Dickson Cheuk Yin Ng
Section 1: How it all began?
From the beginning of the scientific revolution in the 17th century, scientists believed everything in the world isdeterministic, which means that given a well-established theory and sufficient data input, the future is completelypredictable [1]. For instance, the laws of motion and Universal Gravitation derived by Sir Isaac Newton allow the prediction ofthe positions and velocities of different celestial bodies at a particular time in the future, given their initial positions and velocities at a previous time are known.
However, J. Henri Poincaré in the 1880s studied a system which involves three bodies gravitationally attracted. He found the system has non-periodic property. Some other experimental physicists had observed unpredictable behaviour in the turbulence of fluid motion [2]. It started to become clear the widely accepted linear theory at that time was insufficient to explain certain experimental observations. Due to the lack of a fully established theory to explain what had been observed, those phenomena were solely attributed to ‘noise’ or measurement imprecision. The subject of unpredictability was labelled as a special case and left forgotten.
Jules Henri Poincaré (1854–1912) [3]
In 1961, Edward N. Lorenz, a meteorologist and mathematician at MIT, USA, developed twelve differential equations in an attempt to model the weather. He programmed the model into a computer to predict the temperature, pressure, wind speed etc. at a future time. One day, he decided to run the program again to re-examine its output, but to save time, instead of restarting from the beginning, he input a number in the middle of the printout from his first run. He was expecting the same result as last time, but what came out from his computer was surprising after he compared it with the first run. At the early stage of the calculation, both the first and the second runs gave similar outputs. But then at a later stage, the two outputs started to differ, and what was incomprehensible was that the two printouts become totally uncorrelated as if there were completely different inputs.
Edward Norton Lorenz (1917-2008) [4]
The Royal McBee LGP-30. Lorenz used a computer of the same model as above to predict the weather [5].
A schematic diagram similar towhat Lorenz observed, showing the gradual deviation of two predictions having inputs that only differ by a tiny amount [6].
Lorenz initially thought there was a malfunction in his computer. But after checking and evaluating he finally recognised the root of the problem: the outputs of his computer had six decimal places (0.506127), but it rounded them off and printed out the numbers to three decimal places (0.506) [7]. He used this as the input of his second run, thinking that such a small difference wouldn’t be significant.
Lorenz noticed many events in nature could theoretically be determined to high precision, but practically it is impossible since no measurement can be made at infinitely high accuracy and precision. He published a paper in 1963 and gave the name ‘chaos‘to describe phenomena where even a slight difference in an initial condition can lead to a hugely different outcome [8].
The discovery of chaos means that accurate long-term weather forecasting is infeasible. In a talk given in 1972, Lorenz used an analogy that the flap of a wing of a butterfly in Brazil could set off a tornado in Texas to explain the chaotic and unpredictable nature of weather, later known as the ‘Butterfly Effect‘[9].
Section 2: Examples
Chaotic behaviour can easily be seen around us. Even simple dynamical systems can be chaotic. Below are two examples.
Double pendulum
The double pendulum [10].
A double pendulum is composed of a pendulum attached to the end of the other, as shown in the above diagram. The lengths and masses of the inextensible rods L1 and L2 as well as the masses of m1 and m2 may not be equal. The motion can be in two or three dimensions. The system is described by a set of ordinary differential equations (ODEs) (see below) and their solutions are the equations of motion or trajectories of the system, which are sensitive to the initial configuration of the system. The ODEs are often hard to solve and they are done numerically [11].
Swinging Atwood’s machine
The Swinging Atwood’s machine [12].
A massless inextensible string connects a bigger mass M and a smaller mass m which are suspended on two frictionless pulleys of zero radii, as shown above. The smaller mass can swing around the pulley freely in two dimensions, while the bigger mass move up and down in one dimension. For many combinations of M andm, and initial values of θand r, the system behaves chaotically [13].
Section 3: Mathematical description
3.1 Nonlinear ordinary differential equations
An N-dimensional dynamical system can be represented by a set of N ODEs, which include the time derivatives of the coordinates. Their solutions are the time evolution of the system’s states. The ODEs are usually non-linear for the system to be chaotic. A non-linear ODE is defined by the equation
such that f(x) is a function that includes the derivatives with respect to time, C is a constant, and that
where αand βare constants [14].
