. Differently qualitatively.
Logistic Map
Cobweb diagram..
Study more orderly
Growth rate is a parameter.
Break, turning up heat on a stove.
Varying parameters can show bifurcations.
Growth rate of 1
Population vs TIme
Decreases to 0 ,with growth rate less than 1
Than reaches an equilbrium state 1-3
Growth rate over 3 than goes to a period 2 oscillation (2 years)
Repeats ever 2 years
Than > 3.5 repeats every 4 years (period 4) 4 highs and lows.
Thank period 8, period 16, etc.... Bifurcations happen faster and faster.
Infinite period cycles than Chaos sets in...
Nonlinearity makes it impossible to find explicit formulas for the maps long term behavior like we could do in the linear case.
Nonlinearity
Calling a system non-linear is like describing zoology as the study of non-elephant animals
- Stanislaw Ulam
Linear system - very simple ---> whole exactly equal to the parts.
Break problems into parts , add them back and get the same answer.
Principle and interest (you can multiply the whole or the parts and add back together).
Compete, like insects (for food and mates).
Non linear interactions
can be interference or synergistic
Nonlinearity leads to bifurcations and chaos.
Chaos was christened in 1975 by mathematicians Tien-Yien Li and James Yorke.
Fluid Mechanics, laminar flow vs turbulent flow.
Robert May - Biology
Urged educational system to study Nonlinear
Simple nonlinear systems do not possess simple properties.
Logistic Map -
Simple mathematical model with very complicated dynamics
Insect population initially...
By changing the variable r, we can get the insect population... r like the damping or friction.
Any exponential less than 1 goes to zero.
1) Go to zero
2) Stay the same - pure order or equilibrium
transition, damped oscillatory rate but settles in equilibrium.
3) Oscillate in boom and bust cycles... every two years, 2,4,8 and double all the way up to infinity.
Repeats every 2,4, 8 years...
4) Chaos where it bounces around erratically. Non repeating, all over the place.
All this by simply tuning a single knob. Intrinsic growth rate of insect.
Insect population initially...
By changing the variable r, we can get the insect population... r like the damping or friction.
Any exponential less than 1 goes to zero.
1) Go to zero
2) Stay the same - pure order or equilibrium
transition, damped oscillatory rate but settles in equilibrium.
3) Oscillate in boom and bust cycles... every two years, 2,4,8 and double all the way up to infinity.
Repeats every 2,4, 8 years...
4) Chaos where it bounces around erratically. Non repeating, all over the place.
All this by simply tuning a single knob. Intrinsic growth rate of insect.
Orbit Diagram - Mind Blowing
Numbers between zero and one represent the carrying capacity or percent of capacity a population will consume.
First part that is orderly is the fig tree.
Fiegenbaum means fig tree.
Period doublings happening faster and faster.
As we increase the growth rate we see the trunk splitting into a two year cycle of high and low.
And subsequent 4 year, 8 year - numbers
Period doubling cascade, 2,4,8,16,32 and doubling faster and faster.
Looks self similar which is a class of objects called fractals.
As tree
3.57 there is a wall , impenetrable wall... Called the accumulation point.
Infinitely bifurcating tree that converges to an accumulation point, Robert May bewildered.
What happens beyond accumulation point.
Bob May trained as a physicist and went into biology.
One of the worlds great theoretical biologists.
Science advisor to the Queen.
Explosion of dots at accumulation point.
Black stripes interrupt the chaos.
Long term behavior chaotic, unpredictable.
First black stripe - period three window.
Whole orbit diagram is there at the end of the window.
The whole diagram repeats in structure ad infinitum
Next stripe is a period 5 doubling.
What does this orbit diagram mean??
Rosetta Stone
You can predict all the periodicities at any growth rate, where chaos will set in ,everything!
All the cycles
Simpliest system parabolic.
Scroll
Region of periodicity
Accumulation point.
Blackness interrupted by windows.
Further patterns
Precise numerical patterns.
If occurs in differential equations then it occurs in nature because differential equations are the correct physical description of Nature.
First part that is orderly is the fig tree.
Fiegenbaum means fig tree.
Period doublings happening faster and faster.
As we increase the growth rate we see the trunk splitting into a two year cycle of high and low.
And subsequent 4 year, 8 year - numbers
Period doubling cascade, 2,4,8,16,32 and doubling faster and faster.
Looks self similar which is a class of objects called fractals.
As tree
3.57 there is a wall , impenetrable wall... Called the accumulation point.
Infinitely bifurcating tree that converges to an accumulation point, Robert May bewildered.
What happens beyond accumulation point.
Bob May trained as a physicist and went into biology.
One of the worlds great theoretical biologists.
Science advisor to the Queen.
Explosion of dots at accumulation point.
Black stripes interrupt the chaos.
Long term behavior chaotic, unpredictable.
First black stripe - period three window.
Whole orbit diagram is there at the end of the window.
