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Paper text:
nonlinear dynamics is a topic that not only cover all the disciplines , in both natural and social sciences , but also is now becoming part of introductory level undergraduate courses in sciences . searching for affordable , easy to setup , and reconfigurable classroom demonstrations that allow to investigate the physical and mathematical nature of nonlinear dynamical systems has been always a matter of interest for instructors .
this paper describes a simple apparatus used to explore the behavior of one - dimensional chaotic maps , like the logistic map , when different initial conditions are chosen .
the device consists of an inexpensive open - source microcontroller connected to an array of light emitting diodes ( leds ) , and programmed to iterate the logistic map in the one - dimensional interval @xmath0 $ ] .
this interval is divided in ten equal parts and mapped one - to - one to a single led in the array .
when the logistic map produces certain value after an iteration , the corresponding led lights up showing the value in the one dimensional domain approximately .
after several iterations , it is possible to _ visualize _ the trajectory of the map by looking at the blinking leds sequence .
sensitivity to initial conditions , density of periodic orbits , strange attractors , and bifurcations are visualized easily with this device .
the logistic map is , perhaps , the simplest example of how a nonlinear dynamical equation can give rise to very complex , chaotic behavior .
@xcite initially introduced as a mathematical model of population growth ,  @xcite it rapidly found applications in diverse areas like mathematical biology , biometry , demography , condense matter , econophysics , and computation .
@xcite the logistic map function is defined as @xmath1 where the factor @xmath2 is a model - dependent parameter representing _ external _ conditions to the system , and @xmath3 is the population in the @xmath4th - period cycle , scaled so that its value fits in the interval @xmath0 $ ] .
@xcite the function @xmath5 is called an _ iteration function _ because we may find the population fraction @xmath6 in the following period cycle by repeating the mathematical operations expressed in .
@xcite it is also one - dimensional since there is only a single variable @xmath6 , and the resulting curve is a line .
we show a photograph of the apparatus setup in fig .
[ fig01 ] , and its schematics in fig .
[ fig02 ] .    setup of the logistic map device .
an array of ten leds are connected to the arduino microcontroller using the same number of @xmath7 resistors . ]
circuit diagram of the ten leds array connected to the arduino . ]
the apparatus uses an open - source arduino prototyping platform made up of an atmel avr processor ( microcontroller ) .
it has 14 digital input / output pins , 6 analog inputs , a 16 mhz crystal oscillator , a usb connection , a power jack , an icsp header , and a reset button .
@xcite arduino is a registered trademark  only the official boards are named `` arduino''so clones usually have names ending with `` duino . ''
@xcite the arduino can be connected to a computer through the usb port and programmed using a language similar to c++ .
the program is uploaded into the microcontroller using an integrated development environment ( ide ) . in the arduino world , programs are known as _
we connected an array of ten @xmath8 leds to the digital pins of the microcontroller using @xmath7 resistors ( or @xmath9 for dimmer leds ) .
we also used a solderless breadboard to hook up all the electric components to the microcontroller .
listing  [ list01 ] shows a simple program ( sketch ) used to operate the microcontroller , and _ visualize _ the logistic map by looking at the array of blinking leds .
a glowing led represents an iteration of the one - dimensional map , and it is linked with a value in the interval @xmath0 $ ] .
arduino programs require two mandatory functions : ` setup ( ) ` and ` loop ( ) ` .
any variable or constant defined outside these two functions is considered to be _
global_. in the ` setup ( ) ` function , we tell the microcontroller that there are 10 leds connected to the digital pins and that they are intended to be turned on and off . in the ` loop ( ) ` function , the logistic map is iterated , and the visualization process takes place as we observe the leds turning on and off , one after another , following the evolution of the nonlinear system .    using the ` if ( ) ` and ` else ` control structures , we divide the interval @xmath0 $ ] in ten identical segments and associate an led to each one of them .
for example , the first led from the left represents the first interval segment @xmath10 , the second led represents the segment @xmath11 , and so on .
the last led represents the final segment @xmath12 $ ] .
when the microcontroller iterates the logistic map , a value belonging to one of these ten intervals is returned , and the corresponding led turns on for 500 milliseconds .
this process is repeated infinitely , so we can observe the _ orbit _ followed by the logistic map by watching at the blinking leds sequence .
cobweb diagrams for the logistic map for different values of @xmath2 and @xmath13 .
the @xmath14 line , @xmath15 , is used to find the fixed points and follow the iteration process .
plots ( a ) and ( b ) show the presence of attractors at @xmath16 and @xmath17 , respectively .
plot ( c ) displays a period doubling bifurcation , and ( d ) the onset of chaos . ]
the initial conditions for the logistic map can be changed anytime and uploaded again into the microcontroller .
thus , we can explore the behavior of the chaotic map when different values of the parameters @xmath2 and @xmath13 are chosen .
it is amusing to study the bifurcations of the logistic map by looking at the blinking sequence of leds .
we start with the parameter values @xmath18 and @xmath19 ( @xmath20 ) .
the led associated with the interval @xmath21 turns on first .
as the iterations of the logistic map take place , we observe that the sequence of blinking leds moves toward the very first led , which is associated with the segment @xmath10 .
the sequence ends up with this first led left on indefinitely .
this evolution is shown in fig .
