My Thoughts on the book

I though this book was quite interesting at some parts, and then quite dry at other times. It was very interesting to learn about chaos and all the disorder around us. The book presented some concepts which I have never even thought of. I would recommend this book to anyone interested in science and learning about new ideas and concepts. The writing style flows well and is filled with numerous examples to explain complex ideas.  It was difficult to read at times, but overall I feel much more knowledgeable about how different systems work and how disorder plays a part in nature. However, from learning this much more, I know that I know very little of what's out there in the world.

Chapter 11:Chaos and beyond

This last chapter simply summarizes the road chaos has taken and materialized in different aspects of the world. First it summarizes the ways different types of system behave. Stating, ‘Simple systems behave in simple ways’ in terms of the way a pendulum behaves, the way an electrical circuit behaves, and populations fluctuate over time. These systems are only simple as long as they are reduced to few laws predicting their behaviour, which is not the case in the real world. Another concept reviewed is that complex behaviours imply complex causes. That is, an unpredictable system must be subject to random external influences.

We are reminded once again that nature forms patterns, of which some are orderly and some are disorderly. Some patterns are fractal, with structures that are self-similar in scale. The study of chaos in nature has given scientists, over the years, a way of examining nonlinear and chaotic systems in ways that have changed the way they view the world. Looking at all the parts that make up a whole system as opposed to just looking at the outward appearance of a system. We live in a world where order is found in disorder, and understanding this relationship will help further our understanding of life beyond our own. 

Chapter 10:Inner Rhythms

I particularly found this chapter to be one of the most interesting chapters in the book. It looks at the chaos found within the human body. First it takes a look at the heart. Even though the cardiac pattern is periodic, there are numerous nonperiodic pathologies which lead to a person’s death. When a doctor listens to a person’s heartbeat, he hears the pounding of fluid. Blood enters the heart through two large veins, the inferior and superior vena cava, emptying blood without oxyen from

 the body to the right atrium. The pulmonary veins empty oxygen-rich blood from the lungs to the left atrium. Blood flows from the right atrium into the right ventricle, and when the ventricles are full, the tricuspid valve shuts. Blood leaves the heart through the pulmonic valve, and enters the lungs through the pulmonary artery. When an artery is blocked, ventricular fibrillation occurs and a person dies. As an arrhythmia, disorders of the heart's regular rhythmic beating, occurs, the heart’s muscles beat in an ncoordinated fashion as it tries to pump blood. The heart cannot stop fibrillating on its own, it needs a jolt of electricity from a defibrillation device. This is because the chaotic motion is stable over time.

Two unsolved problems many people suffer at some point of their lives are jet lag and insomnia. People have a steady sleep-wake rhythm and body temperature cycle which can restores itself after disturbance. In isolation, without daily resetting cues, the temperature cycle is 25 hours, with low temperatures occurring at night. However, experiments by German researchers found that the sleep-wake cycle became independent of one another, as they became chaotic. The experiment subjects would stay awake for twenty hours, and then sleep for ten hours, while being totally unaware of the time if day. After returning to the real world, the time it took for the participants to return to normal varied greatly.

 

After looking at these chaotic behaviours, it is a wonder how the human body works perfectly in disorder. How does the rhythm of the heart suddenly become chaotic and uncontrolled? How do sleep pattern affect the physiology of the body? It is beyond our understanding how much stress our bodies can take before the inevitable happens.

 

Chapter 9: The Dynamical Systems Collective

This chapter looks at the works of a group called the Dynamic Systems Collective,a small group of physicist that formed in 1977. As a group, they sought ways of connecting theory and experiment. One of thequestions they pondered was could unpredictability itself bemeasured? They found the answer in a Russian concept, the Lyapunov exponent. This number provided a measure of topological quantities that corresponded to unpredictability. The Lyapunov exponent provided a way to measure conflicting effects of a system in phase space. It had the ability to provide a view of properties in a systemthat lead to stability ad instability. All, exponent, which were greater than zero, meant there would be stretching, and all exponents less than zero meant there would becontraction in the system. A Lyapunov  exponent of zero meant the system was at some sort of equilibrium.