The whole diagram repeats in structure ad infinitum
Next stripe is a period 5 doubling.
What does this orbit diagram mean??
Rosetta Stone
You can predict all the periodicities at any growth rate, where chaos will set in ,everything!
All the cycles
Simpliest system parabolic.
Scroll
Region of periodicity
Accumulation point.
Blackness interrupted by windows.
Further patterns
Precise numerical patterns.
If occurs in differential equations then it occurs in nature because differential equations are the correct physical description of Nature.
Universality of Logistic Map
What happens in the logistic map happens universally in a very wide class of systems.
The logistic map features such universal features on route to chaos that we expect they must occur in Nature.
In Reality
The logistic map features such universal features on route to chaos that we expect they must occur in Nature.
In Reality
U-Sequence
Universal sequence in orbit diagram
Burst of yellow/dots is chaos which are interrupted by periodic windows.
The windows are the places where the system appears periodically, repeating.
These windows are called Stable Cycles
Biggest window is the very wide period 3.
Cycles that repeat every 3 years or 3 steps.
U-sequence is what are the periods corresponding to windows.
3,5,6
Starts 2,4,8,16 (period doubling).
Focus on numbers less than 6
1,2,4,
6,5,3,
6,5,6,
4,6,
5,6
Universal sequence...
The same weird sequence has this same pattern of numbers for a humped shaped
Or rounded top logistic map.
Parabola is what we are looking at which we have been calling the logistic map.
Sine Map is ALSO similar but not identical looking.
Only the general shape of the curve matters.
Geometry trumps Algebra
Qualitative Universality
Burst of yellow/dots is chaos which are interrupted by periodic windows.
The windows are the places where the system appears periodically, repeating.
These windows are called Stable Cycles
Biggest window is the very wide period 3.
Cycles that repeat every 3 years or 3 steps.
U-sequence is what are the periods corresponding to windows.
3,5,6
Starts 2,4,8,16 (period doubling).
Focus on numbers less than 6
1,2,4,
6,5,3,
6,5,6,
4,6,
5,6
Universal sequence...
The same weird sequence has this same pattern of numbers for a humped shaped
Or rounded top logistic map.
Parabola is what we are looking at which we have been calling the logistic map.
Sine Map is ALSO similar but not identical looking.
Only the general shape of the curve matters.
Geometry trumps Algebra
Qualitative Universality
2 New Fundamental Constants of Chaos and Nature
Why Two? Because two parameters, a grow rate and the population or state space variable.
Orbit
Parameter
4.699
the graphical representation of the “bifurcation tree“, that have 4.669 length between each new bifurcation.
Shrinkage factor for the sine map, the logistic
Quantitative Universality.
Could not get papers published.
Explained
Used Renormalization Group to explain the numbers. Physics of phase transition.
Take a picture and blow it up and rescale it. That is renormalizing.
Numbers came right
Nobel prize winning work on phase transitions now being applied to the onset of chaos and transitions to chaos.
Common thread was both cases looked self similar RIGHT at the transition point whether it be a phase transition or Chaos.
They must occur in nature.
He confirmed the numbers came out of differential equations, and since differential equations describe nature they should be there.
Precise numbers
Period doubling and are wishbones shrinking at the same rate.
System has to an underlying logistic map that leads to period doubling and leads to chaos. Has an accumulation point followed by chaos.
Not all systems have this.
Shocking... Logistic map has no scientific content, no laws of nature, PURE numerology.
Its like saying I can do something in quantum mechanics by pressing the square root button on my calculator.
Logistic map button...
Experiments on a variety of diff
Old pythagorean dream came true, earth not made of the elements, but rather it is made of number.
Cross Disciplinarity -- He applied a nobel prize winning idea from condensed matter physics, to a math problem that a child could play with on a calculator that came from population biology that could explain the ROUTE to chaos in things as diverse as heart cells, electronic circuits, fluids with convection and chemical reactions going to chaos.
Why Two? Because two parameters, a grow rate and the population or state space variable.
Orbit
Parameter
4.699
the graphical representation of the “bifurcation tree“, that have 4.669 length between each new bifurcation.
Shrinkage factor for the sine map, the logistic
Quantitative Universality.
Could not get papers published.
Explained
Used Renormalization Group to explain the numbers. Physics of phase transition.
Take a picture and blow it up and rescale it. That is renormalizing.
Numbers came right
Nobel prize winning work on phase transitions now being applied to the onset of chaos and transitions to chaos.
Common thread was both cases looked self similar RIGHT at the transition point whether it be a phase transition or Chaos.
They must occur in nature.
He confirmed the numbers came out of differential equations, and since differential equations describe nature they should be there.
Precise numbers
Period doubling and are wishbones shrinking at the same rate.
System has to an underlying logistic map that leads to period doubling and leads to chaos. Has an accumulation point followed by chaos.
Not all systems have this.
Shocking... Logistic map has no scientific content, no laws of nature, PURE numerology.
