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Introduction Chaos theory Summary

Deterministic Chaos

Aleksander Sanjuan Ciepielewski

Uniwersytet WarszawskiWydział Fizyki

25th of November, 2019

Aleksander Sanjuan Ciepielewski Deterministic Chaos


Literatura I

H.G. Schuster.Chaos Deterministyczny. Wprowadzenie.

J.B. Marion, S.T. Thornton.Classical Dynamics of Particles and Systems

K.T. Alligood, T.D. Sauer, J.A. Yorke.Chaos: An Introduction to Dynamical Systems

H. PoincareActa Math. 13, 1(1890)

M. Cattani,I. Caldas,S. de Souza,K. Iarosz.Rev. Bras. Ensino Fıs. vol.39 no.1 Sao Paulo 2017 Epub Oct 17,2016

B.K. NikolićWykład:“Introduction to Deterministic Chaos”



Literatura II

G. Boeing“Chaos Theory and the Logistic Map”

G.A. Leonov, N.V. Kuznetsov“Time-Varying Linearization and the Perron Effects”



Plan

1 IntroductionFirst occurrences of chaosWhy do we need chaos theory?Definition and basic properties

2 Chaos theoryHow does chaos arise?Poincare maps and bifurcation diagramsLyapunov exponents and Kolmogorov entropyPossible criteria for chaos


Introduction Chaos theory Summary First occurrences of chaos Why do we need chaos theory? Definition and basic properties

Plan





The birth of Chaos theoryThree-body problem, Poincare 1887

In 1897 the king of Sweden offered a prize for the anyone able toanswer the question: Is the solar system stable?

The so-called “last universalist” Henry Poincare won the prize,because of his work on the three-body problem, even though hecouldn’t answer the original question.

“..But even if it were the case that the natural laws had no longerany secret for us, we could still only know the initial situationapproximately...A small error in the former will produce an enormouserror in the latter. Prediction becomes impossible, and we have thefortuitous phenomenon”

“It’s so complicated I can’t even draw it.”



Is the three-body problem really chaotic?

https://www.wolframscience.com/nks/notes-7-4–three-body-problem/



Lorenz attractor

In 1963 meteorologist H.Lorenz worked on improvingweather predictions.

(x , y , z) represent theconvection in the atmosphere.

(σ, β, ρ) are the controlparameters, for some values thesystem collapses unambiguouslyinto a so-called strangeattractor for all initialconditions.

“Chaos: When the presentdetermines the future, but theapproximate present does notapproximately determine thefuture.” -Lorenz



Plan





Why bother?

Chaos emerges in many physical systems, it is not an aberration noran interesting anomaly. A few examples:

fluids in turbulent flow,some chemical reactions,lasers and non-linear optics,the trajectories of objects in space,particle accelerators,biological models for population dynamics,excited heart cells.

Arguably the main interest of science is to build models withpredictive capability, if we can’t solve chaotic models mathematicallywe can search for other valuable information about them.Chaos theory can help us gather information, among other things,about:

chaotic and non-chaotic intervals and critical points,dependence on external parameters of the system,the “measure” of chaos in the system,range of uncertainty.



Example: Hurricane Sandy

Almost all models predicted acontinuous route towardsEurope, Sandy turned west.

“In fact, in more than 100 yearsof weather records, ‘this is theonly one that turned back tothe west.’”-“Weather lessonsfrom Superstorm Sandy”, USAToday

117 people died, damagesexceeded $50 billion USD.Countless lives saved thanks tothe European model thatpredicted correctly the turn.

NOAA, Tropical Cycone Report:Hurricane Sandy



Plan





Definition and basic properties

Deterministic chaos

Property of a system, whose solutions are extremely sensitive to smalldifferences in the initial conditions.

Chaos 6= randomness: In stochastic models the evolution of thesystem is always probabilistic, even for the exact same initialconditions. In deterministic models, one unique solution to a givenset of initial conditions ⇒ trajectories in phase space do not overlap.We limit our study of chaos to systems that are self-recurring, thatis:

they are topologically transitive (TT), i.e. C 3 f : U → V is TT iffor any two intervals U,V in phase space there is a positive integer ksuch that

f k(U) ∩ V 6= ∅,they have dense periodic orbits.



Types of chaos

1 Conservative chaosSets in phase space do not change volume in time (Liouvilletheorem).They can be classical or quantum.

2 Dissipative chaosSubsets of phase space reduce their volume.They are thermodynamically open, exchanging energy and matterwith an environment.Examples include convection currents, turbulent flow, cyclones,hurricanes.


Introduction Chaos theory Summary How does chaos arise? Poincare maps and bifurcation diagrams Lyapunov exponents and Kolmogorov entropy Possible criteria for chaos

Plan





Chaotic systems

1 The system has to have at least 3 different independent dynamicvariables.

2 Equations of motion can’t be linear (for a finite set of differentialequations).

3 Why?No trajectories can overlap → there must be twisting and divergingin phase space, so that small intervals expand and contractexponentially.

