The presentation describes the Chaos Theory, its origin, and application in the field of Hydrological Modelling and Analysis.

1. CHAOS THEORY
Smarika Kulshrestha
Roll No: 154406004
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2. Chaos Engineering
◦ The first Chaos engineers may have been the Hindu sages who designed
a method for operating the brain, called yoga.
◦ The Buddhists produced one of the great hands-on do-it-yourself
manuals for operating the brain: The Tibetan Book of the Dying.
◦ Chinese Taoists developed the teaching of going with the flow; not
clinging to idea-structures, but changing and evolving.
◦ The message was: Be cool. Don't panic. Chaos is good. Chaos
creates infinite possibilities.
The Eternal Philosophy of Chaos-
http://erg.ucd.ie/arupa/references/chaos.h
tml
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3. What is Chaos Theory?
◦ Mathematical sub-discipline that studies complex systems.
◦ New paradigm over the Newtonian view of a mechanistic and
predictable universe
Earth's weather system
Behavior of water boiling on a stove
Migratory patterns of birds
Spread of vegetation across a continent.
Chaos Theory for Beginners; An Introduction: http://www.abarim-
publications.com/ChaosTheoryIntroduction.html#.WOY4JG996po 3

4. How Chaos Theory was born and
why?
◦ In 1960 a man named Edward Lorenz created a weather-
model on his computer at the Massachusetts Institute of
Technology.
◦ The machine never seemed to repeat a sequence: Real
Weather
◦ Lorentz decided to start half way down the sequence.
Edward Lorenz
Chaos Theory for Beginners; An
Introduction: http://www.abarim-
publications.com/ChaosTheoryIntroduc
tion.html#.WOY4JG996po 4

5. Butterfly Effect
◦ “A butterfly fluttering its wings in one continent can cause a tornado in
another continent.” -Edward Lorenz
https://media.giphy.com/media/55
PkuDHvG7u36/giphy.gif
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6. Attractors
◦ Complex systems often appear too chaotic
◦ However, they run through some kind of cycle, even though situations
are rarely exactly duplicated and repeated.
◦ A Strange Attractor represents some kind of trajectory upon which a
system runs from situation to situation without ever settling down.
The Lorenz Attractor
Attractor:
https://en.wikipedia.org/wiki/Attractor
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7. Fractals -Benoit Mandelbrot
◦ A fractal is a mathematical set that
exhibits a repeating pattern displayed at
every scale.(not just space though – can
also time)
◦ And self-organized similarity (scale
invariance): new term coined these days.
Fractal:
https://en.wikipedia.org/wiki
/Fractal 7

8. IshaVasya Upanishad
Atharva Veda, verse 8.8.8
(c. 1000 BCE)
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9. River Drainage Network – Very
Fractal
The Multifractal Random Cascade Approach
Prof. M. Anvari Lectures in
Estimating Techniques 9

10. Application in Hydrology
◦ Chaotic Theory
◦ Infinite and noise-free time
series
◦ Hydrological Series
◦ Finite and noisy
◦ Inferences (Sivakumar, 2000):
◦ The problem of data size is not as severe as it was assumed to be.
◦ The presence of noise seems to have much more influence on the nonlinear
prediction method than the correlation dimension method.
Sivakumar, 2000, J. Hydro
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11. Chaos Identification Methods
◦ Metric
◦ the study of distances between points on a
strange attractor (in the phase space)
◦ Topological
◦ the study of the organization of the strange attractor
◦ Phase space reconstruction, Correlation Dimension Method (CDM), false
nearest neighbor algorithm, Lyapunov exponent method, Kolmogorov
entropy method, surrogate data method, Poincaré map, close returns
plot, and nonlinear local approximation prediction method
Sivakumar, 2017, Springer
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12. A chaotic approach to Rainfall-
Runoff Processes
1. Both the rainfall and runoff time series observed in the same
catchment, the Gôta River basin in the south of Sweden, are analysed
to investigate the possibility of the existence of chaos in the rainfall-
runoff process.
2. 131 years of observation
 Employs Correlation Dimension Method for Chaos Identification
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Sivakumar et al., 2001,
Hydrological Sciences Journal

13. Auto Correlation function r(𝜏)
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◦ Xi = Discrete Time
series
Autocorrelation function for (a)
monthly rainfall data, and (b)
monthly runoff data from the
Gota River basin.
Sivakumar et al., 2001,
Hydrological Sciences Journal

