INTRODUCTION

Dynamical systems theory is an area of mathematics used to describe the behaviour of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modelled by mixed operators, such as differential-difference equations.

This theory deals with the long-term qualitative behaviour of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

CHAOS THEORY:

Chaos theory is a branch of mathematics focusing on the study of chaos dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions . Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as : Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. It also occurs spontaneously in some systems with artificial components, such as the stock market and road traffic. This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincare maps. Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, failed verification environmental science, computer science, engineering, economics, ecology, pandemic crisis management, and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.

LORENTZ SYSTEM:

The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the ‘butterfly effect’ stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.

Package deSolve: Solving Initial Value Differential Equations in R

1. A simple ODE: chaos in the atmosphere

The Lorenz equations (Lorenz, 1963) were the first chaotic dynamic system to be described. They consist of three differential equations that were assumed to represent idealized behavior of the earth’s atmosphere. We use this model to demonstrate how to implement and solve differential equations in R. The Lorenz model describes the dynamics of three state variables, x, y and x. The model equations are:

\(\frac{dx}{dt} = \sigma(y-x)\)

\(\frac{dy}{dt} = x(\rho-z)-y\)

\(\frac{dz}{dt} = xy -\beta z\)

with the initial conditions : X(0) = Y(0) = Z(0) =1

The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: x is proportional to the rate of convection, y to the horizontal temperature variation, and z to the vertical temperature variation. The constants σ, ρ, and β are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself

Where \(\sigma\), \(\rho\) and \(\beta\) are three parameters, with values of \(\sigma = 10 , \beta = \frac{8}{3},\rho = 28\). Implementation of an IVP ODE in R can be separated in two parts: the model specification and the model application. Model specification consists of:

• Defining model parameters and their values,

• Defining model state variables and their initial conditions,

• Implementing the model equations that calculate the rate of change (e.g. \(\frac{dx}{dt}\)) of the state variables.

The model application consists of:

• Specification of the time at which model output is wanted,

• Integration of the model equations (uses R-functions from deSolve),

• Plotting of model results.

Below, we discuss the R-code for the Lorenz model.

R Code:

#install.packages(c("ggplot2", "deSolve"))
library(deSolve)
library(ggplot2)
parameters <- c(s=10,r=28,b=8/3) # sigma(s) = 10 , rho (r)= 28 , beta (b) = 8/3
state <- c(x=0,y=1,z=1)
Lorenz<-function(t,state,parameters){
with(as.list(c(state,parameters)),
{
dx <-s*(y-x)
dy <-x*(r-z)-y
dz <-x*y-b*z
list(c(dx,dy,dz))
})
}
times <- seq(0,50,by=0.01)
out <-ode(y=state,times = times ,func =Lorenz,parms = parameters )
head(out)

##      time          x        y         z
## [1,] 0.00 0.00000000 1.000000 1.0000000
## [2,] 0.01 0.09510629 1.003062 0.9741647
## [3,] 0.02 0.18265609 1.030601 0.9499302
## [4,] 0.03 0.26560283 1.080790 0.9272726
## [5,] 0.04 0.34648117 1.152549 0.9062472
## [6,] 0.05 0.42751546 1.245441 0.8869800

par(oma=c(0,0,3,0))
plot(out,xlab = "time",ylab = "-")
plot(out[,"y"],out[,"z"],pch = ".",type="l")
mtext(outer = TRUE,side = 3,"lorenz model",cex = 1.5)

1.1. MODEL SPECIFICATION

Model parameters:

There are three model parameters: s, r, and b that are defined first. Parameters are stored as a vector with assigned names and values:

parameters <- c(s=10,r=28,b=8/3)

State variables:

The three state variables are also created as a vector, and their initial values given:

state <- c(x=0,y=1,z=1)

MODEL EQUATION:

The model equations are specified in a function (Lorenz) that calculates the rate of change of the state variables. Input to the function is the model time (t, not used here, but required by the calling routine), and the values of the state variables (state) and the parameters, in that order. This function will be called by the R routine that solves the differential equations (here we use ode, see below). The code is most readable if we can address the parameters and state variables by their names. As both parameters and state variables are ‘vectors’, they are converted into a list. The statement with (as. list (c(state, parameters)), …) then makes available the names of this list. The main part of the model calculates the rate of change of the state variables. At the end of the function, these rates of change are returned, packed as a list. Note that it is necessary to return the rate of change in the same ordering as the specification of the state variables. This is very important. In this case, as state variables are specified x first, then y and z, the rates of changes are returned as dx, dy, dz.

