Entry
Math: Chaos theory:Equation: Logistic:Can you give BBCBASIC program to show Verhulst logistic curve?
Jul 21st, 2008 09:10
Taksh Verdhan, Knud van Eeden, Deepak Bansal,
----------------------------------------------------------------------
--- Knud van Eeden --- 29 November 2020 - 06:58 pm -------------------
Math: Chaos theory:Equation: Logistic:Can you give BBCBASIC program to
show Verhulst logistic curve?
---
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Here a program written in BBCBASIC which uses the Verhulst logistics
equation to show the
bifurcation.
---
Steps: Overview:
1. -Given the Verhulst logistics equation
newpopulation = constant . (oldpopulation) . (1 - oldpopulation)
or thus here:
x = constant * x * (1 - x)
2. -You use the algorithm
for first to last value of the constant
for first to last generation
calculate the new population
draw a dot at XY position
(parameter, population)
on the screen
next generation
next constant
---
Note:
To possibly see it a bit easier, as less dots are involved,
you could use the following algorithm, which only shows
the last generation
for first to last value of the constant
let the population grow for a while
(e.g. until the first thousand populations)
when finished draw a dot at position
(parameter, population) on the screen
3. -A possible program to implement this could be:
--- cut here ---------------------------------------------------------
REM --- MAIN --- REM
MODE 18 : REM graphical mode
PROCMathDrawCurveLogisticVerhulst
REPEAT UNTIL GET
END
REM --- LIBRARY --- REM
REM library: math: chaos theory: logistic curve: bifurcation: curve:
Verhulst logistic curve (most simple version)
DEF PROCMathDrawCurveLogisticVerhulst
REM e.g. MODE 18 : REM graphical mode
REM e.g. PROCMathDrawCurveLogisticVerhulst
REM e.g. END
LOCAL T%
LOCAL x
LOCAL parameter
REM vary the constant from 0 to 4
FOR parameter = 0 TO 4 STEP 0.01
REM you choose (arbitrarily) your starting population half of total
x = 0.5
REM you first calculate the first (arbitrary) thousand generations
FOR T% = 1 TO 1000
REM this is just the Verhulst logical equation 'constant . x . (1-x)'
x = parameter * x * ( 1 - x )
REM draw dot at transformed XY screen position (parameter, x)
PLOT 69, 100 * parameter, 500 * x
NEXT T%
REM possibly check the x,y values
REM PRINT TAB( 10,
10 ); " "
REM PRINT TAB( 10, 10 ); parameter; TAB( 25 ); x
REM REPEAT UNTIL GET
NEXT parameter
ENDPROC
--- cut here ---------------------------------------------------------
---
4. When you run this program, you will see a graphical screen with
output
similar to the following:
.
. .
.
^ . .
| . . . .
x . . . . .
. . .
. .
...
------------------------------------
| | | |
0 1 3 3.45 ---> parameter
It shows only 1 branch of the bifurcated curve.
