In this paper, we used one-dimensional system that demonstrate chaotic behavior. Equation, analysis and system conditions are presented. Modern software environment LabView was used for analysis of the iteration series with different parameter r. Connection scheme, nominal components and programming code are also presented. For practical realization and visualizing of the logistic map we used Arduino Pro Mini and ten light-emitting diodes (LEDs) with ten resistors for each part of segment of the range [0;1]. 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.
Theory of chaos applied in various fields of research, such as economy [
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: xn+1 = rxn(1-xn) (1)
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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.
The top board is Arduino Pro Mini, the bottom – USB-UART / USB-TTL converter and some connectors (Fig. 5).
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).
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, 6, 7, 8, 9, 10, 11, 12, 13}; const int wait = 500; // Wait for 500 milliseconds // Parameters of Logistic Map const double r = 3.8; // Logistic map constant double X0 = 0.3; // Initial position double X = X0; // X0 - first calculated point // For initializes the LED pins use setup () void setup() { for (int i = 0; i < NbrLEDs; i++) {
pinMode(LEDpin[i], OUTPUT); }
} // For turn on/off LEDs and iterates the Logistic Map use loop () void loop () { if (X < 0.1)
blinkLED(LEDpin [0]) ; else if ((X >= 0.1 ) && (X < 0.2))
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
blinkLED(LEDpin[
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
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