global x,r,y,zinc,lastzinc,counter,drawer,z,soundit,marker
put "0" into iterater
if marker = empty then put "0" into marker
set the hilite of me to not the hilite of me
if the hilite of me = false then exit mouseup
if r = empty then
answer "Please choose a Verhulst number first"
repeat 3
set lockscreen to true
repeat with i = 2 to 8
set hilite of cd btn i to true
end repeat
set lockscreen to false
wait 3
set lockscreen to true
repeat with i = 2 to 8
set hilite of cd btn i to false
end repeat
set lockscreen to false
wait 3
end repeat
end if
put r into cd fld rate
put "0" into cd fld iteration
set linesize to 1
show cd fld stop
repeat until the mouse is down
add 1 to cd fld iteration
set numberformat to ".########"
put r * x* (1-x) into x
put x into cd fld population
put trunc(180-x*100) into y -- moves the hand
get the loc of cd btn plot
put y into item 2 of it
set the loc of cd btn plot to it
put it into zinc
if drawer = 1 then drawit
if soundit = 1 then play harpsichord tempo 500 60 +trunc (y/20)
end repeat
put empty into counter
put empty into zinc
put empty into lastzinc
set the loc of cd btn plot to 226,128
put "0" into marker
choose browse tool
set the hilite of me to false
hide cd fld stop
end mouseup
on drawit
global zinc,lastzinc,counter,marker,drawer
if item 1 of zinc = "226" then add 75 to item 1 of zinc
if lastzinc = empty then put zinc into lastzinc
if marker = 41 then
put "0" into marker
put 0 into counter
put "301" into item 1 of lastzinc
put "301" into item 1 of zinc
choose "select tool"
domenu "select all"
domenu "cut picture"
choose "line tool"
end if
choose "line tool"
add 5 to counter
add counter to item 1 of zinc
drag from lastzinc to zinc
put zinc into lastzinc
add 1 to marker
end drawit
</script>
</part>
<part>
<id>2</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>120</left>
<top>320</top>
<right>167</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>2.7</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>3</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>166</left>
<top>320</top>
<right>213</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>2.9</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>4</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>212</left>
<top>320</top>
<right>259</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>2.96</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>5</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>258</left>
<top>320</top>
<right>305</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>3.4</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>6</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>304</left>
<top>320</top>
<right>351</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>3.55</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>7</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>350</left>
<top>320</top>
<right>397</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>3.99</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>8</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>396</left>
<top>320</top>
<right>443</right>
<bottom>342</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>4.0</name>
<script>on mouseup
global r,x
put the short name of the target into r
put .01 into x
put r into msg
end mouseup
</script>
</part>
<part>
<id>9</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>207</left>
<top>118</top>
<right>245</right>
<bottom>139</bottom>
</rect>
<style>transparent</style>
<showName> <false /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <false /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>19162</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>plot</name>
<script></script>
</part>
<part>
<id>10</id>
<type>field</type>
<visible> <true /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <false /> </lockText>
<rect>
<left>198</left>
<top>181</top>
<right>292</right>
<bottom>197</bottom>
</rect>
<style>rectangle</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>right</textAlign>
<font>Geneva</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<textHeight>16</textHeight>
<name>population</name>
<script></script>
</part>
<part>
<id>11</id>
<type>field</type>
<visible> <true /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <false /> </lockText>
<rect>
<left>120</left>
<top>270</top>
<right>442</right>
<bottom>320</bottom>
</rect>
<style>rectangle</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Geneva</font>
<textSize>12</textSize>
<textStyle>bold</textStyle>
<textHeight>16</textHeight>
<name></name>
<script></script>
</part>
<part>
<id>12</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>302</left>
<top>199</top>
<right>407</right>
<bottom>217</bottom>
</rect>
<style>radiobutton</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <false /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>Show Graph</name>
<script>on mouseUp
global drawer
set the hilite of me to not the hilite of me
if the hilite of me = true then
put "1" into drawer
else
put "0" into drawer
end if
end mouseUp
</script>
</part>
<part>
<id>13</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>382</left>
<top>220</top>
<right>433</right>
<bottom>240</bottom>
</rect>
<style>shadow</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>Erase</name>
<script>on mouseUp
global r,marker
if r = empty then exit mouseup
choose "select tool"
domenu "select all"
domenu "cut picture"
choose "browse tool"
put "0" into marker
end mouseUp
</script>
</part>
<part>
<id>14</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>0</left>
<top>0</top>
<right>169</right>
<bottom>51</bottom>
</rect>
<style>transparent</style>
<showName> <false /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>ABOUT</name>
<script>on mouseUp
if the visible of cd fld "about" = true then
hide cd fld "about"
show cd btn "ACCCK!"
else
hide cd btn "ACCCK!"