The Lyapunov  exponent is

                                                   defined as:

 

After learning about this exponent, the Dynamic Systems Collective group related the exponent to other important properties. They looked at the Information theory and its relation to entropy. This came Second law of Thermodynamics, which states that entropy, or disorder, is constantly increasing. For example, if you divided a tub of water, with barriers, in half and filled one half of the tub with ink and the other half with water, and lifted the barrier, the two liquids would start to mix. However, the mixing will never reverse itself, because it continues to drift towards disorder. To evaluate the level of mixing at a particular point in time would require knowledge of the Information theory, which involves quantifying information. With the information theory, results are measured in bits, as opposed to digits or letters. A bit is either zero or one. The Dynamic Systems Collective was able

I found it very interesting how the information theory could be used to quantify and relay none numerical situations in ways that many people can comprehend.

 

 

Chapter 8: Images of Chaos

This next chapter deals with a complex concept which I have not encountered thus far in my education. I find it really interesting and will do my best to explain the idea. First, the complex plan is examined. In a complex plane, numbers from negaive infinite to positive infinite lie on a line stretching from east to west. These are real numbers. There is also a north-south latitude made up of imaginary numbers. These complex numbers are written in the following manner: 2+3i, I signifying imaginary number. Historically, imaginary numbers were invented to answer complex questions, such as “what is the square root of a nrgative number?” Then, the square root of -1 is i, and the square root of -4 would be 2i. This is an example of a complex plane, which is a geometric representation on the complex numbers in the real axis and in the imaginary axis.

In the complex plane, a complex object, the Mandelbrot set is found. TheMandelbrot, named after Benoit Mandelbrot set is a fractal. A fractal is a rough geometric shape which displays self-similarities at various scales, meaning it can be split in

to parts which are all reduced-size copies of the whole object. To graph the Mandelbrot set in a complex plane, first we need to find numerous numbers in the set. To do this, the equation, Z = Z² +  C, is used. Where C is the complex number corresponding to the point being tested, and Z is a number starting at zero. Only the magnitude of Z is used, and not the direction. To calculate themagnitude of a complex number, as add the square of the number’s distance from the x-axis to the square of the number’s distance from the y-axis, and take the square root of the result. As many complex numbers are tested, we can begin to graph the Mandelbrot set.

When thousands of points are plotted an image such as this one with form:

 

After taking time to understand this new idea, I found it really fascinating. There were some aspects of its make up that I didn’t fully understand, but for the most part the concept is not too complex. 

Chapter 7: The Experimenter

This chapter looks at different experiments performed by Albert Libchaber. I will focus on one of his experiments on convection, which I found to be quite interesting.

Libchaber was an intelligent experimenter who spent his days in laboratories. One of his early inventions, which was he used to measure a liquid’s temperature, was called “Helium in a box”. It contained a cell with liquid helium, that was about the size of a lemon seed”. Liquid helium, which was chilled to 4degrees above absolute zero, was put into this cell. There was also a tiny sapphire “bolometer” inside which measured the fluid’s temperature. The cell was embedded in a casing in order to prevent noise ad vibrations to skew his results. Linchaber used electric heating coils and Teflon gaskets to as heat conductors. The liquid helium would enter the system through a reservoir. The entire system was contained in liquid nitrogen to help stabilize temperature. His plan with the experiment was to create convection, the rising of hot gas or liquid, in the liquid helium as he made the bottom plate warmer than the top. 

 The reason Libchaber used liquid helium in his experiment is because it has low viscosity, meaning it will roll with a slight push.  To cause convection in his tiny cell he had to create a temperature difference of a thousandth of a degree between the top and bottom surfaces. Through his design, Libchaber chose dimensions that allowed enough room for only two rolls.  The heated liquid helium would rise in the center turn up and over to the right and descend on the outside edges of the cell. To record his results, he inserted two microscopic temperature probes in the sapphire, ad their output would be recorded by a pen plotter. 

His results corresponded with the Navier-Stokes equation, which relates a fluid’s velocity, pressure, density, and viscosity, is used to represent the nonlinear relationship. The equation describes the process by which when a liquid is heated from below, and the fluid moves in cylindrical rolls. Hot fluid will rise on one side, lose heat, and descend on the other side. However, when the heat is turned up further, there is instability, and the movement along the cylindrical shape is no longer uniform. At even higher temperature, the motion becomes turbulent and chaotic. These are just a few examples of where chaos exists in our world. With nonlinear systems, this chaos can be further observed and interpreted.