---
---
Another more general program would be:
--- cut here ---------------------------------------------------------
REM --- MAIN ---
PROCCurve1DifferenceEquation1Parameter( 0.6876, 2.5, 6, 50,
1000, "parameter * x * (1 - x)" )
END
:
:
:
REM --- LIBRARY ---
REM: library: math: curve: chaos theory: bifurcation: DIFFERENCE
EQUATION with 1 PARAMETER [kn, zoe, su, 31-12-2020 18:16:31]
DEF PROCCurve1DifferenceEquation1Parameter( xbegin, parameterbegin,
parameterend, parameterstepmaxT%, iterationmaxT%, fx$ )
REM e.g. REM PROCScreenGraphicResolutionHigh
REM e.g. PROCCurve1DifferenceEquation1Parameter( 0.6876, 2, 6, 50,
1000, "parameter * x * (1 - x)" )
REM e.g. REM PROCScreenText
REM e.g. END
LOCAL parameter, parameterstep, parameterstepT%, iterationT%, x,
trialT%
parameterstep = FNStep( parameterbegin, parameterend,
parameterstepmaxT% )
PRINT; "xold starts at the begin value = "; xbegin
PRINT
FOR parameterstepT% = 0 TO parameterstepmaxT%
parameter = parameterbegin + parameterstepT% * parameterstep
x = xbegin
FOR iterationT% = 1 TO iterationmaxT%
x = FNEval( x, fx$ )
IF iterationT% > (iterationmaxT% - 16) THEN
PROCCurve1DifferenceEquation1ParameterOutput( parameter, x )
NEXT iterationT%
REM PROCPlotPointAt( parameter, x )
PRINT
PRINT
PRINT "---"
PRINT
REPEAT UNTIL GET
NEXT parameterstepT%
ENDPROC
:
DEF PROCCurve1DifferenceEquation1ParameterOutput( parameter, x )
PRINT; "parameter = "; parameter; " : xnew after "; iterationT%; "
generations = "; x
ENDPROC
:
:REM library: plot a point at the given coordinates [kn, zoe, su, 31-
12-2000 17:29:56]
DEF PROCPlotPointAt( x, y )
PLOT 69, (1260/2) + 100 * x, (1000/2) + 100 * y
ENDPROC
:
REM: library: screen: graphic [kn, zoe, su, 31-12-2020 17:38:11]
DEF PROCScreenGraphicResolutionHigh
MODE(18)
ENDPROC
:
REM: library: screen: graphic [kn, zoe, su, 31-12-2020 17:38:11]
DEF PROCScreenText
MODE(3)
ENDPROC
:
DEF FNStep( xbegin, xend, xstepmaxT% )
LOCAL xdelta
xdelta = xend - xbegin
IF xdelta = 0 THEN PRINT "do not choose begin interval and end
interval the same"
IF xstepmaxT% = 0 THEN PRINT "do not choose xstepmaxT% equal to
zero" : = 0
= ABS( xdelta / xstepmaxT% )
:
DEF FNEval( x, s$ )
IF s$ = "" THEN PRINT "eval: no empty string allowed" : = 0
= EVAL s$ : REM here should the output of your function be generated
(given your x). For example if your function is ln(x) + x, and x is
given to be 3, then the output here should be ln(3) + 3.
REM
REM so here you should write, if your computer language does not have
an EVAL function,
REM something like:
REM
REM RETURN( EXP(x) )
REM
REM ---
REM
REM or use e.g. a global variable, and write something similar to:
REM y = EXP(x)
REM RETURN( y )
REM
:
--- cut here ---------------------------------------------------------
---
Here you see some results for the total amount of animals, after
985 and 1000 generations, given the value for that constant parameter
---
>RUN
xold starts at the begin value = 0.6876
parameter = 2.5 : xnew after 985 generations = 0.6
parameter = 2.5 : xnew after 986 generations = 0.6
parameter = 2.5 : xnew after 987 generations = 0.6
parameter = 2.5 : xnew after 988 generations = 0.6
parameter = 2.5 : xnew after 989 generations = 0.6
parameter = 2.5 : xnew after 990 generations = 0.6
parameter = 2.5 : xnew after 991 generations = 0.6
parameter = 2.5 : xnew after 992 generations = 0.6
parameter = 2.5 : xnew after 993 generations = 0.6
parameter = 2.5 : xnew after 994 generations = 0.6
parameter = 2.5 : xnew after 995 generations = 0.6