Lorenz<-function(t,state,parameters){

with(as.list(c(state,parameters)),

{dx <-s*(y-x)

dy <-x*(r-z)-y

dz <-x*y-b*z

list(c(dx,dy,dz))})}

1.2. MODEL APPLICATION:

Time specification

We run the model for 100 days, and give output at 0.01 daily intervals. R’s function seq()

creates the time sequence:

times <- seq(0, 500, by = 0.01)

Model integration

The model is solved using deSolve function ode, which is the default integration routine. Function ode takes as input, a.o. the state variable vector (y), the times at which output is required (times), the model function that returns the rate of change (func) and the parameter vector (parms). Function ode returns an object of class deSolve with a matrix that contains the values of the state variables (columns) at the requested output times.

out <-ode(y=state,times = times ,func =Lorenz,parms = parameters )

head(out)

Plotting Results:

Finally, the model output is plotted. We use the plot method designed for objects of class deSolve, which will neatly arrange the figures in two rows and two columns; before plotting, the size of the outer upper margin (the third margin) is increased (oma), such as to allow writing a figure heading (mtext). First all model variables are plotted versus time, and finally Z versus X:

par(oma=c(0,0,3,0))

plot(out,xlab = "time",ylab = "-")

plot(out[,"Y"],out[,"Z"],pch = ".",type="l")

mtext(outer = TRUE,side = 3,"lorenz model",cex = 1.5)

BUTTERFLY EFFECT:

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.The term is closely associated with the work of mathematician and meteorologist Edward Norton Lorenz. He noted that the butterfly effect is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of butterfly and tornado by 1972.

R Code:

library(deSolve)
library(ggplot2)
parameters <- c(s = 10, b = 8/3)
r <- seq(0, 30, by = 0.05)
state <- c(X = 0, Y = 1, Z = 1)
times <- seq(0, 50, by = 0.01)

Lorenz <- function(t, state, parameters) {
  with(as.list(c(state, parameters)), {
    dX <- s * (Y - X)
    dY <- X * (r - Z) - Y
    dZ <- X * Y - b * Z
    list(c(dX, dY, dZ))
  })
}
out <- lapply(r,
              function(x){
                ode(y = state,
                    times = times,
                    func = Lorenz,
                    parms = c(r = x, parameters),
                    method = "euler")
              }
)

imageStorePath <- "C:/Users/Avhirup Moitra/Videos/LorentzR"

plots_XZ <- list()
for(i in 1:length(r)) {
  p <- ggplot(data.frame(out[[i]])) + 
    geom_path(aes(x = X, y = Z, colour = Y)) +
    ggtitle(paste0('Lorenz attractor r = ', r[i])) +
    theme(
      axis.text.x = element_blank(),
      axis.title.x = element_blank(),
      axis.text.y = element_blank(),
      axis.title.y = element_blank(),
      axis.ticks = element_blank(),
      panel.background = element_blank(),
      legend.position = "none")
  
   ggsave(filename = paste0(imageStorePath, sprintf('%05.2f', r[i]), '.png'),
          plot = p, 
          device = 'png') 
  
  plots_XZ[[i]] <- p
}

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Above code chunks is executed to create the plots and save it into users file directory. The animated version of this simulation provided here.

https://drive.google.com/file/d/1hHOd-qRyWhygfGJQ1Ol_1A86K5baQPUs/view?usp=sharing

Limitations:

The animation of the butterfly effect is done using online gif converter. We’re fail to execute the R package like gganimate( ) , gifski ( ) which helps to present image format plots into animated or gif format which helps us to build up better conception about how a variable is changing with respect to another variable.

Future Work:

We’ll try to figure out the problem of the R package like gganimate( ) , gifski ( ) that why we’re keep failing to execute our animated plots using this kind of package and how should we approach for that. Once we do that , our next aim will be use this theory to execute some real world phenomenon.

References:

Chaos: Making a New Science by James Gleick
https://www.forbes.com/sites/startswithabang/2018/02/13/chaos-theory-the-butterfly-effect-and-the-computer-glitch-that-started-it-all/?sh=6de3cfc469f6
Nonlinear dynamics and Chaos by Steven Strogatz
https://fs.blog/the-butterfly-effect/

Chaos and Butterfly Effect

Abhirup Moitra

2022-09-03

INTRODUCTION

CHAOS THEORY:

LORENTZ SYSTEM:

Package deSolve: Solving Initial Value Differential Equations in R

1. A simple ODE: chaos in the atmosphere

R Code:

1.1. MODEL SPECIFICATION

Model parameters:

State variables:

MODEL EQUATION:

1.2. MODEL APPLICATION:

Time specification

Model integration

Plotting Results:

BUTTERFLY EFFECT:

R Code:

Limitations:

Future Work:

References: