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Glimpses of Chaos

Bharath Raj G N. Glimpses of Chaos. Why Chaos?. Chaos theory provides an astounding array of tools to make short-term predictions, to get a flavor of the long-term trends in the data set, to estimate which of the variables are driving the

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Glimpses of Chaos

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  1. Bharath Raj G N Glimpses of Chaos

  2. Why Chaos? Chaos theory provides an astounding array of tools to make short-term predictions, to get a flavor of the long-term trends in the data set, to estimate which of the variables are driving the dynamics of a system and moreover this analysis is often possible without any prior knowledge of an underlying model or the set of equations governing the system.

  3. What is Chaos? An ancient term originally meaning “utter lawlessness” Now, informally refers to all those systems whose variation “only looks random” but “actually not random” Chaos is a proxy to the more appropriate term: “DETERMINISTIC CHAOS”

  4. The 3-Body Problem Configuration of the simplest 3-Body Problem Solution to 2-Body Problem

  5. A particular solution to the 3-Body problem Analytical Solution is impossible, we can at best use some numerical schemes and implement the Newton's Laws and a very small “time-step” to get more accurate solutions

  6. Logistic Maps

  7. Feigenbaum's Number the rate at which the bifurcations occur is governed by a constant number. If, say, L(n) is the value of L for which the n-th bifurcation occurs, and L(n + 1) is the value where the next bifurcation occurs, then at all levels L(n + 1) divided by L(n) is always the same value. It turns out that this number, now called the Feigenbaum number, is irrational and approximately 4.6692016090... It also turns out that the Feigenbaum number is associated with all chaotic systems. For all chaotic systems, the infinite bifurcations preceding the transition to chaos are characterised by the Feigenbaum number.

  8. Lorenz Attractor “Deterministic non-periodic flow” -Edward N Lorenz (J.Atmos.Sci,20, 130-141,1963) dX/dt = s*(X-Y) dY/dt = r*X – X*Z – Y dZ/dt = X*Y - b*X

  9. Characteristics of ChaoticSystems 1) All Chaotic Systems exhibit, Sensitive Dependence on Initial Condition. 2) For all Chaotic Systems, the trajectory never repeats. 3) For all Chaotic Systems, the transition to Chaos is preceded by infinite levels of bifurcation. 4) The infinite levels of bifurcation preceding the transition are characterized by the Feigenbaum Number. 5) All Chaotic Systems exhibit fractional dimensionality.

  10. Applications of Chaos Theory Much like Physics, Chaos Theory provides a foundation for the study of all other scientific disciplines. It is actually a tool-box of methods for incorporating non-linear dynamics. In Mathematics, the use of strange attractors, fractals and cellular automata are used for studying data that was previously thought of as random. In Physics, Chaos is being applied to the study of turbulence, non-periodic phenomenon (Eg: “ElNino on a Devil's Staircase”:Neelin,Jin and Ghil, Science, 1994) and to model such complex systems as the COSMOS. In Biology, It is used for better understanding of evolutionary processes, genetic algorithms, artificial life simulations, understanding the functionalities of brain. In Computer Science, it is being used to find better compression and encryption algorithms, fractals promise an image compression of 600:1.

  11. Thank You & Happy Deepavali

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