14. Auto Correlation function
◦ For a purely random process, the ACF fluctuates about zero,
indicating that the process at any certain instance has no "memory" of
the past at all.
◦ For a periodic process, the ACF is also periodic, indicating the strong
relationship between values that repeat over and over again.
◦ For a chaotic process, the ACF decays exponentially with increasing
lag, because the states of a chaotic process are neither completely
dependent (i.e. deterministic) nor completely independent (i.e.
stochastic) of each other.
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The autocorrelation function is neither a necessary nor a
sufficient tool to characterize a process, whether stochastic
or chaotic.
Sivakumar et al., 2001,
Hydrological Sciences Journal

15. Correlation Dimension Method
◦ Correlation dimension is a measure of the extent to which the presence
of a data point affects the position of the other points lying on the
attractor.
◦ Uses a Correlation Integral: to distinguish chaotic and stochastic systems
(depending on value for the dimension)
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Sivakumar et al., 2001,
Hydrological Sciences Journal

16. Phase Space Reconstruction
◦ Method of Delays (Takens, 1980)
◦ 𝜏 is a delay time (multiple of ∆𝑡)
◦ j =1, 2, …,N-(m-1)𝜏/∆𝑡
◦ m = dimension of phase space
◦ The correlation function C(r)
◦ H= Heaviside step function H(u) =1 for u>0; H(u) = 0 for u<0
◦ U =
◦ N = Number of Data Points
◦ ν = correlation exponent, α = constant
◦ r = radius of sphere centred on Yi or Yj
16Sivakumar, 2017, Springer

17. Phase space diagram
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Stochastic system (left) versus chaotic system (right) (source Sivakumar,
2017)
The basic idea in the method of delays is that the evolution of
any single variable of a system is determined by the other
variables with which it interacts
(m = 2)
(𝜏=1)
Sivakumar, 2017, Springer

18. Correlation dimension
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Stochastic system
(left) versus
chaotic system
(right)
Saturation Value =
Correlation
Dimension of
Attractor

19. Chaos analysis of monthly rainfall
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a. time series;
b. phase space;
c. Log C(r) versus Log r;
d. relationship between
correlation exponent and
embedding dimension;
e. relationship between
correlation coefficient and
embedding dimension;
and
f. comparison between time
series plot of predicted
and observed values

20. Correlation dimension analysis of
monthly runoff coefficient
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a. time series;
b. phase space;
c. Log C(r) versus Log r;
d. relationship between correlation
exponent and embedding
dimension;
e. relationship between correlation
coefficient and embedding
dimension; and
f. comparison between time series
plot of predicted and observed
values

21. Conclusions
◦ The study (Sivakumar et al., 2001) suggested that the dynamics of
rainfall-runoff could be understood from ideas gained from theory of
chaos.
◦ Recommended further investigation to confirm existence of a dominant
chaotic component.
◦ Recommended employment of additional variables (e.g. evaporation
and infiltration), to understand rainfall-runoff dynamics.
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22. Thank You
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23. References
◦ The Eternal Philosophy of Chaos: http://erg.ucd.ie/arupa/references/chaos.html
◦ Chaos Theory for Beginners; An Introduction: http://www.abarim-
publications.com/ChaosTheoryIntroduction.html#.WOY4JG996po
◦ Attractor: https://en.wikipedia.org/wiki/Attractor
◦ Prof. M. Anvari Lectures in Estimating Techniques:
http://cbafaculty.org/10_RM/Chaos%20Theory%20Anvari.ppt
◦ Sivakumar, B. (2000), Chaos theory in hydrology: Important issues and interpretations,
Journal of Hydrology, 227, 1-20.
◦ Sivakumar, B., Berndtsson, R., Olsson, J and Jinno, K. (2001) Evidence of chaos in the rainfall-
runoff process, Hydrological Sciences Journal, 46:1,
◦ 131-145, DOI: 10.1080/02626660109492805
◦ Elshorbagy, A., Simonovic, S.P., and Panu, U.S. (2002) Estimation of missing streamflow data
using principles of chaos theory. Journal of Hydrology, 255:123–133
◦ Sivakumar, B., Kim, H. S., and Berndtsson, R. (2017). Chaos in Hydrology : Bridging
Determinism and Stochasticity. Springer.
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24. Fractals
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