parameter = 2.5 : xnew after 996 generations = 0.6
parameter = 2.5 : xnew after 997 generations = 0.6
parameter = 2.5 : xnew after 998 generations = 0.6
parameter = 2.5 : xnew after 999 generations = 0.6
parameter = 2.5 : xnew after 1000 generations = 0.6
---
parameter = 2.57 : xnew after 985 generations = 0.610894941
parameter = 2.57 : xnew after 986 generations = 0.610894942
parameter = 2.57 : xnew after 987 generations = 0.610894941
parameter = 2.57 : xnew after 988 generations = 0.610894942
parameter = 2.57 : xnew after 989 generations = 0.610894941
parameter = 2.57 : xnew after 990 generations = 0.610894942
parameter = 2.57 : xnew after 991 generations = 0.610894941
parameter = 2.57 : xnew after 992 generations = 0.610894942
parameter = 2.57 : xnew after 993 generations = 0.610894941
parameter = 2.57 : xnew after 994 generations = 0.610894942
parameter = 2.57 : xnew after 995 generations = 0.610894941
parameter = 2.57 : xnew after 996 generations = 0.610894942
parameter = 2.57 : xnew after 997 generations = 0.610894941
parameter = 2.57 : xnew after 998 generations = 0.610894942
parameter = 2.57 : xnew after 999 generations = 0.610894941
parameter = 2.57 : xnew after 1000 generations = 0.610894942
---
parameter = 2.64 : xnew after 985 generations = 0.621212121
parameter = 2.64 : xnew after 986 generations = 0.621212121
parameter = 2.64 : xnew after 987 generations = 0.621212121
parameter = 2.64 : xnew after 988 generations = 0.621212121
parameter = 2.64 : xnew after 989 generations = 0.621212121
parameter = 2.64 : xnew after 990 generations = 0.621212121
parameter = 2.64 : xnew after 991 generations = 0.621212121
parameter = 2.64 : xnew after 992 generations = 0.621212121
parameter = 2.64 : xnew after 993 generations = 0.621212121
parameter = 2.64 : xnew after 994 generations = 0.621212121
parameter = 2.64 : xnew after 995 generations = 0.621212121
parameter = 2.64 : xnew after 996 generations = 0.621212121
parameter = 2.64 : xnew after 997 generations = 0.621212121
parameter = 2.64 : xnew after 998 generations = 0.621212121
parameter = 2.64 : xnew after 999 generations = 0.621212121
parameter = 2.64 : xnew after 1000 generations = 0.621212121
---
parameter = 2.71 : xnew after 985 generations = 0.63099631
parameter = 2.71 : xnew after 986 generations = 0.63099631
parameter = 2.71 : xnew after 987 generations = 0.63099631
parameter = 2.71 : xnew after 988 generations = 0.63099631
parameter = 2.71 : xnew after 989 generations = 0.63099631
parameter = 2.71 : xnew after 990 generations = 0.63099631
parameter = 2.71 : xnew after 991 generations = 0.63099631
parameter = 2.71 : xnew after 992 generations = 0.63099631
parameter = 2.71 : xnew after 993 generations = 0.63099631
parameter = 2.71 : xnew after 994 generations = 0.63099631
parameter = 2.71 : xnew after 995 generations = 0.63099631
parameter = 2.71 : xnew after 996 generations = 0.63099631
parameter = 2.71 : xnew after 997 generations = 0.63099631
parameter = 2.71 : xnew after 998 generations = 0.63099631
parameter = 2.71 : xnew after 999 generations = 0.63099631
parameter = 2.71 : xnew after 1000 generations = 0.63099631
---
parameter = 2.78 : xnew after 985 generations = 0.64028777
parameter = 2.78 : xnew after 986 generations = 0.64028777
parameter = 2.78 : xnew after 987 generations = 0.64028777
parameter = 2.78 : xnew after 988 generations = 0.64028777
parameter = 2.78 : xnew after 989 generations = 0.64028777