3.2 Attractors
The time evolution of a dynamical system can be represented geometrically by plotting its trajectories in phase space. Each point in the phase space corresponds to a particular state of the system (positions, momenta, energies, temperature, pressure, volume etc.). Attractors are a set of points or curves in the phase space of a system that nearby states would eventually attract to them asymptotically as the system evolves. The trajectories and attractors of dynamical systems depend on the ODEsthat represent them. Note not all solutions of these ODEs have chaotic behaviour. The Lyapunov exponent is the key to tell whether a system is chaotic or not.
3.3 Lyapunov exponent
Mathematically, the Lyapunov exponent, λ, is used to measure how fast two closely separated states diverge:
where δΖ0 is the initial separation of two states in phase space, δΖ(t) is the separation at time t. The inverse of the Lyapunov exponent is the Lyapunov time, which indicates the amount of time that the system has evolved such that it becomes chaotic, and any predictions beyond this time is unreliable. At the Lyapunov time, the separation between two states has grown by a factor of e. The number of Lyapunov exponents equals to the number of dimensions the system has, andusually the largest one is used [15].
3.4 Regular attractors
If the Lyapunov exponent is not positive, the system is not regarded as a chaotic one. When it equals to zero, the system’s trajectories in its phase space either converge to a point or periodically circle around a limiting path which they asymptotically approach. The attractor is called a regular attractor or limit cycle [16].
3.5 Strange attractors
If the Lyapunov exponent is positive, the system is then chaotic. The attractor has afractal structure (see below), and it is called a strange attractor [14]. Since it is chaotic, it is very sensitive to the initial conditions. Two points in the phase space that are close together will eventually be far apart. However, they never leave the attractor. It is said that the system is locally unstable but globally stable [17]. One cannot tell exactly in what state the system will be on a strange attractor, and it is non-periodic, which means that the same state never repeats itself. Below are some examples of strange attractors of dynamical systems that show chaotic behaviour.
Lorenz attractor
The Lorenz attractor [18].
where σ, ρ and β are the system’s parameters. When σ=10, ρ=28 and β=83, the system is chaotic, as well as some other combinations of nearby values. The plot is shown in the figure above. This the first attractor found to behave chaotically, when Lorenz derived a simplified model from his twelve equations to describe atmospheric convection [7].
Rössler attractor
The Rössler attractor [19].
where a, b and c are the system’s parameters. Some combinations of values that show chaotic behaviour include a=0.2, b=0.2, c=5.7; a=0.2, b=0.2, c=8.0; a=0.1, b=0.1, c=14.0 etc [20].
Rabinovich–Fabrikant attractor
The Rabinovich–Fabrikant attractor [21].
where α and γ are the system’s parameters. An example of chaotic behaviour is observed at α=0.87, γ=1.1 [22].
Thomas’ cyclically symmetric attractor
The Thomas’ cyclically symmetric attractor [23].
where b is the system’s parameter such that the system is chaotic when it approaches 0.21 [24].
3.6 Fractals
Some simple examples that show the construction of fractals [25,26,27,28,29,30].
A fractal is an object that displays self-similarity. In other words, the same type of structure must be shown if the object is zoomed in at all scales (see the above animations). The figure at the topillustrates four different fractals, constructed by continually dividing the original shapes on the left into fragments of equal size. If one keeps zooming in to see the object on the right, he would see the same image repeating itself. Some fractals have a finite area but infinite perimeter, like the one on the second row in the above figure (it is called the ‘Koch Snowflake’). All strange chaotic attractors have a fractal structure.
A fractal is characterised by its fractal dimension (also called similarity dimension, capacity dimension or Hausdorff dimension), and it is calculated by the following equation:
where D is the fractal dimension, N is the number of segments that a line is divided into during the construction of the fractal at each iteration, r is the scale of each fragment compared to the original line [31]. For the Koch Snowflake, at each iteration, one side of the equilateral triangle is divided into four segments with each segment having one-third the length of the original side. Thus, N=4, r=1/3 and therefore, from the equation,D=1.26. The fractal dimension for the Lorenz attractor is around 2.06 [32].
Section 4: Chaos or Randomness?
