# Logistic Equation

### Introduction

In this report we shall be investigating the behaviour of the logistic equation; developing our understanding further by then investigating other dynamic systems with two- and three-dimensional properties, namely the Henon and Lorenz attractors.

### The Logistic Equation

The equation we will be observing can be written in the form:

PN+1 = PN + kPN(1-PN)

where ‘k' is an independent variable that we will define as being between 0.500 and 3.000. The starting value of P1 is set at 0.300. We shall investigate the graphs, starting by incrementing the value of k by 0.1 to observe the behaviour of the output diagrams.

Analysing the graphs, we can see that when k has a value of between 0.5 and 1, the values of PN tend to 1 exponentially. We call this value of 1 the ‘steady state'. The gradient of the graphs before reaching the steady state increases with each iteration of k. They reach the steady state quicker for each k iteration. Between k being 1 and 2, the population results start to oscillate above and below 1, gradually settling out at 1. This oscillation is called ‘hunting'. Between k of 2 and 2.5 there are two steady states. These states gradually move further apart from each other, and after k values of 2.5 and above, the number of steady states begin to increase rapidly.

We will need to analyse the graphs in more detail after k of 2.5 in order to observe the behaviour of the logistics maps in more detail. Smaller increments of k will be used between the maps, using more decimal places. To view the graphs more clearly, more values of N are needed and so for ease, we shall now use ‘Matlab' (as opposed to Microsoft Excel) to plot the maps.

### Four steady states are starting to appear.

The previous steady state values are now merging into two more steady states, above and below the previous value. This is doubling the total amount of steady states in the graph, compared to the previous critical behaviour, making a total of eight steady states.

The numbers of steady states begin to double rapidly. They start to double with increasingly smaller increments in k; the logistic equation is non-linear. The graphs become larger and more complicated with the increasing numbers of steady states. They become unmanageable and it is not feasible to determine any sensible results from them. The graphs enter a phase of chaos where the iterations covers all values within the limits. It would be unpractical to represent them all in this report, and so we shall display only are few more to justify the logistic behaviour.

Here we can see that for a short time window approximately between k(2.83) and k(2.85), the system stabilises and there are a number of steady states, in this case three (0.298, 0.211, and 0.683). These periods of reduced chaos are called ‘islands of stability', and will discuss these in further detail later.

To have a better understanding of the behaviours of the maps and steady states, we shall plot a graph of k against Psteady state .This will show us exactly how many steady states are occurring at each value of k.

This is known as the Bifurcation Diagram and shows the attractors between k and Psteady state . We were able to obtain the steady states from Microsoft Excel (see 1) but were able to get a better representation that was more accurate using Matlab.

We can now observe more clearly the relationships between the steady states. The diagram begins to fork at about k(1.935), dividing from one steady state of ‘1', into two steady states. The numbers of steady states then double with increasingly less increments between k values. This is known as the Period Doubling Effect, and describes the bifurcations as multiplying by twice that of the previous state. In this dynamical system the numbers of steady states are powers of 2, before chaos, explaining why very quickly the system bifurcates into such high values that it leads to chaos.

From 2, we can now also observe the islands of stability within the chaos region. This is indicated by the thin windows of non-chaos where there are a number of steady states. The thicker, more prominent island is of that which we discovered earlier at k(2.83). Zooming in on this area allows us to view what is happening more clearly (see 3).

We can see that within these islands the steady states then bifurcate into chaos again, having the same behaviour as the original graph. These states also follow the period of doubling effect, although in this instance they combine and multiply in powers of 3. The Bifurcation Diagram is known as a ‘fractal', which means that each bifurcation phase can be looked at separately, and seen that they are similar to the whole graph. The smaller representations of the graph, within the stable islands of chaos, show that there is a link between chaos and fractals.

There are also other windows within the chaos region; the noticeable ones can be seen to have five or six steady states.

There is a pattern between the system's bifurcation points. If we look at the distance between successive bifurcation points, we can see that they become shorter and shorter. That is, the period of doubling occurs with increasingly less increments of k. If we look at the ratio between the length of one bifurcation, and the length of the next successive bifurcation, it can be observed that this is a constant value. This is known as ‘Feigenbaum's Constant': 4.669 (to three decimal places).

### 2D and 3D Attractors

An attractor is a set of points to which a dynamical systems evolves after a certain period of time. Points that get close enough to the attractor remain in the attracters orbit. An attractor can be presented as a point, a curve, and a manifold. In this part of the coursework, we will be looking at two certain attractors, Henon (2D) and Lorenz (3D). Henon's attractor is a good example of an attractor that is in a form of a curve, whereas the Lorenz attractor has properties of a manifold. Both of these attractors are said to have a complicated fractal structure known as a strange attractor. A strange attractor is an attractor which contains strange non-integer dimension or if it has chaotic behaviour; the same chaotic behaviour observed in the bifurcation graph obtained in the Logistic Equation, 2.