parameter = 2.78 : xnew after 990 generations = 0.64028777
parameter = 2.78 : xnew after 991 generations = 0.64028777
parameter = 2.78 : xnew after 992 generations = 0.64028777
parameter = 2.78 : xnew after 993 generations = 0.64028777
parameter = 2.78 : xnew after 994 generations = 0.64028777
parameter = 2.78 : xnew after 995 generations = 0.64028777
parameter = 2.78 : xnew after 996 generations = 0.64028777
parameter = 2.78 : xnew after 997 generations = 0.64028777
parameter = 2.78 : xnew after 998 generations = 0.64028777
parameter = 2.78 : xnew after 999 generations = 0.64028777
parameter = 2.78 : xnew after 1000 generations = 0.64028777
---
parameter = 2.85 : xnew after 985 generations = 0.649122806
parameter = 2.85 : xnew after 986 generations = 0.649122808
parameter = 2.85 : xnew after 987 generations = 0.649122806
parameter = 2.85 : xnew after 988 generations = 0.649122808
parameter = 2.85 : xnew after 989 generations = 0.649122806
parameter = 2.85 : xnew after 990 generations = 0.649122808
parameter = 2.85 : xnew after 991 generations = 0.649122806
parameter = 2.85 : xnew after 992 generations = 0.649122808
parameter = 2.85 : xnew after 993 generations = 0.649122806
parameter = 2.85 : xnew after 994 generations = 0.649122808
parameter = 2.85 : xnew after 995 generations = 0.649122806
parameter = 2.85 : xnew after 996 generations = 0.649122808
parameter = 2.85 : xnew after 997 generations = 0.649122806
parameter = 2.85 : xnew after 998 generations = 0.649122808
parameter = 2.85 : xnew after 999 generations = 0.649122806
parameter = 2.85 : xnew after 1000 generations = 0.649122808
---
parameter = 2.92 : xnew after 985 generations = 0.657534245
parameter = 2.92 : xnew after 986 generations = 0.657534248
parameter = 2.92 : xnew after 987 generations = 0.657534245
parameter = 2.92 : xnew after 988 generations = 0.657534248
parameter = 2.92 : xnew after 989 generations = 0.657534245
parameter = 2.92 : xnew after 990 generations = 0.657534248
parameter = 2.92 : xnew after 991 generations = 0.657534245
parameter = 2.92 : xnew after 992 generations = 0.657534248
parameter = 2.92 : xnew after 993 generations = 0.657534245
parameter = 2.92 : xnew after 994 generations = 0.657534248
parameter = 2.92 : xnew after 995 generations = 0.657534245
parameter = 2.92 : xnew after 996 generations = 0.657534248
parameter = 2.92 : xnew after 997 generations = 0.657534245
parameter = 2.92 : xnew after 998 generations = 0.657534248
parameter = 2.92 : xnew after 999 generations = 0.657534245
parameter = 2.92 : xnew after 1000 generations = 0.657534248
---
parameter = 2.99 : xnew after 985 generations = 0.665550898
parameter = 2.99 : xnew after 986 generations = 0.665552772
parameter = 2.99 : xnew after 987 generations = 0.665550916
parameter = 2.99 : xnew after 988 generations = 0.665552753
parameter = 2.99 : xnew after 989 generations = 0.665550935
parameter = 2.99 : xnew after 990 generations = 0.665552735
parameter = 2.99 : xnew after 991 generations = 0.665550953
parameter = 2.99 : xnew after 992 generations = 0.665552717
parameter = 2.99 : xnew after 993 generations = 0.665550971
parameter = 2.99 : xnew after 994 generations = 0.6655527
parameter = 2.99 : xnew after 995 generations = 0.665550988
parameter = 2.99 : xnew after 996 generations = 0.665552683
parameter = 2.99 : xnew after 997 generations = 0.665551005
parameter = 2.99 : xnew after 998 generations = 0.665552666
parameter = 2.99 : xnew after 999 generations = 0.665551021