B.K.NikolićAleksander Sanjuan Ciepielewski Deterministic Chaos


Plan





Poincare maps

Consider a lower-dimensionalsubspace of the system which istransversal to the orbits in agive interval.

Mark out the intersectionpoints, P0,P1, ...,Pn, of suchsubspace with the orbits.

From a continuoustime-dependent function weobtain a discrete iterativefunction Pn+1 = f (Pn).

The correlations betweenneighboring points provideinformation about the behaviorof the system.



Example: Henon map

Introduced by Michal Henon as a simplified model for the Poincaresection of the Lorenz attractor.

xn+1 = a+ byn − x2n ,

yn+1 = xn.(1)

a = 1.45, b = 0.2→ periodic orbits. a = 0.2, b = 1.48→ chaoticattractor.



Bifurcation diagrams

A bifurcation diagram shows the values visited or approachedasymptotically of a system as a function of a parameter in thesystem.

Very useful to obtain chaotic and non-chaotic intervals, criticalpoints, stable and non-stable points and range of uncertainty.



Example: Logistic map

Initially presented by mathematician Pierre Verhulst but popularized bybiologist Robert May as a discrete-time model of population growth in aclosed environment.

xn+1 = rxn(1− xn), (2)

r ∈ [0, 4].

For r ∈ [0, 1] the solution is trivial, the population extinguishes aftersome time.

For r ∈ (1,≈ 3.6) the solution is stable, the population either leadsto a certain value or oscillates between a number of solutions.

What happens afterwards?




Source: G.Boeing




Source: Wiki



Period doubling: Feigenbaum constants

limn→∞

an−1 − an−2

an − an−1= 4.669201609...,

bnb∗n+1

= 2.502907875096...



Plan





Lyapunov exponents

A Lyapunov exponent is a quantity that characterizes the rate ofseparation of infinitesimally closed trajectories.

Remember, that chaos ⇒ exponential deformation of phase-spacesubsets.

Chaos ⇒ ∃i : λi > 0, necessary but not sufficient condition.



Lyapunov exponent: Logistic map

Source: Rev.Bras.Aleksander Sanjuan Ciepielewski Deterministic Chaos


Kolmorogov entropy

Kolmorogov entropy (K ) can be interpreted as the amount ofinformation lost in time.

Consider the trajectory of the system ~x(t) = [x1(t), x2(t), ..., xd(t)]and divide the phase-space of dimension d into small boxes ofvolume Ld , and the time into intervals of length τ .

Claude Shannon (1948) argued, that the quantity

Kn = −∑i0...in

Pi0...in lnPi0...in

. Where:Pi0...in is the joint probability of trajectory~x(t = mτ),m = 0, 1, 2, ..., n passing through boxes i0, i1, ..., in,

is proportional to the information needed to locate the system on aspecial trajectory with precision L.



Then, the Kolmorogov entropy is defined as

K = limτ,L→0

limN→∞

1Nτ

N−1∑n=0

(Kn+1 − Kn), (3)

and it represents the average loss of information of the system,(notethat Kn+1 − Kn is the additional information needed to predict statein+1 knowing that the system is in in).



K = limτ,L→0

limN→∞

1Nτ

N−1∑n=0

(Kn+1 − Kn)

Source: Nikolić



Plan





Possible criteria for chaos

1 The time-dependent signal “looks” chaotic, (simple but not precise,some signals can be complicated yet periodic or quasi-periodic).

2 Conduct a Fourier transform and then plot the spectrumP(ω) = |x(ω)|2.

If the solution is periodic or quasi-periodic we should see discretelines at certain frequencies, for a chaotic system we observebroadband noice

3 In the Poincare map the points cover the space (not always true,curves can arise from both chaotic and quasi-periodic solutions).Analogous in the case of the bifurcation diagram. Again, an analysisof the spectrum of the Fourier transform yields more information(equivalently, one can compute the autocorrelation function).

4 The Lyapunov exponent of at least one dynamical variable is greaterthan zero (for most deterministic systems enough but there areexceptions, [Leonov,Kuznetsov]). Careful with stochastic systems,they can also have positive exponents.

5 The Kolmorogov entropy is positive, (again careful with stochasticsignals)



Summary

Chaos is inherent in our universe, it emerges from a very wide rangeof phenomena and we need a theory to deal with its complexity.

Chaos theory allows us to estimate, among other things:Chaotic, stable and periodic intervals.Uncertainty range and diffusion of information.When, how and under which conditions we can expect chaos.

Strangely enough, with the tools of chaos theory we find that thereis a great deal of order in chaos (dependence on externalparameters, period doubling, fractal structure, and much more)



Summary

Chaos is inherent in our universe, it emerges from a very wide rangeof phenomena and we need a theory to deal with its complexity.Chaos theory allows us to estimate, among other things:

Chaotic, stable and periodic intervals.Uncertainty range and diffusion of information.When, how and under which conditions we can expect chaos.

Strangely enough, with the tools of chaos theory we find that thereis a great deal of order in chaos (dependence on externalparameters, period doubling, fractal structure, and much more)



Thank you for your attention!


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