[ fig03](a ) , the cobweb diagram of the logistic map for that selection of parameters . for this case
, @xmath16 is an _ attractor _ , and the domain @xmath0 $ ] forms a _ basin of attraction_.    another example of an attractor happens when @xmath22 and @xmath23 ( @xmath24 ) .
we see that , after few iterations and blinking leds , there is only one led that remains on indefinitely .
this led corresponds to the interval @xmath25 . in this case
, we have a _ fixed point
_ occurring at @xmath17 . as it is shown in fig .
[ fig03](b ) , the orbit of the map follows a square spiral that converges into that fixed point .
now , if @xmath26 and @xmath23 ( @xmath27 ) , we can observe a 2-_period cycle _ ; the value of @xmath6 oscillates between 0.558 and 0.765 , and the two corresponding leds turn on and off intermittently . here , the fixed point @xmath28 becomes a _
repeller_. figure  [ fig03](c ) shows the periodic behavior of the logistic map for this set of parameters .
unfortunately , higher periodicity ( like 4-period , 8-period cycles , or higher ) can not be observed clearly with this device because of the way the leds are mapped into the @xmath0 $ ] interval .
more leds are required to improve the `` resolution '' of the device
. however , the microcontroller has a limit of 13 digital pins , and other electric components ( like an extra microchip ) need to be incorporated into the circuit if we want more leds controlled by the device .    finally , when @xmath29 and @xmath30 ( @xmath31 ) , chaos onsets.@xcite an `` apparently '' randomness of blinking leds is observed .
after many iterations , there is no single led that remains on indefinitely , and no single led that has not been turned on at least once .
the orbit of the logistic map covers all the domain of the interval @xmath0 $ ] .
figure  [ fig03](d ) shows all points visited by the logistic map orbit after @xmath32 iterations .    ....
// blinking logistic map      //
choose the pin for each led    const int nbrleds   = 10 ;    const int ledpin [ ] = { 2,3,4,5,6,7,8,9,10,11 } ;      const int wait = 500 ;   //
wait for 500 milliseconds      //
logistic map parameters    const double a   = 3.7 ;   // logistic map constant      double x0 = 0.2 ;
// initial position ( 0 < x0 < 1.0 )    double x   = x0 ;    // use x0 as your first calculated point      // setup ( ) initializes the led pins    void setup ( ) {        for ( int i = 0 ; i < nbrleds ; i++ ) {            pinmode(ledpin[i ] , output ) ;        }    }      // loop ( ) iterates the logistic map and turn on / off leds    void loop ( )    {        if ( x < 0.1 )            blinkled(ledpin[0 ] ) ;        else if ( ( x > = 0.1 ) & & ( x < 0.2 ) )            blinkled(ledpin[1 ] ) ;        else if ( ( x > = 0.2 ) & & ( x < 0.3 ) )            blinkled(ledpin[2 ] ) ;        else if ( ( x > = 0.3 ) & & ( x < 0.4 ) )            blinkled(ledpin[3 ] ) ;        else if ( ( x > = 0.4 ) & & ( x < 0.5 ) )            blinkled(ledpin[4 ] ) ;        else if ( ( x > = 0.5 ) & & ( x < 0.6 ) )            blinkled(ledpin[5 ] ) ;        else if ( ( x > = 0.6 ) & & ( x < 0.7 ) )            blinkled(ledpin[6 ] ) ;        else if ( ( x > = 0.7 ) & & ( x < 0.8 ) )            blinkled(ledpin[7 ] ) ;        else if ( ( x > = 0.8 ) & & ( x < 0.9 ) )            blinkled(ledpin[8 ] ) ;        else            blinkled(ledpin[9 ] ) ;          // iterates the logistic map function        x0 = x ;        x   = a * x0 * ( 1.0 - x0 ) ;    }      // blinkled function    // turn on / off leds    void blinkled(const int pin ) {        digitalwrite(pin , high ) ;   //
turn led on        delay(wait ) ;               // wait 500 milliseconds        digitalwrite(pin , low ) ;    //
turn led off    } ....
with this apparatus , students may understand more easily the behavior of one - dimensional chaotic maps looking at an _ actual _ one - dimensional array of leds ; every point in the interval [ 0,1 ] produces another point within the same interval after the logistic map is iterated . by controlling the parameter @xmath2 and the initial value @xmath13
, the instructor and students can observe under what conditions periodic and chaotic behavior may occur , providing a better understanding of periodicity , bifurcations , and the route to chaos of a nonlinear dynamical system .
the setup described here is inexpensive , easy , and fun to assemble , also enhances computer programming and electronics assembly skills .
the logistic map is one of the simplest nonlinear dynamical systems that clearly exhibit the route to chaos . in this paper 
 , we explored the evolution of the logistic map using an open - source microcontroller connected to an array of light emitting diodes ( leds ) . 
 we divided the one - dimensional interval @xmath0 $ ] into ten equal parts , and associated and led to each segment . every time an iteration took place a corresponding led turned on indicating the value returned by the logistic map . by changing some initial conditions of the system 
 , we observed the transition from order to chaos exhibited by the map .