Introduction

Theory of chaos applied in various fields of research, such as economy [1][2][3], biology [4], mathematics [5][6][7][8][9][10][11][12], optics [13], robotics [14], memristors [15][16][17][18][19][20], electronics schemes [21][22][23][24][25][26][27][28][29][30][31][32][33][34], etc.

The logistic map is a one-dimensional discrete-time map that, despite its formal simplicity, exhibits an unexpected degree of complexity. Historically it has been one of the most important and paradigmatic systems during the early days of research on deterministic chaos. The map was popularized in a 1976 paper by the biologist Robert May [35] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

The logistic map is defined by the following equation:

x n+1 = rx n (1-x n ) (1)

The logistic map is a very simple system, which can produce chaotic behavior with the right values of the parameter r. Sensible values of r range from 0 to 4; also, the values of x range from 0 to 1.

Fig. 1 shows plot the behavior of the orbits for r = 2.6, r = 3.3, and r = 3.9. The software implementation of logistic map using LabView software environment shows in Fig. 2.

ARDUINO board

The top board is Arduino Pro Mini, the bottom -USB-UART / USB-TTL converter and some connectors (Fig. 5).

Fig. 5. Set of the Arduino board, converter and some connectors

Now about the Arduino Pro Mini board / platform. Structurally, it is a board with a microcontroller soldered on it, a RESET button, a power chip, and other peripherals that are not essential for us at this stage.

There are two versions of the Pro Mini: one running at 3.3V at 8 MHz, the other at 5V at 16 MHz. The board is based on the Atmega 168 or Atmega 328 microcontroller -the difference between them is the volume of the system-programmable Flash memory -16 or 32 kB. This is the so-called "program memory", i.e. the memory in which the program will be recorded and its contents will not change during operation. Atmega is built on the so-called Harvard architecture in which the "program memory" and "data memory" are implemented separately, for greater speed and reliability. The "data memory" is divided into 2 parts: operational SRAM, which for 168 and 328 is 1 KB, and a permanent EEPROM (Electrically Erasable Programmable Read-Only Memory) of 512 bytes, data from which is not "lost" when the power is turned off.

The connection scheme is quite simple and it's rather difficult to make a mistake (Fig. 6).

Program realization

The Arduino integrated development environment is an environment in which an Arduino board can be programmed.

A written program or code is called SKETCH. In this work, the Arduino IDE is used as an environment in which the Arduino Pro Mini program is written, compiled and uploaded on the Arduino board as depicted in Fig. 7.

The Arduino can be connected to a computer through the USB port and programmed using a language similar to C++.

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 [0;1].

As example, the first LED from the left represents the first interval segment [0;0.1), the second LED represents the segment [0.1; 0.2), and so on. The last LED represents the final segment [0.9; 1.0]. 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 ms. This process is repeated infinitely, so we can observe the orbit followed by the logistic map by watching the blinking LEDs sequence.

The initial conditions for the logistic map can be changed and uploaded again into the microcontroller in any time. Thus, we can explore the behavior of the chaotic map when different values of the parameters r and X0 are chosen.

Fig. 7. Arduino IDE

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 ten light-emitting diodes 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.

The software to connect the LEDs using the following code:

// Logistic Map // Select of the pin for each light-emitting diode (LED) const int NbrLEDs = 10; const int LEDpin[] = {4, 5,

() 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]); // Logistic Map function with iterates X0 = X; X = r * X0 * (1.0 -X0); } // Function for blinkLED // 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 }

Experimental realization

Fig. 8 shows connection scheme for experimental demonstration of chaotic behavior that realized one-dimensional system.

We connected an array of ten light-emitting diodes (HL1-HL10) to the digital pins (D4-D13) of the microcontroller using 220 Ohm resistors (R1-R10).

Conclusions

The Arduino Pro Mini, ten light-emitting diodes (LEDs) and ten resistors for each part of segment of the range [0;1] was used for practical realization and visualizing of the logistic map. It is one-dimensional system that demonstrate chaotic behavior.

Equation, system conditions and analysis of the iteration series with different parameter r are shown. For this analysis was used modern software environment Lab-View. Connection scheme, nominal components and programming code are also presented.

The Arduino was connected to a computer through the USB port and programmed using a language similar to C++. Sketch was uploaded into Arduino using program software ArduinoIDE.