In everyday language, the word ‘chaos’ gives people a rather negative impression of randomness, disorder or confusion. But mathematically, chaos and randomness are different. Chaos is deterministic, meaning that by knowing the initial condition, one can calculate the outcome by following some rules and the result is always the same if the initial conditions remain exactly unchanged. Whereas, randomness is stochastic, that is there is no way to calculate or predict a system’s final state. One can distinguish between the two by comparing the time evolution of two nearby states. If the difference between them remains constant or increases exponentially i.e. the Lyapunov exponent is non-negative, then the system is deterministic, specificallyit is chaotic for a positive Lyapunov exponent, otherwise it is a stochastic system [33].
References
[1] L. Bradley,Laplace’s Demon, Chaos & Fractals, (2010), URL: http://www.stsci.edu/~lbradley/seminar/laplace.html Accessed in February 2017.
[2] Kolmogorov, A. N., Local structure of turbulence in an incompressible fluid for very large Reynolds numbers, Doklady Akademii Nauk SSSR, Vol. 30 No. 4 (1941), p. 301-305.
[3]https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9#/media/File:Henri_Poincar%C3%A9-2.jpg
[4] https://en.wikipedia.org/wiki/Edward_Norton_Lorenz#/media/File:Edward_lorenz.jpg
[5]https://en.wikipedia.org/wiki/LGP-30#/media/File:LGP-30_Manhattan_College.rjf.jpg
[6] L. Bradley,The Butterfly Effect, Chaos & Fractals, (2010), URL: http://www.stsci.edu/~lbradley/seminar/butterfly.html Accessed in February 2017.
[7] Gleick, J., Chaos: Making a New Science, Penguin Books (2008).
[8] Lorenz, E. N., Deterministic non-periodic flow, Journal of the Atmospheric Sciences, Vol. 20 No. 2 (1963), p. 130-141.
[9] Lorenz, E. N., The Essence of Chaos, University of Washington Press (1996).
[10] https://en.wikipedia.org/wiki/Double_pendulum
[11] Levien, R. B. and Tan, S. M., Double Pendulum: An experiment in chaos, American Journal of Physics, Vol. 61 No. 11 (1993), p. 1038-1044.
[12] https://en.wikipedia.org/wiki/Swinging_Atwood’s_machine
[13] Tufillaro, N. B. et al., Swinging Atwood’s Machine, American Journal of Physics, Vol. 52 No. 10 (1984), p. 895-903.
[14] Boeing, G., Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction, Systems, Vol. 4 No. 4 (2016), p. 37.
[15] Bezruchko, B. P. and Smirnov, D. A., Extracting Knowledge From Time Series: An Introduction to Nonlinear Empirical Modeling, Springer, (2010), p. 56-57.
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[18] http://www.stsci.edu/~lbradley/seminar/attractors.html
[19] http://www.stsci.edu/~lbradley/seminar/attractors.html
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[23] Sprott, J. C., Strange Attractors, The Online Journal of the Harvard Extension School Environmental Club, Vol. 1 (2008), p. 56-61.
[24] Thomas, R., Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’, International Journal of Bifurcation and Chaos, Vol. 9 No. 10 (1999), p. 1889-1905.
[25] Fractal, WolframMathWorld (2017), URL: http://mathworld.wolfram.com/Fractal.html Accessed in February 2017.
[26] http://wifflegif.com/gifs/406927-cinema-4d-fractals-gif
[27]https://upload.wikimedia.org/wikipedia/commons/6/6a/Fractal.gif
[28]https://upload.wikimedia.org/wikipedia/commons/9/91/Sphinx_rep-tile_fractal.gif
[29]http://giphy.com/gifs/submission-math-fractal-hIZ5TAcS7Q9s4
[30]http://giphy.com/gifs/fractal-apollonian-gasket-NyAH7qnSa4HbG
[31] L. Bradley, Fractals, Chaos & Fractals, (2010), URL: http://www.stsci.edu/~lbradley/seminar/fractals.html Accessed in February 2017.
[32] Grassberger, P. and Procaccia, I., Measuring the strangeness of strange attractors, Physica D.: Nonlinear Phenomena, Vol. 9 No. 1-2 (1983), p. 189-208.
[33] Provenzale, A. et al., Distinguishing between low-dimensional dynamics and randomness in measured time-series, Physica D.: Nonlinear Phenomena, Vol. 58 No. 1-4 (1992), p. 31-49.
