<script>on mouseupglobal x,r,y,zinc,lastzinc,counter,drawer,z,soundit,markerput "0" into iteraterif marker = empty then put "0" into markerset the hilite of me to not the hilite of meif the hilite of me = false then exit mouseupif r = empty thenanswer "Please choose a Verhulst number first"repeat 3set lockscreen to truerepeat with i = 2 to 8set hilite of cd btn i to trueend repeatset lockscreen to falsewait 3set lockscreen to truerepeat with i = 2 to 8set hilite of cd btn i to falseend repeatset lockscreen to falsewait 3end repeatend ifput r into cd fld rateput "0" into cd fld iterationset linesize to 1show cd fld stoprepeat until the mouse is downadd 1 to cd fld iterationset numberformat to ".########"put r * x* (1-x) into xput x into cd fld populationput trunc(180-x*100) into y -- moves the handget the loc of cd btn plotput y into item 2 of itset the loc of cd btn plot to itput it into zincif drawer = 1 then drawitif soundit = 1 then play harpsichord tempo 500 60 +trunc (y/20)end repeatput empty into counterput empty into zincput empty into lastzincset the loc of cd btn plot to 226,128put "0" into markerchoose browse toolset the hilite of me to falsehide cd fld stopend mouseupon drawitglobal zinc,lastzinc,counter,marker,drawerif item 1 of zinc = "226" then add 75 to item 1 of zincif lastzinc = empty then put zinc into lastzincif marker = 41 thenput "0" into markerput 0 into counterput "301" into item 1 of lastzincput "301" into item 1 of zincchoose "select tool"domenu "select all"domenu "cut picture"choose "line tool"end ifchoose "line tool"add 5 to counteradd counter to item 1 of zincdrag from lastzinc to zincput zinc into lastzincadd 1 to markerend drawit</script>
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<name>Show Graph</name>
<script>on mouseUpglobal drawerset the hilite of me to not the hilite of meif the hilite of me = true thenput "1" into drawerelseput "0" into drawerend ifend mouseUp</script>
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<name>Erase</name>
<script>on mouseUpglobal r,markerif r = empty then exit mouseupchoose "select tool"domenu "select all"domenu "cut picture"choose "browse tool"put "0" into markerend mouseUp</script>
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<script>on mouseUpif the visible of cd fld "about" = true thenhide cd fld "about"show cd btn "ACCCK!"elsehide cd btn "ACCCK!"Show cd fld "about"end ifend mouseUp</script>
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<name>userrate</name>
<script>on mousedownrepeat until the mouse is upif cd fld userrate ‚â• 4 thenput "0" into cd fld userrateend ifadd .1 to cd fld userrateend repeatend mousedown</script>
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<name>Use MY growth rate >></name>
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<name>Sound On</name>
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<content>
<layer>card</layer>
<id>11</id>
<text>Verhulst's numbers.Select a growth rate number below and then click the Chaos button.</text>
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<content>
<layer>card</layer>
<id>15</id>
<text>ABOUT THIS STACKLet me be clear from the outset. I am not a mathematician nor an authority on this topic. I found it fascinating and wanted to share it.This stack is an introduction to Chaos Theory and Verhulst numbers. It borrows heavily from scripts in "The Hypertalk Bible" by the Waite Group. Hopefully they will forgive me, this stuff is too important to be buried in scientific archives and minds. Understanding just enough about this to be dangerous, I have already thought of some interesting applications for chaos theory.Simply put, Chaos Theory attempts to find order where there appears to be none! Could it be useful in predicting the Dow Jones Market indicator? Verhulst, in 1845, discovered a formula which is used here to identify stability, pattern or chaos fluctuations resulting from specified growth rates (iterations of the growth rate).The formula is:X(iteration) = rx(1-x)r = rate of growthx = "population", as a range from 0 to 1,where 1 is maximum possible obtainable and 0 is minimum possible obtainable.I believe that Verhulst's formula was directed primarily at Malthusian population theory.So what? Good question.It appears to be applicable to many other things as well. This is not to suggest that this is the only formula used in Chaos Theory. I'm confident that there must be dozens.Applying certain growth rates to the Verhulst formula has yielded a magic number. The number is ".6522" according to the "HyperTalk Bible". I found a slightly different number.The thin line in the graph represents the highest possible static population