This is the most general form of the Navier–Stokes equat

ion :

 

 

where v is the velocity of the flow, ρ is the density of the fluid, p is the pressure, T and  represents forces acting on the body (per unit volume) acting on the fluid and D is the delta operator.

 

 

 

 

 

 

Chapter 6: Universality

One interesting topic presented in this chapter is a colourful debate between Newton and Goethe about the nature of colour. Goethe’s ideas about colour opposed Newton’s in that Newton thought of colour as a static quantity that could be measured in a spectrometer.Goethe argued that colour was a matter of perception. Newton’s experiment using a prism broughthim to his theory. Through his experiment, he witnessed how a prism breaks a beam of white light into rainbow colour, andconcluded that these

 were the components that add up to produce white. He also proposed that these colours corresponded to frequencies. In Goethe’s experiment, he held a prism to his eye and looked through it. He sawno colour at all. Heobserved the same results when looking at a white surface, as he did when looking at a blue sky. He saw uniformity. However, Goethe did witness

 spots on his white surface or blue sky that he concluded to be  “the interchange of light and shadow” that causes colour.  He then went on to explore the ways people perceive shadows cast by different sources of coloured light. One way in which Goethe and Newton differed was that Newton broke light apart and found physical explanations for colour, while Goethe studying paintings and nature looking for holistic explanations of colour. In a sense, I believe that Goethe’s theories were worth more attention than they actually received. Newton’s theory was very much legitimate, but should not be used alone in defining the nature of light. The colours we perceive vary from time to time, and person to person. The scientific theory doesn’t completely describe the real life brightness of colour, which we see, although it did explain the components of the colours we see. Does that mean that there can be a happy medium between the scientific theory and the philosophical idea? 

Chapter 5: Strange Attractors

This chapter presented numerous concepts that I found interesting. It is very detailed and long, so I will split the topics into two sections.   

The first topic presented dealt with turbulence, and the way it presents interest that is usually one-sided. Many people are in favour of its disappearance, while few others give supporting evidence for its importance. In some cases turbulence is desirable- for example, inside a jet engine, where efficient burning depends on rapid mixing. However, in most cases turbulence leads to

disaster. For example, turbulent airflow above an airplane’s wings destroys its lift. So, what really is turbulence? It is a mess of disorder that drains energy and creates drag. The process by which flow changes to turbulence is quite interesting. When something shakes a fluid, the fluid starts to move vigorously, as energy drains out of it.  As the liquid shakes, energy is added at low frequency, or large wavelengths, and the large wavelengths decompose into small ones. This is when eddies form, as they dissipate the fluid’s energy. An eddy is the swirling of a fluid and the reverse current created. Knowing all this, there is the question of what happens just when turbulence begins. The theory to understanding this question came from Lev D. Landau, a Russian scientist. His theory on fluid dynamics explained how when more energy comes into a system, new frequencies begin one at a time, such as how a violin responds to harder bowing by vibrating with a second, dissonant tone, and goes on until the sound gets louder. In Landau’s view, wild motion, such as oscillation, simply accumulate one on top of the other, creating rhythms with overlapping speeds.

The rest of this chapter goes on to explain a different idea- phase space and the strange attractor. The strange attractor is a complex attractorwith chaotic motion, which lives in phase space. Phase space 

is a space in which all possible states of a system are represented. It gives way to obtaining essential information from a system of moving parts and making a road map to all its possibilities, and predicting future behaviour. One point in phase space contains all the information about the state of a dynamic system at any instant. In phase space, when looking at a pendulum, with friction as a factor, a central point, where velocity is zero, “attracts” the orbits, as friction takes away some of the energy from the system. This is an example of an attractor, which exhibits predictable behaviour. A strange attractor in phase space is a very interesting concept. The situation is presented when a bounded chaotic system has a long-term pattern that is not a simple periodic orbit. If this situation is plotted on a graph over an extended period, patterns that were not obvious in the short term will be apparent as the system attempts to reach equilibrium. A strange attractor represents a path where a system changes from system to system without settling down. Like snow flakes, they come in great variety and no two are alike.