parameter = 2.99 : xnew after 1000 generations = 0.665552649
---
parameter = 3.06 : xnew after 985 generations = 0.582751879
parameter = 3.06 : xnew after 986 generations = 0.744045507
parameter = 3.06 : xnew after 987 generations = 0.582751879
parameter = 3.06 : xnew after 988 generations = 0.744045507
parameter = 3.06 : xnew after 989 generations = 0.582751879
parameter = 3.06 : xnew after 990 generations = 0.744045507
parameter = 3.06 : xnew after 991 generations = 0.582751879
parameter = 3.06 : xnew after 992 generations = 0.744045507
parameter = 3.06 : xnew after 993 generations = 0.582751879
parameter = 3.06 : xnew after 994 generations = 0.744045507
parameter = 3.06 : xnew after 995 generations = 0.582751879
parameter = 3.06 : xnew after 996 generations = 0.744045507
parameter = 3.06 : xnew after 997 generations = 0.582751879
parameter = 3.06 : xnew after 998 generations = 0.744045507
parameter = 3.06 : xnew after 999 generations = 0.582751879
parameter = 3.06 : xnew after 1000 generations = 0.744045507
---
parameter = 3.13 : xnew after 985 generations = 0.542694153
parameter = 3.13 : xnew after 986 generations = 0.776794665
parameter = 3.13 : xnew after 987 generations = 0.542694153
parameter = 3.13 : xnew after 988 generations = 0.776794665
parameter = 3.13 : xnew after 989 generations = 0.542694153
parameter = 3.13 : xnew after 990 generations = 0.776794665
parameter = 3.13 : xnew after 991 generations = 0.542694153
parameter = 3.13 : xnew after 992 generations = 0.776794665
parameter = 3.13 : xnew after 993 generations = 0.542694153
parameter = 3.13 : xnew after 994 generations = 0.776794665
parameter = 3.13 : xnew after 995 generations = 0.542694153
parameter = 3.13 : xnew after 996 generations = 0.776794665
parameter = 3.13 : xnew after 997 generations = 0.542694153
parameter = 3.13 : xnew after 998 generations = 0.776794665
parameter = 3.13 : xnew after 999 generations = 0.542694153
parameter = 3.13 : xnew after 1000 generations = 0.776794665
---
parameter = 3.2 : xnew after 985 generations = 0.51304451
parameter = 3.2 : xnew after 986 generations = 0.79945549
parameter = 3.2 : xnew after 987 generations = 0.51304451
parameter = 3.2 : xnew after 988 generations = 0.79945549
parameter = 3.2 : xnew after 989 generations = 0.51304451
parameter = 3.2 : xnew after 990 generations = 0.79945549
parameter = 3.2 : xnew after 991 generations = 0.51304451
parameter = 3.2 : xnew after 992 generations = 0.79945549
parameter = 3.2 : xnew after 993 generations = 0.51304451
parameter = 3.2 : xnew after 994 generations = 0.79945549
parameter = 3.2 : xnew after 995 generations = 0.51304451
parameter = 3.2 : xnew after 996 generations = 0.79945549
parameter = 3.2 : xnew after 997 generations = 0.51304451
parameter = 3.2 : xnew after 998 generations = 0.79945549
parameter = 3.2 : xnew after 999 generations = 0.51304451
parameter = 3.2 : xnew after 1000 generations = 0.79945549
---
parameter = 3.27 : xnew after 985 generations = 0.817084375
parameter = 3.27 : xnew after 986 generations = 0.488726023
parameter = 3.27 : xnew after 987 generations = 0.817084375
parameter = 3.27 : xnew after 988 generations = 0.488726023
parameter = 3.27 : xnew after 989 generations = 0.817084375
parameter = 3.27 : xnew after 990 generations = 0.488726023
parameter = 3.27 : xnew after 991 generations = 0.817084375
parameter = 3.27 : xnew after 992 generations = 0.488726023
parameter = 3.27 : xnew after 993 generations = 0.817084375
