IMAGE#1: DOUBLE PENDULUM CHAOS 

IMAGE#2: LORENZ DETERMINISTIC CHAOS BY ROBERT GHRIST

IMAGE#3: CHAOTIC BEHAVIOR IN THE REAL WORLD [LINK] 

A HISTORY OF CHAOS THEORY by Christian Oestreicher [LINK]  

1. THE DETERMINISTIC SCIENCE OF LAPLACE:  Determinism is predictability based on scientific causality. Local determinism concerns a finite number of elements as in ballistics, where the trajectory of a projectile can be precisely predicted on the basis of the propulsive force, the angle of shooting, the projectile mass, and air resistance. This is local determinism. In contrast, universal determinism gets complicated – the universe for example or even just the solar system. 
2. In a whirlwind of dust, raised by elemental force, confused as it appears to our eyes. the classical mathematical scientists knew exactly the different energies acting on each particle of dust, with the properties of the particles moved, could demonstrate that after the causes given, each particle acted precisely as it ought to act, and that It could not have acted otherwise than It did.
3. This is the principle of universal determinism assumed in climate science where it is thought that the complexity of the large number of particles moving in different directions can be captured in a climate model of sufficient complexity. It follows that given a computer model of the universe of sufficient complexity we ought to be able to exactly describe the motions of the greatest bodies of the universe and those of the lightest atom such that nothing would be uncertain and the future, as the past, would be present would be exactly reproduced on the computer model.
4. THE PHASE SPACE OF POINCARE: Henri Poincaré developed another point of view as follows: in order to study the evolution of a physical system over time, one has to construct a model based on a choice of laws of physics and to list the necessary and sufficient parameters that characterize the system (differential equations are often in the model). One can define the state of the system at a given moment, and the set of these system states is named phase space.
5. This view persisted until the phenomenon of sensitivity to initial conditions was discovered by Poincaré in his study of the the n-body problem. Thus, a century after Laplace, Poincaré discovered a very small cause, which eludes us, determines a considerable effect that we cannot fail to see, and so we say that this effect Is due to chance.
6. If we knew exactly the laws of nature and the state of the universe at the initial moment, we could accurately predict the state of the same universe at a subsequent moment. But this is not always so because small differences in the initial conditions may generate very large differences in the final phenomena. Prediction then becomes impossible, and we have a random phenomenon. This was the discovery of of chaos in nature.
7. ANDREI KOLMOGOROV is one of the most Important mathematicians and statisticians of the 20th century. He is the creator of probability theory, turbulence theory, Information theory, and topology. In going over the work of Poincaré, he showed further that a quasiperiodic regular motion can persist in an integrable system even when a slight perturbation is Introduced Into the system. This Is known as the KAM theorem which Indicates limits to integrability.
8. The theorem describes a progressive transition towards chaos within an Integrable system. All trajectories are regular and quasiperlodlc. As levels of perturbation are introduced, the probability of quasiperiodic behavior decreases and an increasing proportion of trajectories becomes chaotic, until a completely chaotic behavior is reached. In terms of physics, in complete chaos, the remaining constant of motion is only energy and the motion is called ergodic.
9. In a linear system, the sum of causes produces a corresponding sum of effects and it suffices to add the behavior of each component to deduce the behavior of the whole system. Phenomena such as a ball trajectory, the growth of a flower, or the efficiency of an engine can be described according to linear equations. In such cases, small modifications lead to small effects, while Important modifications lead to large effects as in reductionism.
10. The nonlinear equations concern specifically discontinuous phenomena such as explosions, sudden breaks In materials, or tornadoes. Although they share some universal characteristics, nonlinear solutions tend to be individual and peculiar. In contrast to regular curves from linear equations, the graphic representation of nonlinear equations shows breaks, loops, recursions and turbulence. Using nonlinear models, one can identify critical points in the system at which a minute modification can have a disproportionate effect.
11. EDWARD LORENZ: Edward Lorenz, from the Massachusetts Institute of Technology (MIT) is the official discoverer of chaos theory. He first observed the phenomenon as early as 1961 while making calculations to predict weather as in weather forecasting.  Lorenz considered, as did many mathematicians of his time, that a small variation at the start of a calculation would Induce a small difference in the result, of the order of magnitude of the initial variation. This was turned out to be not the case. The sensitivity to initial conditions was addressed by meteorologist Philip Merilees, who organized the 1972 conference where Lorenz presented his results. The title of the presentation by Lorenz was “Predictability: does the flap of a butterfly’s wing in Brazil set off a tornado in Texas?”. This title has become famous and a popular way of describing chaos theory.
12. Lorenz had rediscovered the chaotic behavior of a nonlinear system, that of the weather, but the term chaos theory was only later given to the phenomenon by the mathematician James Yorke, in 1975. Lorenz also gave a graphic description of his findings using his computer. This graphic was his second discovery: the attractors.
13. Strange Attractors: The Belgian physicist David Ruelle studied the Lorenz graphic and he coined the term strange attractors in 1971. The clearly recognizable trajectories in the phase space never cut through one another, but they seemed to form cycles that are not exactly concentric, not exactly on the same plan. It is also Ruelle who developed the thermodynamic formalism. The strange attractor is a representation of a chaotic system in a specific phase space, but attractors are found in many dynamical systems that are nonchaotic. There are four types of attractors. They are fixed point, limit-cycle, limit-torus, and strange attractor.
14. The four types of attractors are displayed in the graphic above as 1a Fixed Point=a point that a system evolves towards, such as the final states of a damped pendulum.  1b=Limit Cycle a periodic orbit of the system that is isolated as in the swings of a pendulum clock and heartbeat at rest. 1c=Limit Taurus, In a Limit Taurus there may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. If two of these frequencies form an irrational ratio, the trajectory is no longer closed, and the limit cycle becomes a limit torus.  1d=Strange Attractor. The strange attractor characterizes the behavior of chaotic systems in a phase space. The dynamics of satellites in the solar system is an example.
15. We can describe future trajectories of our planet with Newton’s laws  but how do we know that these laws work at at the dimension of the universe? These laws concern only the solar system and exclude all other astronomical parameters. Therefore, while the earth is indeed to be found repetitively at similar locations in relation to the sun, these locations will ultimately describe a strange attractor of the solar system.
16. ALEKSANDR LYAPUNOV:   Chaos amplifies initial distances in the phase space. Two trajectories that are initially at a distance D will be at a distance of 10 times the value of D after a sufficient delay described as Characteristic Lyapunov Time. If the characteristic Lyapunov time of a system is short, then the system will amplify its changes rapidly and be more chaotic.  It is within the amplification of small distances that certain mathematicians, physicists, or philosophers consider that one can find randomness. The characteristic Lyapunov time of the solar system is thought to be in the order of 10 million years.
17. NEGATIVE AND POSITIVE FEEDBACK:  Negative and positive feedback mechanisms are ubiquitous in living systems, in ecology, in daily life psychology, in climate, as well as in mathematics. A feedback does not greatly influence a linear system, while it can induce major changes in a nonlinear system. Thus, feedback participates in the frontiers between order and chaos.
18. FELGENBAUM AND THE LOGISTIC MAP: Mitchell Jay Feigenbaum proposed the scenario called period doubling to describe the transition between a regular dynamics and chaos. His proposal was based on the logistic map introduced by the biologist Robert M. May in 1976. The logistic map is a function of the segment [0,1] within itself defined by: xn+1=rxn(1-xn) where n = 0, 1, … describes the discrete time, the single dynamical variable, and 0≤r≤4 is a parameter. The dynamic of this function presents very different behaviors depending on the value of the parameter r.
19. For 0≤r≤3, the system has a fixed point attractor that becomes unstable when r=3.For 3<r<3,57…, the function has a periodic orbit as attractor, with a period of 2n where n is an integer that tends towards infinity when r tends towards 3,57. When r=3,57, the function has a Feigenbaum fractal attractor. When r>4 the function goes out of the interval [0,1] as seen in the graphic below.
20. This function of a simple beauty in the eyes of mathematicians.  It has numerous applications. For example, for the calculation of populations taking into account only the initial number of subjects and their growth parameter r (as birth rate). When food is abundant, the population increases, but then the quantity of food for each individual decreases and the long-term situation cannot easily be predicted.

