## Introduction and Overview (and Conclusion)

__Chaos__is about order and disorder in deterministic systems that are nonlinearWhat is the science of Chaos all about?

In ordinary language, chaos means disorder, utter confusion and randomness. If that were true, it would put chaos inside the study of probability theory which it really isn't.

Chaos deals exclusively with

__Deterministic systems. The present state completely and uniquely determines the future. Nothing random about it.__

**deterministic systems.**Chaotic system however can impersonate random behavior, masquerade random behavior because of their long term unpredictability. That comes from the

__the hallmark of Chaos which is the extreme sensitivity of a chaotic system to imperceptible, tiny changes in its initial conditions.__

**butterfly effect**,Definition attempt #2.

Chaos is the science of disorder in deterministic systems. Poincare 3 body problem. Lorenz artificial weather.

But it soon came to be realized that there is a tremendous amount of

__. Magnificent order and structure in chaos. Order in an unfamiliar guise. Of course short term order... we can predict a few moments ahead (no surprise). Long term predictability... but long term order in an average sense as the system moved through an abstract phase space or state space. Imaginary space showing its motion as an imaginary point moving from state to state telling what the system is doing throughout its history, throughout its time course. Patterns in long term evolution of a system.__

**order in chaos**When we trace out the motion in state space we found out that chaotic systems don't just make a random blob, just fill up their state space with a mess or a snarl of spaghetti. They form very delicate and beautiful structures called

__Strange because their geometry seemed strange to the early pioneers. Rossler attractor, Lorenz attractor, etc. Long Term order encapsulated in the strange attractor.__

**strange attractors.**Also a medium term order in the way the system hopped around the strange attractor. Also unforseen.

How the system hopped around state space when viewed from a strobe light.

Ways of Looking at Non-linear and Complex systems...

1)

__is like a movie, a cinematic picture of reality unfolding instant by instant. Newtons great contribution.__

**Differential Equation**2) Time Lapsed Photography, like an overnight exposure, seeing the accumulated shape of chaos in state space, which gave us the strange attractor.

3) Flash a strobe light... in dark cannot see... tracking its motion from flash to flash we created an iterated mapping or

__embodying the medium term order in chaotic systems. And that tracked this motion under the strobe light.__

**iterated map**Having realized that iterated maps were a short cut through the order we were interested in. We can go farther with iterated maps than differential equations, because it allowed us to uncover even more order.

a. There was order with respect to parameters.

__are knobs that can be manually turned to change the characteristics of a system. Intrinsic growth rate of insects. Break on the water-wheel. Flow rate in chemical reactions.__

**Parameters**By changing the parameters we could send the system into different long term behavior.

It could be steady equilibrium, sometimes it would oscillate periodically OR it would oscillate wildly in a way we associate with chaos.

Those transitions which we called

__, the splitting in the behavior of a system. Those bifurcations and all the order that was captured in them in two magnificent diagrams.__

**bifurcations**One showed the order in the logistic map as we changed the growth parameter. The

**(with all is bifurcations and windows and U numbers and Fiegenbaum numbers.). The scroll.**

__orbit diagram__Then the order in chaos intermingled with periodicity was illustrated even more vividly in the

__with its fantastic richness of structure.__

**Mandelbrot set**Those diagrams were rosetta stones for us where we could decode chaos where we could see all the possible behavior a system could have. Saw periodicity to chaos to periodicity again. All stretched out on these beautiful charts allowing us to see everything at once.

The charts were all about bifurcations, where do the transitions occur. Equilibrium to perio

Orbit diagram had

__as did Mandelbrot set.__

**fractal structure**Tiny copies of orbit diagram in each window. Fractal structure even more spectacular in Mandelbrot.

Strange attractors had this mica-like structure with infinitely many sheets. Like Filo (leaf) dough with many sheets of dough.

__- numbers that controlled the ratio of successive spacings in the fig tree. Showing where equilbrium split to period 2, repeating every 2 years, then splitting to 4 years, then 8 years (in insect problem) approaching an impenetrable wall which is kind of an accumulation point. Fractal branching structure controlled by these Feigenbaum numbers. These are universal.__

**Feigenbaum Numbers**They were in insect population

Chaos of fluids undergoing convection

Chemical reactions

Electronic circuits

Heart cells standard rhythmicity

Lasers

Fractals took on a like of their own.