Chapter 4: Geometry of Nature

Benoit Mandelbrot examined data drawn from rivers. He obtained data from Egyptians on the height of the Nile River for millennia. He found that the Nile had unusually great variation, as it flooded in some years, and subsidized in others. Mandelbrot classified the variations he observed using the Noah and Joseph Effects. The Noah effect meant discontinuity: when a quantity changes, it was change very fast. The Joseph Effect meant persistence: change over time was constant. These Effects explained how trends in nature are real and can vanish very quickly. Mandelbrot also studied geometry, which when applied to the real world is not perfect. It mirrors a rough universe, that doesn’t perfectly mirror the accepted geometric shapes. For example, mountains are not cones and lightning does not travel in a straight line.  

Since Euclid measure

ments – lengths, depth, thickness- failed to capture irregular shapes, Mandelbrot turned to idea of dimensions, then fractional dimensions. A fractional dimension is a way of measuring quantities that do not otherwise have a clear definition. For example, measuring the degree or definition of roughness or brokenness, or irregularity in an object. Fractal dimensions, as they were soon called, found applications on a series of problems that connected to surfaces that were in contact with each other. Several examples include the contact between tire treads and concrete, the contact between joints, and contact in electricity. Then there are the equations of fluid flow, which are dimensionless, meaning they do not require a set scale. Blood vessels, from aorta to capillaries, form a continuum as they branch and divide until they have to move in a single file. With the nature of its structure, the circulatory system performs dimension magic, as they it manages to squeeze a large surface area into a small volume. Yet, the blood only takes up five percent of the body. This is one thing I found very fascinating. How did the body evolve to form such a perfectly complex system?

Chapter 3: Life's ups and downs

This chapter looks at fluctuations in nature, and the work of population biologist as they started incorporating chaos as a factor in changing wildlife populations.  It also looks how hard it is to come up with an equation complex enough to represent real phenomena.

First when analyzing a population a few factors pollution biology looks at are connected to the history of life, looking at how predators and prey interact, and how the spread of a disease is affected by population density. 

 Another way of analyzing a population over time is through modeling the data with an appropriate function. The ideal function to model the growth of a population over time would give way to many factors and allowthe population to settle into its long-term behaviour. One equation that was derived to represent population growth is X_{next=rx(1-x). The parameter r represents the rate of growth, (1-x) keeps the growth within reasonable bounds, since rises, (1-x) falls. Population is then expressed as a fraction between one and zerp. Zero representing extinction, and one representing the largest possible population in the particular environment. After calculating for high parameters, biologists began to notice a higher fluctuation of populations. Many scientists at this time just thought this chaotic behavior was because of malfunctioning calculators, as they noticed that populations tend to rise sharply and fall dramatically before reaching an equilibrium. But, this fluctuation of results could have been attributed to the oversimplified equations that were not capable to take into account all the factors of nature that influence a system. It would require nonlinear equations that would have the necessary variables, but would be extremely difficult to solve. In secondary school, students are only taught how to solve linear systems that are solvable. Non-linear systems with real chaos are rarely understood, as people are constantly trying to make sense of the world and not account for all the disorder amongst them. Robert May, a biologist, further reasoned for the necessities of students to learn about chaos. May believed proclaimed, “ The mathematical intuition so developed ill equips the students to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems.” One example May looked at that explained the need for chaos to be understood by scientists in all fields was the debate of population change.  Some ecologists argued that populations are regulated and steady, while others argued that populations fluctuate erratically. May believed that the answer to this argument could be explained though chaos theory. The message that simple models could produce what looked like random behaviour, but was in fact behaviour with fine structure. It was like seeing disorder in an ordered system.}

Chapter 2: Revolution

The next chapter discusses the continuous path of chaos, and the difficulties of communicating new ideas to skeptical scientists. As more people were involved in exploring the new science, scientists were willing to test and apply the theory to other applications in the world.