parameter = 3.27 : xnew after 994 generations = 0.488726023
parameter = 3.27 : xnew after 995 generations = 0.817084375
parameter = 3.27 : xnew after 996 generations = 0.488726023
parameter = 3.27 : xnew after 997 generations = 0.817084375
parameter = 3.27 : xnew after 998 generations = 0.488726023
parameter = 3.27 : xnew after 999 generations = 0.817084375
parameter = 3.27 : xnew after 1000 generations = 0.488726023
---
parameter = 3.34 : xnew after 985 generations = 0.831548319
parameter = 3.34 : xnew after 986 generations = 0.467852879
parameter = 3.34 : xnew after 987 generations = 0.831548319
parameter = 3.34 : xnew after 988 generations = 0.467852878
parameter = 3.34 : xnew after 989 generations = 0.831548319
parameter = 3.34 : xnew after 990 generations = 0.467852879
parameter = 3.34 : xnew after 991 generations = 0.831548319
parameter = 3.34 : xnew after 992 generations = 0.467852878
parameter = 3.34 : xnew after 993 generations = 0.831548319
parameter = 3.34 : xnew after 994 generations = 0.467852879
parameter = 3.34 : xnew after 995 generations = 0.831548319
parameter = 3.34 : xnew after 996 generations = 0.467852878
parameter = 3.34 : xnew after 997 generations = 0.831548319
parameter = 3.34 : xnew after 998 generations = 0.467852879
parameter = 3.34 : xnew after 999 generations = 0.831548319
parameter = 3.34 : xnew after 1000 generations = 0.467852878
---
parameter = 3.41 : xnew after 985 generations = 0.843791216
parameter = 3.41 : xnew after 986 generations = 0.449463916
parameter = 3.41 : xnew after 987 generations = 0.843791215
parameter = 3.41 : xnew after 988 generations = 0.449463917
parameter = 3.41 : xnew after 989 generations = 0.843791216
parameter = 3.41 : xnew after 990 generations = 0.449463916
parameter = 3.41 : xnew after 991 generations = 0.843791215
parameter = 3.41 : xnew after 992 generations = 0.449463917
parameter = 3.41 : xnew after 993 generations = 0.843791216
parameter = 3.41 : xnew after 994 generations = 0.449463916
parameter = 3.41 : xnew after 995 generations = 0.843791215
parameter = 3.41 : xnew after 996 generations = 0.449463917
parameter = 3.41 : xnew after 997 generations = 0.843791216
parameter = 3.41 : xnew after 998 generations = 0.449463916
parameter = 3.41 : xnew after 999 generations = 0.843791215
parameter = 3.41 : xnew after 1000 generations = 0.449463917
---
parameter = 3.48 : xnew after 985 generations = 0.831680119
parameter = 3.48 : xnew after 986 generations = 0.487159279
parameter = 3.48 : xnew after 987 generations = 0.869426203
parameter = 3.48 : xnew after 988 generations = 0.395064496
parameter = 3.48 : xnew after 989 generations = 0.831680119
parameter = 3.48 : xnew after 990 generations = 0.487159279
parameter = 3.48 : xnew after 991 generations = 0.869426203
parameter = 3.48 : xnew after 992 generations = 0.395064496
parameter = 3.48 : xnew after 993 generations = 0.831680119
parameter = 3.48 : xnew after 994 generations = 0.487159279
parameter = 3.48 : xnew after 995 generations = 0.869426203
parameter = 3.48 : xnew after 996 generations = 0.395064496
parameter = 3.48 : xnew after 997 generations = 0.831680119
parameter = 3.48 : xnew after 998 generations = 0.487159279
parameter = 3.48 : xnew after 999 generations = 0.869426203
parameter = 3.48 : xnew after 1000 generations = 0.395064496
---
parameter = 3.55 : xnew after 985 generations = 0.887370897
parameter = 3.55 : xnew after 986 generations = 0.354800449