**- a small part of a structure looks like the original, like the whole. We could see fractals in both shape (space) and time. Heartbeats, brainwaves.**

__Self-similarity__Fractals not just in state space or orbit diagrams but REALLY out there in the world.

Brocolli

Coastlines

Mountains

Clouds

Ups and downs stock market

Patterns of earthquakes

Fluctuations in human HRV.

__Controlling Chaos__Order embodied in fractals and chaos could be harnessed. Quantum electronics.

Quantum dots... Space travel...harnessing chaos by surfing the chaos in the solar systems gravitational fields.

Epilepsy, arrythmias.

Behind all this hidden order there was a puppeteer pulling all the strings.

__**Nonlinearity**__Linearity and

__- simple proportionality between cause and effect. The idea that the whole is nothing more than the some of the parts, they are just exactly equal. If we lived in a linear world all that structure would be gone, all of the richness of the world would be gone. Life and everything beautiful around us depends on Non-linearity.__

**Reductionism**The Whole can be more than the sum. The Parts are interacting, cooperating, sometimes competing and interfering... But the whole is not just the sum of the parts of all the things that matter to us.

You can only give a definition at the end when you understand something.

__Chaos__is about order and disorder in deterministic systems that are nonlinearVirtually all the unsolved problems in science are nonlinear.

Think about the orchestration of millions of chemical reactions inside a living cell and the disruption of that orchestration when the cell turns cancerous.

Consider the battle of the immune system with its hundreds of components, antibodies, T-cells and the aids.

Emergence of chaos from billions of neurons in the brain.

Think about the origins of life from a meshwork of chemical reactions in the primordial soup.

Decentralized dynamics in which enormous numbers of components keep changing their states from moment to moment looping back on each other in ways that cannot be understood by looking at a component in isolation. That is why reductionism is not strong enough to help us to tackle these mysteries.

Our minds are not wired for nonlinear systems. We are accustomed to thinking in terms of top down hierarchies , chains of command, and straight forward logic of cause and effect.

In huge interconnected systems where every part affects every other part, our standard ways of thinking fall apart. Cannot use verbal reasoning alone.

Thanks to chaos theory we have mastered the simplest nonlinear systems.

Need to master next, complex nonlinear system.

__means an enormous # of parts. These parts are not identical, but diverse and interact with one another in bewildering networks.__

**Complexity**Genes proteins, and enzymes reactions... SEE CHARTS \http://biochemical-pathways.com/#/map/1

P53 - break on cell division... to suppress tumors... Like driving without a break

Yeast Eukaryotes like us... Math models of Yeast cell cycle..

Models based on methods of Chaos theory. Combines biology w/ bifurcations. Avant Garde.

Neurons not linear, you stimulate and nothing might happen but hit a certain threshold and it fires.

Newtonian, Linear, Determintistic, Static, materialism, classical, mechanistic, random, accidental

Age of Enlightenment - STASIS - Things have always been the same.

But really it's a nineteenth century idea, and in order to see how it developed let's go back to about 1790 or 1800; so at the end of the Century of the Enlightenment.

**Genome Project - Problem of STATIC Models**Sequencing of genome gave us a huge list of PARTS of

the 23,000 gene and the proteins they encode.

Science still has no clue about how the interlocking activities of all these genes and proteins as choreographed in the living cell and how they go awry when the cell turns cancerous. These are problems of dynamics, or how things unfold in time.

We usually think of the genome as a blueprint, a blueprint of life. That is a very static metaphor, the metaphor of an assembly line or if you are building a house or something (which is static).

It is true to some extent that genes do give instructions for building proteins which are the molecular machines that carry out all the functions of life. But that is not ALL it is.. That is too linear.

Some of the proteins act back on other genes... Regulatory genes create proteins that turn genes on or off forming feedback loops.

Once you have the picture of genes making proteins that affect other genes.

Now it is seeming to look less like a blueprint and more like a COMPUTER (Universe as a quantum computer).

A very complicated circuit. To understand things like cancer, we need to understand the DYNAMICS of regulatory genes and proteins and gene networks and the dynamics of it.

Newton founded laws of inanimate dynamics, but we are still looking for the laws of life.

Systems Biology is trying to understand these networks

Synthetic Biology - building living versions of circuits out of genes and proteins.

Computational Neuroscience - computer people collaborating with Neuroscientists

Econophysics - economy ... data analysis

Consciousness - who is in there? How can 100 billion neurons fall in love.. recognize a face...