Show cd fld "about"
end if
end mouseUp
</script>
</part>
<part>
<id>15</id>
<type>field</type>
<visible> <true /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <true /> </lockText>
<rect>
<left>0</left>
<top>55</top>
<right>169</right>
<bottom>258</bottom>
</rect>
<style>scrolling</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>left</textAlign>
<font>Geneva</font>
<textSize>10</textSize>
<textStyle>plain</textStyle>
<textHeight>13</textHeight>
<name>about</name>
<script>on mouseup
hide me
end mouseup
</script>
</part>
<part>
<id>16</id>
<type>field</type>
<visible> <true /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <true /> </lockText>
<rect>
<left>351</left>
<top>244</top>
<right>380</right>
<bottom>263</bottom>
</rect>
<style>rectangle</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>left</textAlign>
<font>Geneva</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<textHeight>16</textHeight>
<name>userrate</name>
<script>on mousedown
repeat until the mouse is up
if cd fld userrate ‚â• 4 then
put "0" into cd fld userrate
end if
add .1 to cd fld userrate
end repeat
end mousedown
</script>
</part>
<part>
<id>17</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>198</left>
<top>244</top>
<right>352</right>
<bottom>263</bottom>
</rect>
<style>rectangle</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <true /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>Use MY growth rate >></name>
<script>on mouseUp
global r, x
put cd fld userrate into r
put .01 into x
put r into msg
end mouseUp
</script>
</part>
<part>
<id>18</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>405</left>
<top>199</top>
<right>512</right>
<bottom>217</bottom>
</rect>
<style>radiobutton</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <false /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>Sound On</name>
<script>on mouseUp
global soundit
set the hilite of me to not the hilite of me
if the hilite of me = true then
put "1" into soundit
else
put "0" into soundit
end if
end mouseUp
</script>
</part>
<part>
<id>19</id>
<type>button</type>
<visible> <false /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>19</left>
<top>231</top>
<right>153</right>
<bottom>253</bottom>
</rect>
<style>rectangle</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <false /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>ACCCK!</name>
<script>on mouseUp
hide me
show cd fld about
end mouseUp
</script>
</part>
<part>
<id>20</id>
<type>button</type>
<visible> <true /> </visible>
<reserved5> 0 </reserved5>
<reserved4> 0 </reserved4>
<reserved3> 0 </reserved3>
<reserved2> 0 </reserved2>
<reserved1> 0 </reserved1>
<enabled> <true /> </enabled>
<rect>
<left>455</left>
<top>321</top>
<right>512</right>
<bottom>342</bottom>
</rect>
<style>rectangle</style>
<showName> <true /> </showName>
<highlight> <false /> </highlight>
<autoHighlight> <false /> </autoHighlight>
<sharedHighlight> <true /> </sharedHighlight>
<family>0</family>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Chicago</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<name>Quit</name>
<script>on mouseUp
domenu "quit hypercard"
end mouseUp
</script>
</part>
<part>
<id>21</id>
<type>field</type>
<visible> <true /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <true /> </lockText>
<rect>
<left>361</left>
<top>181</top>
<right>395</right>
<bottom>197</bottom>
</rect>
<style>rectangle</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>right</textAlign>
<font>Geneva</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<textHeight>16</textHeight>
<name>iteration</name>
<script></script>
</part>
<part>
<id>22</id>
<type>field</type>
<visible> <true /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <true /> </lockText>
<rect>
<left>447</left>
<top>181</top>
<right>481</right>
<bottom>197</bottom>
</rect>
<style>rectangle</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>right</textAlign>
<font>Geneva</font>
<textSize>12</textSize>
<textStyle>plain</textStyle>
<textHeight>16</textHeight>
<name>rate</name>
<script></script>
</part>
<part>
<id>23</id>
<type>field</type>
<visible> <false /> </visible>
<dontWrap> <false /> </dontWrap>
<dontSearch> <false /> </dontSearch>
<sharedText> <false /> </sharedText>
<fixedLineHeight> <true /> </fixedLineHeight>
<autoTab> <false /> </autoTab>
<lockText> <true /> </lockText>
<rect>
<left>120</left>
<top>270</top>
<right>441</right>
<bottom>320</bottom>
</rect>
<style>rectangle</style>
<autoSelect> <false /> </autoSelect>
<showLines> <false /> </showLines>
<wideMargins> <false /> </wideMargins>
<multipleLines> <false /> </multipleLines>
<reservedFamily> 0 </reservedFamily>
<titleWidth>0</titleWidth>
<icon>0</icon>
<textAlign>center</textAlign>
<font>Geneva</font>
<textSize>24</textSize>
<textStyle>bold</textStyle>
<textHeight>32</textHeight>
<name>stop</name>
<script>on mouseup
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<text>Verhulst's numbers.
Select a growth rate number below and then click the Chaos button.</text>
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<text>ABOUT THIS STACK
Let 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 growth
x = "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 STACK
1) 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 EKG
A 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 3.4
At 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 3.55
At 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 Atanasio
WildCard Software
2233 N. Laurel Ave.
Phoenix, AZ 85007
WCTV BBS
(602) 252-7928
ABOUT WILDCARD SOFTWARE
Other WildCard Software available:
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$50
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Under Development:
WildCard Software has several modem to modem games under development.
Business simulation modeling for production or process oriented businesses.</text>