21. Mandelbrot and fractal dimensions

    In 1973, Benoît Mandelbrot, who first worked in economics, wrote an article about new forms of randomness in science. He listed situations where, in contrast to the classical paradigm, incidents do not compensate for each other, but are additive, and where statistical predictions become invalid. He described his theory in a book where he presented what is now known as the Mandelbrot set. This is a fractal defined as the set of points c from the complex plane for which the recurring series defined by z_n+1 = z_n ² + c, with the condition z₀ = 0, remains bounded.

22. A characteristic of fractals is the repetition of similar forms at different levels of observation (theoretically at all levels of observation). Thus, a part of a cloud looks like the complete cloud, or a rock looks like a mountain. Fractal forms in living species are for example, a cauliflower or the bronchial tree, where the parts are the image of the whole. A simple mathematical example of a fractal is the so-called Koch curve, or Koch snowflake. Starting with a segment of a straight line, one substitutes the two sides of an equilateral triangle to the central third of the line. This is repeated for each of the smaller segments obtained. At each substitution, the total length of the figure increased by 4/3, and within 90 substitutions, from a 1 -meter segment, one obtains the distance from the earth to the sun.
23. Shown above are the first four iterations of the Koch snowflake. Fractal objects have the following fundamental property: the finite (in the case of the Koch snowflake, a portion of the surface) can be associated with the infinite (the length of the line). A second fundamental property of fractal objects, clearly found in snowflakes, is that of self similarity, meaning that parts are identical to the whole, at each scaling step. A few years later, Mandelbrot discovered fractal geometry and found that Lorenz’s attractor was a fractal figure, as are the majority of strange attractors. He defined fractal dimension. Mandelbrot quotes, as illustration of this new sort of randomness, the French coast of Brittany; its length depends on the scale at which it is measured, and has a fractal dimension between 1 and 2. This coast is neither a one-dimensional nor a two-dimensional object. For comparison the dimension of Koch snowflake is 1.26, that of Lorenz’s attractor is around 2.06, and that of the bifurcations of Feigenbaum is around 0.45.