40 Hz - in sync.. Not just electrical chatter or idle state of the brain.. its a chorus.

Parts of the brain start singing in unison

Chorus arises spontaneously out of chaos. Self organizing thing when we have consciousness.

Fleeting processes of synchronization in our brain.

Be Well and Stay Chaotic.

## Big Ideas

## Butterfly Biology

Strange Attractor of Health

## Introduction Chaos

Chaos

By having fun we can learn better.

Chaos Theory - Gleick

Zooming on Mandelbrot

Mandelbrot

Magnified by factors of itself

Factor of 1000

People got interested… catchy

Genuine reason for excitement chaos, people sense correctly that there is something revolutionary happening in science.

The erratic side of Nature, the irregular and unpredictable.

Jagged side of Nature, the discontinuous,

Science obsessed with the other side of Nature, the predictable, deterministic, lawful and orderly side especially in

Ecology - unpredicable wild flucuation

Insect populations

Arrthyrias

Epilepsy

Turbent sea and atmosphere,

Pathologies monsters…

Geologists , Earthquakes

Electronic circuits

chemical reactions

double pendulum

damped driven pendulum

lorenz waterwheel

convection and weather models

30 years

Strange connections

Heart rate variability, structures reminiscent of the whole.

Rules of chaos turned out to be universe.

Turbulence annoying in science.

We know equations of turbulence, but we still do

Leornardo , swirls within swirls

Laws of chaos intersect all disciplines.

What is Chaos - Paradoxical state , unpredictable, but on the other hand.

Deterministic - future is determined by the present.

Only one possible future.

Double pendulum newtons laws of motion.

Small angles

Large angles unpredictable.

Chaos

1) Breaks a lot of the rules of convential science by asking unusual questions. Everyday scale of normal life, not unusual regimes of small and fast.

2) Chaos takes the focus off the laws of nature and shifts it onto their consequences. Goal of physics - Find the new laws, the fundamental laws.

There is another game… Having found the laws, what the consequences of implications.

3) Way it uses the computer, not just a number cruncher.

An intuition telescope.

4) Emphasizes a different side of nature.

Not reductionism, take a problem that is hard and reduce it by breaking it up into smaller parts, try to analyze each part separately and then that should be easier, then put them back together. (Taylor Series).

Some problems do not permit reductionism, that is if you break them up into parts, you lose something essential. There are some systems that the interaction between the parts creates something that the whole is greater than the sum.

That is where there are cooperative effects taking place or some kind of interference.

In systems like that we have to find a way to do scientific holism to go from the parts to the whole.

Rigorous scientific.

5) Chaos is radically interdisciplinary … Usually walls between disciplines.

Makes a lot of progress by specializing.

Economist and biologists and physicists and mathematician.

Astronomy, zoology, the arts, commerce, humanities, internet, medicine, all of it. Condensed matter physics, population biology.

6) Chaos paints a topsy turvy picture of the world. We thought Simple laws lead to simple consequences. Simple system can show complex behavior.

Very complex things can have simple patterns

7) The change in thinking came suddenly.

Pictures can be much more important then formulas.

Visualizing in pictures.

Chaos is the science of how things change.

Purely mathematical side of chaos.

Isaac Newton gave a view of the universe so regular and so orderly that it seemed to outlaw chaos once and for all.

200 years later a chink in Newtons armor.

Henri Poincare… Horrifying picture.

Forgotten for over 70 years.

Ed Lorenz Butterfly effect .. no effect another 10 years.

Fractals shapes whose structures repeat ad-infinitum like the mandelbrot set.

Tiniest parts look like whole. Fractal structure breaks down in a real system, eventually branches to twigs.

Fractals are the remnants of chaos, the footprints that chaos leaves behind.

Vision - cancer to the mystery of consciousness.

By having fun we can learn better.

Chaos Theory - Gleick

Zooming on Mandelbrot

Mandelbrot

Magnified by factors of itself

Factor of 1000

People got interested… catchy

Genuine reason for excitement chaos, people sense correctly that there is something revolutionary happening in science.

The erratic side of Nature, the irregular and unpredictable.

Jagged side of Nature, the discontinuous,

Science obsessed with the other side of Nature, the predictable, deterministic, lawful and orderly side especially in

Ecology - unpredicable wild flucuation

Insect populations

Arrthyrias

Epilepsy

Turbent sea and atmosphere,

Pathologies monsters…

Geologists , Earthquakes

Electronic circuits

chemical reactions

double pendulum

damped driven pendulum

lorenz waterwheel

convection and weather models

30 years

Strange connections

Heart rate variability, structures reminiscent of the whole.