parameter = 3.55 : xnew after 987 generations = 0.81265567
parameter = 3.55 : xnew after 988 generations = 0.540474833
parameter = 3.55 : xnew after 989 generations = 0.881684347
parameter = 3.55 : xnew after 990 generations = 0.37032556
parameter = 3.55 : xnew after 991 generations = 0.827805116
parameter = 3.55 : xnew after 992 generations = 0.506030511
parameter = 3.55 : xnew after 993 generations = 0.887370897
parameter = 3.55 : xnew after 994 generations = 0.354800449
parameter = 3.55 : xnew after 995 generations = 0.81265567
parameter = 3.55 : xnew after 996 generations = 0.540474833
parameter = 3.55 : xnew after 997 generations = 0.881684347
parameter = 3.55 : xnew after 998 generations = 0.37032556
parameter = 3.55 : xnew after 999 generations = 0.827805116
parameter = 3.55 : xnew after 1000 generations = 0.506030511
---
parameter = 3.62 : xnew after 985 generations = 0.778562727
parameter = 3.62 : xnew after 986 generations = 0.62409816
parameter = 3.62 : xnew after 987 generations = 0.84925072
parameter = 3.62 : xnew after 988 generations = 0.463446642
parameter = 3.62 : xnew after 989 generations = 0.900163144
parameter = 3.62 : xnew after 990 generations = 0.325327437
parameter = 3.62 : xnew after 991 generations = 0.794551975
parameter = 3.62 : xnew after 992 generations = 0.590925666
parameter = 3.62 : xnew after 993 generations = 0.875071734
parameter = 3.62 : xnew after 994 generations = 0.395742723
parameter = 3.62 : xnew after 995 generations = 0.865652121
parameter = 3.62 : xnew after 996 generations = 0.421000665
parameter = 3.62 : xnew after 997 generations = 0.88240796
parameter = 3.62 : xnew after 998 generations = 0.375626231
parameter = 3.62 : xnew after 999 generations = 0.849002819
parameter = 3.62 : xnew after 1000 generations = 0.464073256
---
parameter = 3.69 : xnew after 985 generations = 0.701885704
parameter = 3.69 : xnew after 986 generations = 0.772103579
parameter = 3.69 : xnew after 987 generations = 0.649291079
parameter = 3.69 : xnew after 988 generations = 0.840257921
parameter = 3.69 : xnew after 989 generations = 0.49528858
parameter = 3.69 : xnew after 990 generations = 0.922418091
parameter = 3.69 : xnew after 991 generations = 0.264067308
parameter = 3.69 : xnew after 992 generations = 0.717098973
parameter = 3.69 : xnew after 993 generations = 0.748583052
parameter = 3.69 : xnew after 994 generations = 0.69448186
parameter = 3.69 : xnew after 995 generations = 0.782932415
parameter = 3.69 : xnew after 996 generations = 0.627112728
parameter = 3.69 : xnew after 997 generations = 0.862878288
parameter = 3.69 : xnew after 998 generations = 0.436598395
parameter = 3.69 : xnew after 999 generations = 0.907667073
parameter = 3.69 : xnew after 1000 generations = 0.309249887
---
parameter = 3.76 : xnew after 985 generations = 0.928952379
parameter = 3.76 : xnew after 986 generations = 0.248159461
parameter = 3.76 : xnew after 987 generations = 0.701527049
parameter = 3.76 : xnew after 988 generations = 0.78729455
parameter = 3.76 : xnew after 989 generations = 0.629656524
parameter = 3.76 : xnew after 990 generations = 0.876791339
parameter = 3.76 : xnew after 991 generations = 0.40618636
parameter = 3.76 : xnew after 992 generations = 0.906908243
parameter = 3.76 : xnew after 993 generations = 0.317440563
parameter = 3.76 : xnew after 994 generations = 0.814686915
parameter = 3.76 : xnew after 995 generations = 0.567655266
parameter = 3.76 : xnew after 996 generations = 0.922789596
