Fortran90 Examples of Dynamic Systems

Table of contents

1. Introduction
2. Usage
3. Examples
   1. Oscillator with non-linear forcing components
   2. Logistic map
   3. Henon map
   4. Lorenz model
   5. Multifractal measure
4. Online resources
5. carepackage.f90
6. System requirements

Introduction

Fortran90 examples of simple dynamic systems, for an introductory university class. Aside from the code, all examples include a:

1. Makefile for ease of cleaning/compilation
2. *_plotter.plt (sub the * with the program name) Gnuplot script for the plots required by the class assignments
3. *_runner.sh (see above for the *) bash helper script to create data folders and run both the compiled example and gnuplot

In addition, examples 3 and 4 include a parameters.txt file from which the *_runner.sh script takes the needed input values. To use these examples stand-alone, just uncomment the values i left in the code and comment all the get_command_argument related lines.

Feel free to alter the code to fit your needs, just don't keep it to yourself, fortran90 is already hard enough to use and/or get accustomed to, people need all the help they can get! I sure did.

Usage

All examples come already compiled and with data generated.
In order to run them from scratch:

1. Open your terminal of choice and navigate to the example folder
2. Either use the make clean comand to remove the compiled example and all data and plots, or do so manually
3. Either use the make comand to compile, or do so with preferred/required method of choice
4. Type ./*_runner.sh to run the helper script (you may need to run chmod +x *_runner.sh to make it executable) in order to execute both the example and Gnuplot, or comment the gnuplot line if other plotting/fitting methods are preferred/required.

Important: should you choose not to use the included Makefile and/or *_runner.sh script, make sure to create the needed data folders and to feed the examples the right input values (like for examples 3 and 4).

Examples

Oscillator with non-linear forcing components

$$\ddot{x} + 2\lambda\epsilon\dot{x} + \omega_0^2x +\epsilon gx^3 = \epsilon f \cos [(\omega_0 + \epsilon \sigma) \cdot t]$$

1. Integration using Runge-Kutta 4th order algorithm in two ways:
   • Increasing values of $\sigma$. Starting from the second $\sigma$ value onward, for each evolution the starting conditions are the last values of the previous one.
   • Decreasing values of $\sigma$. Same procedure as above.
2. Study of maxima behavior in function of $\sigma$ once the forcing effects have subsided for the two kinds of evolution above mentioned
3. Comparison between the maxima behavior in function of $\sigma$ for the increasing and decreasing $\sigma$ evolutions

Logistic map

$$x(n+1) = \mu x(n)\cdot [1-x(n)]$$

1. Integration using Euler method
2. Calculation of all possible Lyapunov coefficients for different $\mu$ values

Henon map

$$\begin{align} x(n+1) &= y(n) + 1 - 1.4\cdot x^2(n)\\ y(n+1) &= 0.3 \cdot x(n) \end{align}$$

1. Integration using Euler method
2. Lyapunov coefficient calculation with Benettin algorithm, for single and double trajectory
3. Fractal dimension calculation with Grassberger-Procaccia algorithm and comparison with Kaplan-Yorke conjecture

Lorenz model

$$\begin{align} \dot{x} &= \sigma\cdot (y - x)\\ \dot{y} &= rx - y - xz\\ \dot{z} &= xy - bz \end{align}$$

1. Integration with Runge-Kutta 4th order
2. Lyapunov coefficient calculation with Benettin algorithm, for single and triple trajectory
3. Fractal dimension calculation with Grassberger-Procaccia algorithm and comparison with Kaplan-Yorke conjecture

Multifractal measure

Generation and study of a multifractal measure with the multiplicative process

Online resources

1. Runge-Kutta algorithm (Wikipedia)
2. Euler method (Wikipedia)
3. Lyapunov exponent (Wikipedia)
4. Fractal dimension (Wikipedia)
5. Benettin algorithm (CERN)
6. Kaplan-Yorke conjecture (Wikipedia)
7. Grassberger-Procaccia algorithm (Scholarpedia)
8. Logistic map (Wikipedia)
9. Henon map (Wikipedia)
10. Lorenz model (Wikipedia)

carepackage.f90

The file carepackage.f90 is a homemade compilation of useful subroutines not present by default in fortran90. Right now it contains

1. Ports of arange and linspace from python
2. A subroutine for linear fitting (without error calculation) with efficiency comparable with Gnuplot's fitting function, developed with antonio-evangelista
3. A subroutine for searching the max value of a 1D vector, can be easily manipulated to find the minimum

System requirements

All Work was done on Ubuntu 20.04.5 LTS and Arch Linux using the following tools, available for all linux platforms (to my knowledge). Gnuplot and Make are optional, depending on user preference.

1. gfortran (gcc fortran compiler)

   • for Ubuntu/Debian based systems

     sudo apt install gfortran  
     

   • For Arch based systems

     sudo pacman -S gfortran
     

2. Gnuplot (powerful plotting software, also useful for data fitting)

   • for Ubuntu/Debian based systems

     sudo apt install gnuplot  
     

   • For Arch based systems

     sudo pacman -S gnuplot
     

3. Make (build system, usually already present on system)

   • for Ubuntu/Debian based systems

     sudo apt install make  
     

   • For Arch based systems

     sudo pacman -S make