Daniel Anderson

Chaos and the Logistic Map

2/19/01

Math 0450

 

 

Task 1.

 

If p(x) = ax(1-x), then p(½) = ½a(1-½) = a/4 –max

 

If 0 R a R 4, and p(x) is [0,1]

 

Then:

 

Case 1: a = 0, then p(x) = 0

 

Case 2: a = 4, then p(x) = 4x(1-x), then p maps [0,1] back into [0,1], with max=1 and min=0

 

Case 3: 0PaP 4 for the domain [0,1] the range is [0,a/4]

 

Hence for a in [0,4], p maps [0,1] back into [0,1].

 

If x1 = 1 then the 1-tail of (xn­­) = 0, because p(1)= 1a(1-1) = 0

[if !supportEmptyParas]> 

If x1 = 0 then the 1-tail of (xn) = 0, because p(0)= 0Ea(1-0) = 0

 

Hence: x1: (0,1) and ai (0,4).

 

 

Task 2.

 

If 0PaP1, then (xn) is decreasing, because axn­ is decreasing and (1-xn­) converges to 1, (xn­) is decreasing

 

Since aP1 xn becomes linearly smaller, hence (xn) converges to 0

 

If 1PaP3, the (xn) converges to 1-1/n, (I can’t figure out why??)

 

Task 3.

 

(1) When a is between Y [3.7,4] (xn) is chaotic

 

 

 

Task 4.

 

a1=3                             ÕZ.42

a2Y3.4496                   Õily:StarMath'>ÕZ.175

a3Y3.5442                   ÕZ.07

a4Y3.5644                   ÕZ.03

a5Y3.5688                   ÕZ.01

a'Y3.5701

 

Çan ­­= 3. ­­= 3.4496-3 = 0.4496 Çan+1 = 3.5442-2.4496 = .0946 Ôn­ Z .4496/.0946 = 4.7526

 

Õn = .42/.175 = 2.4

 

 

Task 5.

s=MsoNormal>Task 5.

 

p(x) = ax(1-x) = ax-ax2

 

p“(x) = a-2ax or a(1-2x)

 

p“ (1-1/a) = a-2a+2 = s2-as

 

If 1PaP3, the p“(1-1/a)P1

 

If 1Px1P3, then x1 is near the lim (xn) (because what is above).

  

Hence (xn) converges if 1Px1P3

 

 

Task 6.

 

Closest to a1=1

 

Closest to a2=1

 

Task 7.

 

Xn+1 = sin2 (àyn+1) = 4Esin2(àyn)E¢1-sin2(àyn)£ = 4Esin2(àyn)Ecos2(àyn­) =

 

=¢2Esin(àyn)Ecos(àyn­)£2 = sin2(2àyn)

 

Hence: sin2 (àyn+1) = sin2(2àyn)

 

 

Chaos Theory in Weather
            Chaos theory was first discovered by a meteorologist, named Edward Lorenz in 1963 (Orrell, 1).  Many of the environmental systems that govern the earth’s weather are chaotic systems.  In a chaotic system the precision outcome is exponentially dependent on the precision of the initial conditions, therefore a slight error in initial measurements can lead to drastic differences further along the system.  Due to these chaotic systems meteorologists have difficulty predicting the weather far in advance.

Lorenz encountered a problem while running a computer program which used twelve equations to modeled the weather.  There was a discrepancy in the number of digits used between the computer’s data and the printed out data (Rae, 1).  The computer used six decimal points, but the printouts rounded to three.  Typically Lorenz’s program was run starting with the six decimal place accuracy, but Lorenz decided to re-run a certain scenario and entered the beginning data from the printout (which only had three decimal places).  The final result of the second run was strikingly different from that of the same scenario run from the numbers with six decimal places (fig. 1).  In the years after this discover Lorenz investigated chaotic systems, simplifying the complex weather systems to fairly simple equations which still behaved in a chaotic manner.  Lorenz had difficulty publishing his findings in a suitable journal because he was trained as a meteorologist.  Since his work  could only be published in meteorological journal, it was not discovered until years latter (Rae, 2-3).

            Today there are n>Today there are two fields of thought on the mathematics of weather and environmental forecasting.  The oldest approach involves the collection of historical weather patterns and data, and correlating what will proceed.  This imperial-statistical method can prove to be very labor intensive and result in vague, inaccurate forecasts.  The empirical-statistical method, also called the correlation method is best suited for forecasting long-term trends in environmental weather, but falls short when trying to pinpoint the path a particular storm.  This method is still used today for forecasting a few environmental factors months in advance, for example: average rainfall, or sea surface temperature (Hunt, 272)  Typical weather forecasting also used the empirical-statistical method until several years ago, it was then replaced by the reductionist approach.

            The second, more resent approach, has evolved from the work of Lorenz and the development of computers.  The “reductionist” approach strives to reduce weather to several simple, often chaotic, forces or systems, often represented by mathematical, and chaotic equations.  Since many of these systems are chaotic, their use in forecasting becomes less accurate with advance forecasts.  Like many chaotic equations, the systems of weather patterns follow a “normal” path for a period of time before becoming chaotic.  Therefore the reductionist method is useful in predicting specific weather patterns for as long as 5 days in advance.  After this point the chaotic nature of the systems become apparent, and different scenario runs of similar data lead to completely different results.

            When it was first discovered that weather had chaotic factors, some became skeptical  of meteorology, saying that it would be impossible to ever forecast the weather with accuracy.  With advancements and better understanding of chaos theory, weather forecasting is becoming more accurate.  Although it may never be possible to forecast precise weather conditions years in advance, the advancements made in the last century lead one to believe that better weather forecasting is eminent.  In the future more precise measurement of data and more advance computer power may lead to a better understanding of the chaotic equations which drive our weather and other environmental factors.

References:

Hunt, J.C.R. Environmental forecasting and turbulence modeling, Physica D 133 (1999) 270-295, Elsevier.

Orrell, David Model Error in Weather Forecasting, Does Chaos Matter?,

http://www.beatrizl.freeserve.co.uk/AGUposter.htm. (2/16/01)

Rae, Gregory. Chaos Theory: A Brief Introduction, http://www.imho.com/grae/chaos/chaos.html. (2/16/01)

chaos.html. (2/16/01)