Its like saying I can do something in quantum mechanics by pressing the square root button on my calculator.
Logistic map button...
Experiments on a variety of diff
Old pythagorean dream came true, earth not made of the elements, but rather it is made of number.
Cross Disciplinarity -- He applied a nobel prize winning idea from condensed matter physics, to a math problem that a child could play with on a calculator that came from population biology that could explain the ROUTE to chaos in things as diverse as heart cells, electronic circuits, fluids with convection and chemical reactions going to chaos.
Experiments and proof of Chaos
Borderland between order and chaos... Limited Domain... Doesn't apply to everything.
Small number of variables ... Otherwise state space becomes too abstract (100 variables = 100 dimensions.
Oscillating Chemical Reactions... Change
Oscillating reaction between yellow and blue... Chemical oscillations.
Reactions can oscillate on route to equilibrium.
Doesn't
Far from equilbrium
Growth Rate, Tighten break on waterwheel, dampening in oscillator.
Bromide Ion Potential.
There is an iterated map... Lorenz had a witches hat.
And a strange attractor (one wing of lorenz).
Variable flow parameter is the flow.
They saw period doubling in the chemical experiment.
Period 1 is a pure oscillation
Than Period 2, than pushing further 4 and period doubling Cascade
Chaos interrupted by periodic windows.
SEEN IN CHEMISTRY.
6,5,3... Great confirmation of the theory.
Fluids undergoing convection
Water and Chemistry
Electronic circuits showing oscillations that became chaotic.
Diodes
Transistors
Josephson Junctions
4.669 - Shrinkage factor as we apporach chaos.
Universal Number also appeared as did periodic windows.
Hard to measure more than 4 or 5 period doublings.
Hard to vary control parameter
Flow rate
Flow in fluid
growth rate
dampening
Control must be taken to control other instabilities.
Starts to vary in space ... Chemical... mixed to avoid spacial patterns.
Hands and care of a watch maker
Mercury in convection rolls. If you drive system harder and harder, you get instabilities and you get turbulence.
Liquid mercury used because responsive to a magnetic. Suppressed instabilities.
Feigenbaum cannot handle turbulence. Delicate state (like quantum coherence).
Complexity Therometer
Order - simple in time and space - Laws known
Chaos - complex in time, but not space - Laws known
(LIVES IN A VERY COOL PART OF THE THERMOMETER.)
Turbulence - complex in time and space, Laws Know
Fibrillation - complex in time and space , laws unknown
Complex systems - billions of interconnected variables - complex space and time, laws unknown.
That kind of overreaching provides a backlash.
Hardest problem in classical physics.
Time complex, Space Simple
Chaos revolution launched science in a new journey it needed to take... Into the world around us, the world of non-linearity and complexity.
Small number of variables ... Otherwise state space becomes too abstract (100 variables = 100 dimensions.
Oscillating Chemical Reactions... Change
Oscillating reaction between yellow and blue... Chemical oscillations.
Reactions can oscillate on route to equilibrium.
Doesn't
Far from equilbrium
Growth Rate, Tighten break on waterwheel, dampening in oscillator.
Bromide Ion Potential.
There is an iterated map... Lorenz had a witches hat.
And a strange attractor (one wing of lorenz).
Variable flow parameter is the flow.
They saw period doubling in the chemical experiment.
Period 1 is a pure oscillation
Than Period 2, than pushing further 4 and period doubling Cascade
Chaos interrupted by periodic windows.
SEEN IN CHEMISTRY.
6,5,3... Great confirmation of the theory.
Fluids undergoing convection
Water and Chemistry
Electronic circuits showing oscillations that became chaotic.
Diodes
Transistors
Josephson Junctions
4.669 - Shrinkage factor as we apporach chaos.
Universal Number also appeared as did periodic windows.
Hard to measure more than 4 or 5 period doublings.
Hard to vary control parameter
Flow rate
Flow in fluid
growth rate
dampening
Control must be taken to control other instabilities.
Starts to vary in space ... Chemical... mixed to avoid spacial patterns.
Hands and care of a watch maker
Mercury in convection rolls. If you drive system harder and harder, you get instabilities and you get turbulence.
Liquid mercury used because responsive to a magnetic. Suppressed instabilities.
Feigenbaum cannot handle turbulence. Delicate state (like quantum coherence).
Complexity Therometer
Order - simple in time and space - Laws known
Chaos - complex in time, but not space - Laws known
(LIVES IN A VERY COOL PART OF THE THERMOMETER.)
Turbulence - complex in time and space, Laws Know
Fibrillation - complex in time and space , laws unknown
Complex systems - billions of interconnected variables - complex space and time, laws unknown.
That kind of overreaching provides a backlash.
Hardest problem in classical physics.
Time complex, Space Simple
Chaos revolution launched science in a new journey it needed to take... Into the world around us, the world of non-linearity and complexity.