parameter = 3.76 : xnew after 997 generations = 0.26789608
parameter = 3.76 : xnew after 998 generations = 0.737440415
parameter = 3.76 : xnew after 999 generations = 0.728018905
parameter = 3.76 : xnew after 1000 generations = 0.744507745
---
parameter = 3.83 : xnew after 985 generations = 0.957416597
parameter = 3.83 : xnew after 986 generations = 0.156149317
parameter = 3.83 : xnew after 987 generations = 0.50466649
parameter = 3.83 : xnew after 988 generations = 0.957416597
parameter = 3.83 : xnew after 989 generations = 0.156149317
parameter = 3.83 : xnew after 990 generations = 0.50466649
parameter = 3.83 : xnew after 991 generations = 0.957416597
parameter = 3.83 : xnew after 992 generations = 0.156149317
parameter = 3.83 : xnew after 993 generations = 0.50466649
parameter = 3.83 : xnew after 994 generations = 0.957416597
parameter = 3.83 : xnew after 995 generations = 0.156149317
parameter = 3.83 : xnew after 996 generations = 0.50466649
parameter = 3.83 : xnew after 997 generations = 0.957416597
parameter = 3.83 : xnew after 998 generations = 0.156149317
parameter = 3.83 : xnew after 999 generations = 0.50466649
parameter = 3.83 : xnew after 1000 generations = 0.957416597
---
parameter = 3.9 : xnew after 985 generations = 0.270586601
parameter = 3.9 : xnew after 986 generations = 0.76974102
parameter = 3.9 : xnew after 987 generations = 0.69123515
parameter = 3.9 : xnew after 988 generations = 0.832373558
parameter = 3.9 : xnew after 989 generations = 0.54415849
parameter = 3.9 : xnew after 990 generations = 0.967395108
parameter = 3.9 : xnew after 991 generations = 0.12301307
parameter = 3.9 : xnew after 992 generations = 0.420735333
parameter = 3.9 : xnew after 993 generations = 0.950496739
parameter = 3.9 : xnew after 994 generations = 0.183505484
parameter = 3.9 : xnew after 995 generations = 0.584341762
parameter = 3.9 : xnew after 996 generations = 0.947257222
parameter = 3.9 : xnew after 997 generations = 0.194847813
parameter = 3.9 : xnew after 998 generations = 0.611840356
parameter = 3.9 : xnew after 999 generations = 0.926217765
parameter = 3.9 : xnew after 1000 generations = 0.266519824
---
parameter = 3.97 : xnew after 985 generations = 0.937922881
parameter = 3.97 : xnew after 986 generations = 0.231147494
parameter = 3.97 : xnew after 987 generations = 0.70554177
parameter = 3.97 : xnew after 988 generations = 0.824777746
parameter = 3.97 : xnew after 989 generations = 0.573742081
parameter = 3.97 : xnew after 990 generations = 0.970911559
parameter = 3.97 : xnew after 991 generations = 0.112121945
parameter = 3.97 : xnew after 992 generations = 0.395215941
parameter = 3.97 : xnew after 993 generations = 0.948910595
parameter = 3.97 : xnew after 994 generations = 0.192462734
parameter = 3.97 : xnew after 995 generations = 0.617020694
parameter = 3.97 : xnew after 996 generations = 0.938135444
parameter = 3.97 : xnew after 997 generations = 0.230408212
parameter = 3.97 : xnew after 998 generations = 0.703961463
parameter = 3.97 : xnew after 999 generations = 0.827346895
parameter = 3.97 : xnew after 1000 generations = 0.567090721
---
---
Internet: see also:
---
Tony Tooth's Mathematical Diversions (inverted pendulum, 3 planets in
motion)
http://www.btinternet.com/~tony.tooth/index.html
---
Math: Chaos theory: Overview: Can you give me an overview of chaos
theory?
http://www.faqts.com/knowledge_base/view.phtml/aid/26929/fid/867
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