One tool that scientists used to apply the new science was a pendulum, which can be applied to many real life situations. Galileo observed a church lampswaying back and fourth, and Christian Huygens used the pendulum as a way of timekeeping. Foucault, the Pantheon of Paris, used a pendulum to demonstrate the earth’s rotation. Also,

 every clock, until the era of 

the vibrating quartz, relied on a pendulum. Basic electronic circuits are described by the same equations that describe a swinging pendul

um. Galileo saw regularity in the pendulum, and contended that a pendulum not only keeps precise time, but keeps the same time no matter h

ow wide the angle of its swing. In other words, a wide-swinging pendulum has farther to travel, but happens to travel that much faster.  Galileo’s theory was so convincing, that it is still taught in some physics books, but it is not correct, as was proved later on. The consistency Galileo saw was only an approximation. The changing angle creates equations that are slightly nonlinear. At low amplitudes, the error is almost nonexistent, but as the amplitude increases, the error margin increases. In his experiments, Galileo disregarded friction and air resistance, which play a role in the motion of a pendulum. For example, when looking at a playground swing, the swing accelerates on its way down, and decelerates on its way up, while losing some speed to friction. As the swing gets a push from behind, it accelerates and 

eventually settles back into steady motion. There is friction acting on the swing pulling it down. Also, it can be observed that on a swing as the angle decreases, the swing slows down and goes back and fourth less times that when is at a higher angle.

 

The second part of chapter one looks at the mystery of the Great Red Spot of Jupiter, which is a vast, chaotic, swirlig oval. Astronomers noticed a blemish on the planet, and there were multi

ple theories on what the coloration was. Here are a few examples:

The Lava Flow theory: Scientists imagined a huge oval lake of molten lava flowing out of a volcano.

The New Moon Theory: A German Scientist suggested that the spot was a new moon emerging from the planet’s surface.

The Egg Theory: Some scientists thought the spot was a solid body floating in the atmosphere just like an egg floats in water

When the Voyager orbited in 1978, astronomers saw the spot as a hurricane-like system. However, it possessed some characteristics that were unlike a hurricane.

The final conclusion was that The Great Red Spot was a great anti-cyclonic storm. As hot gases on Jupiter’s atmosphere swirl, and cooler gas falls through the atmosphere, the Corolis effect causes the region to start swirling. These small swirling storms move around and eventually come together, combining their energies and forming the Great Red Spot. The Great Red Spot persists over time because there is no solid ground to create frictions, and slow down the storms. The chaotic flow created takes up much energy, and although the spot itself is a self-organizing system, created by the same factors that create he unpredictable twists, it is stable chaos. It is very interesting how the storm has been going on for over 40 years. I wonder when it will finally die 

down. And will it ever die down?

Chapter 1: The butterfly Effect

The discovery of the new science, Chaos, brought on much skepticism from established scientists. This new notion, Chaos, can be defined as the existence of unpredictable or random behaviour. The study of chaos was the beginning of analyzing disorder in the atmosphere, in turbulent seas, in fluctuations of wildlife populations, and in oscillations of the heart and the brain. In the first chapter of Chaos, by James Gleick, the reader learns about Lorenz,a meteorologist who built a weather simulation program on his computer. This program mirrored real world weather trends, until one day Lorenz made a small rounding error while entering his data, and skewed all his results. From his small miscalculation, Lorenz discovered that the weather cannot be predicted long term because small disturbances, such as rounding to a wrong decimal or the disturbance of the air by a butterfly, was enough to change weather patterns.  

From this, he went on with his extensive research and discovered the phenomenon known as “sensitive dependence on initial condition”, which is referred to as the butterfly effect. The term butterfly effect refers to the idea that a butterfly flapping its wings can create small changes in the atmosphere, which may alter the path of a tornado. The butterfly flapping its wings represents a small change in the initial state of a system, which causes a chain of events leading to larger chaotic events. While the butterfly itself doesn’t case the tornado, the flapping of its wings is part of the initial condition of the system 

I find this very interesting because it shows how small actions, such as a butterfly flapping its wings, or a person making a decision to tie his shoe, can have larger results that influence the lives of many people.  How is it possible that such a small animal and such small movements in the air can drastically change events on ourplanet?

 

Figure 1: Lorenz Attractor: Example of butterfly effect representation