sustainable. This is the magic ".6522" optimum.The highest optimum I was able to narrow to, after several hundred iterations was between .66312 and .67017. I assume that the optimum will be .666667.HOW TO USE THIS STACK1) Choose a growth rate(either one of the rates provided or your own) If you want to use your own growth rate rather than one provided click the field next to the "Use MY growth rate" button. This will run through the range of rates available. These rates range from .1 to 4.0. Once you see a rate that you wish to use, then click the "USE MY GROWTH RATE BUTTON". This will insert that rate into the formula.You must select a growth rate before you start the CHAOS.2) Start by clicking the CHAOS button. 3) To stop, just click anywhere.If you have toggled the SHOW GRAPH button ON then it will draw the resulting iterations for you. If you have toggled the SOUND ON button, it will play notes for you that correspond the the value of the population.You can toggled each of these on or off.Speed. Since it does a lot for each iteration, if you have the graphing ON it will be slow, otherwise its pretty fast.The graphing option shows the visual effect of 41 iterations at a time. If you dont stop.. it continues on, clears the graph and picks up where it left off.The rest of the card should be self evident.WHAT HAPPENS?A STATIC growth rate, will return a population that will change several times and then settle down into a recurring population number that will not change again or very very little. Sort of like a flat EKGA STABLE PATTERN growth rate, will provide a recurring pattern of population ups and downs that bounce above and below the magic number, but in a discernable pattern.An UNSTABLE (chaos) growth rate will oscillate in a non-pattern manner.Growth rates from .1 to 1.2 or so will provide a population increase so small that it will not even register for many iterations.Growth rates from 1.3 to 2.7 or so show gradually increasing populations that plane off fairly quickly, but never reach the Optimum.The MAGIC part.It appears that a growth rate from 2.9 through 3.0 will eventually return a STATIC STABLE population.At a growth rate of 3.1 there appears to be a continuous repeating pattern, with very very small differences in the high and the low of the repeating pattern. This holds true for growth rates up to 3.4At 3.5 there is a curious break down in the repeating cycles. there is still a repeating pattern, but it now involves repeating a series of two cycles rather than 1.This hold true for 3.55At 3.6 there is furthr breakdown, the recurring pattern changes a bit each time and involves more cycles. This holds true for 3.7 as well.At 3.8 the permutation of each cycle begins to obscure the pattern.At 3.9 it appears chaos sets in. there is no longer any visually observable pattern.At 3.99 and 4.0 it looks like a complete breakdown of any structure. The cycles are wild fluctuations.There is no provision for any growth rate above 4.0 as this results almost immediately in the formula reaching for infinity.These were my visual observations, and as such hold no scientific truth.I was fascinated by this and hope you enjoy the stack and concept.Mike AtanasioWildCard Software2233 N. Laurel Ave. Phoenix, AZ 85007WCTV BBS (602) 252-7928ABOUT WILDCARD SOFTWAREOther WildCard Software available:Ten Key calculator - allows use of the keyboard to do ten key calculations (has scrolling tape) FreewareTeleFix - allows Hypercard developers and consultants to FIX or change scripts in remote stacks via modem.$50MedSystem- A comprehensive Medical Office Patient Tracking and Billing System (Call)Under Development:WildCard Software has several modem to modem games under development.Business simulation modeling for production or process oriented businesses.</text>
</content>
<content>
<layer>card</layer>
<id>16</id>
<text>.1</text>
</content>
<content>
<layer>card</layer>
<id>23</id>
<text>Click to stop</text>
</content>
<name></name>
<script>on idleput the mouselocend idleon opencardset the loc of cd btn plot to 226,128end opencardon closecardsend mouseup to cd btn "erase"put empty into cd fld populationset the hilite of cd btn "show graph" to falseset the hilite of cd btn "sound on" to falseset the hilite of cd btn "chaos" to falseput empty into cd fld iterationput empty into cd fld rateput empty into cd fld populationput ".1" into cd fld userratechoose "select tool"domenu "select all"domenu "cut picture"choose "browse tool"domenu compact stackend closecard</script>
 (January 1993).iso/Files/Hyper/C-Cl/Chaos Theory.cpt/Chaos Theory/BMAP_3499.png){kind=link}
 (January 1993).iso/Files/Hyper/C-Cl/Chaos Theory.cpt/Chaos Theory/PATs.png){kind=link}