Rules of chaos turned out to be universe.

Turbulence annoying in science.

We know equations of turbulence, but we still do

Leornardo , swirls within swirls

Laws of chaos intersect all disciplines.

What is Chaos - Paradoxical state , unpredictable, but on the other hand.

Deterministic - future is determined by the present.

Only one possible future.

Double pendulum newtons laws of motion.

Small angles

Large angles unpredictable.

Chaos

1) Breaks a lot of the rules of convential science by asking unusual questions. Everyday scale of normal life, not unusual regimes of small and fast.

2) Chaos takes the focus off the laws of nature and shifts it onto their consequences. Goal of physics - Find the new laws, the fundamental laws.

There is another game… Having found the laws, what the consequences of implications.

3) Way it uses the computer, not just a number cruncher.

An intuition telescope.

4) Emphasizes a different side of nature.

Not reductionism, take a problem that is hard and reduce it by breaking it up into smaller parts, try to analyze each part separately and then that should be easier, then put them back together. (Taylor Series).

Some problems do not permit reductionism, that is if you break them up into parts, you lose something essential. There are some systems that the interaction between the parts creates something that the whole is greater than the sum.

That is where there are cooperative effects taking place or some kind of interference.

In systems like that we have to find a way to do scientific holism to go from the parts to the whole.

Rigorous scientific.

5) Chaos is radically interdisciplinary … Usually walls between disciplines.

Makes a lot of progress by specializing.

Economist and biologists and physicists and mathematician.

Astronomy, zoology, the arts, commerce, humanities, internet, medicine, all of it. Condensed matter physics, population biology.

6) Chaos paints a topsy turvy picture of the world. We thought Simple laws lead to simple consequences. Simple system can show complex behavior.

Very complex things can have simple patterns

7) The change in thinking came suddenly.

Pictures can be much more important then formulas.

Visualizing in pictures.

Chaos is the science of how things change.

Purely mathematical side of chaos.

Isaac Newton gave a view of the universe so regular and so orderly that it seemed to outlaw chaos once and for all.

200 years later a chink in Newtons armor.

Henri Poincare… Horrifying picture.

Forgotten for over 70 years.

Ed Lorenz Butterfly effect .. no effect another 10 years.

Fractals shapes whose structures repeat ad-infinitum like the mandelbrot set.

Tiniest parts look like whole. Fractal structure breaks down in a real system, eventually branches to twigs.

Fractals are the remnants of chaos, the footprints that chaos leaves behind.

Vision - cancer to the mystery of consciousness.

## Amazing Mandelbrot Zoom - Comes from a simple equation!

The Equation

The equation of the Mandelbrot Set is really one of the simplest equations that you can write down. In fact, it is even simpler than Einstein’s equation e = mc2 which has three terms (E, M, and C) whereas the equation Z = Z2 + C only has two terms (Z and C).

You will also notice (in the case of the Mandelbrot Set equation)… that Z is on both sides of the equation. What this means is that the output of the EQUATION… is being fed back INTO the EQUATION… thus setting up a positive feedback loop or system. It is this feedback system that gives rise the endless variety of forms and images that I will be presenting to you.

The Mandelbrot Set has been described as the most complex object in mathematics. It has been proven to be the absolute maximum space filling curve possible in 2+ dimensions. It is said that, if its boundary were even one QUANTA more curved inward on itself… … it would HAVE TO overlap thus popping into the third dimension.

The equation of the Mandelbrot Set is really one of the simplest equations that you can write down. In fact, it is even simpler than Einstein’s equation e = mc2 which has three terms (E, M, and C) whereas the equation Z = Z2 + C only has two terms (Z and C).

You will also notice (in the case of the Mandelbrot Set equation)… that Z is on both sides of the equation. What this means is that the output of the EQUATION… is being fed back INTO the EQUATION… thus setting up a positive feedback loop or system. It is this feedback system that gives rise the endless variety of forms and images that I will be presenting to you.

The Mandelbrot Set has been described as the most complex object in mathematics. It has been proven to be the absolute maximum space filling curve possible in 2+ dimensions. It is said that, if its boundary were even one QUANTA more curved inward on itself… … it would HAVE TO overlap thus popping into the third dimension.