### The Henon 2D Attractor

The Henon Attractor is modelled by two general equations. The two general equations are applied to a starting point, x(0) and y(0). Points are then mapped to new points and this is done repeatedly for an amount of points until the results are graphed. Below are the two equations that are used.

\begin{align} x_{n+1} &= y_n+1-a x_n^2,\\ y_{n+1} &= b x_n. \end{align}The map depends on two parameters,aandb. From the literature, we have learnt that for an ideal representation of the graph, a and b have the values 1.4 and 0.3 respectively. During the graphs creation some points close to x(0) and y(0) will be attracted and some will be repelling away. This map is often used to test the dynamical system theory as it produced one of the simplest systems showing chaotic behaviour.

Changing the starting values of x and y will not alter the general formation of the graphs; so we will keep the starting points constant at zero for simplicity. Using Matlab, we shall start with the ideal values for the Henon Attractor, and start by decreasing the value of b, keeping a the same, to observe the system behaviour:

We can see that by increasing b from zero to three, the formation of the inside and inner arcs are created, with a maximum possible value of b=0.314 (to three decimal places). Beyond this value the graph cannot be mapped.

We shall now begin to decrease a, keeping b constant, in order to observe the attractor's behaviour:

We can see that by increasing the value of a, again contributes to the formation of the inner arcs. It is also apparent that at certain values of a, the graph is not complete, and appears to have broken arcs. Only certain values can be mapped. Values below 1.1 cause general deterioration of the entire graph and only a few short arcs are shown. The maximum possible value for a when b=0.3 is 1.426 (to three decimal places).

Observing the maximum values further, we have discovered that by decreasing one variable by a small amount enables the other variable to increase further to a higher maximum value. We shall start by decreasing a (enabling an increased value of b) to observe the behaviours:

The two arcs are repelled away from each other (repelled from the point (1,0)) and the inner arcs begin to represent chaos. The ideal values for displaying chaos are when a=1 and b=0.54. The graph, again, begins to deteriorate with values of a less than one.

We shall now decrease b in order to increase values of a:

The inside arcs begin to merge with the outer arcs, which are in turn attracted together until a single arc is formed.

We can see that the Henon map is formed of two arcs, with the inner arcs representing stable and chaotic behaviours. If we zoom in on one section of the graph, the arcs in the local region look like a series of lines, running almost parallel to each other (see 4). If we zoom in closer on the arcs, these in turn are made up of a number of almost-parallel lines. If we zoom in further, these lines are also made up of a number of more lines.

### The Lorenz 3D Attractor

The Lorenz Attractor is the outcome of first-order differential equations, whose solutions tend towards a strange attractor with a fractal structure. The equations and the attracter were introduced by Edward Lorenz in 1963. He derived them from studying the long term behaviour of the Lorenz oscillator. Below are the three equations that are used.

\frac{dy}{dt} = x (\rho - z) - y \frac{dx}{dt} = \sigma (y - x)We can see that the equations are time variant and depend upon the three variables β, σ (the Prandtl number) and ρ (the Rayleigh number). From the literature, we have learnt that for an ideal representation of the graph, β is defined as 8/3, σ as 10, and ρ as 28.

\frac{dz}{dt} = xy - \beta zWe are now going to look at how manipulating the values of these variables will change the graphical representation of the Lorenz attractors. Using Matlab, we shall begin by altering the value of ρ to observe the behaviours of the graphs.

By changing the value of ρ the shape of the Lorenz Attractor changes dramatically. With values below 19.661 (to three decimal places when x(0)=y(0)=z(0)=10) the graph is stable and terminates at a fixed point. Above this value, the graphs become unstable and begin to locate around two fixed points, having two manifolds. The shape of this can be known as a ‘Torus Knot'. Changing the value of the Prandtl number, σ, has a similar effect on the outcome of the graphs.

Changing the value of β has an effect on the number of revolutions that the attractor oscillates about it's fixed point. Beyond a certain value, the system stabilises and the graph reverts back to one manifold.

By changing the starting points of the equations the manifolds tend to a different fixed point, but the effect of changing the three main variables is the same. However, the value of these variables creates a different graph trajectory than if the same values were used for a graph with different starting points. That is, the values of ρ for which the graph changes from being stable, to being unstable, would be different for that of a graph with different start points.

e.g.: Using the same critical value earlier of 19.661, using start points of x(0)=3, y(0)=15, and z(0)=1

The critical value of these start points is actually 26.485:

Comparing two different starting points with the same value of ρ:

It can be seen from the above graphical representations that a small variation in the value of the variables can produce a large change in the graph formed. This was coined the ‘Butterfly Effect' by Lorenz, and he came up with this idea whilst researching weather conditions. Lorenz was attempting to recreate weather conditions he had seen previously by entering in values the computer had calculated and produced on the earlier occasion. He found that the results were different from the original. After much consideration, he realised that the numbers used to do the second run were to 3 significant s, but the numbers in the program were to 6 significant s. This difference, one part in a thousand, had been thought by Lorenz to be unsubstantial, but as a result of the recursive nature of the equations, caused a major change.

This lead to another thought experiment where it is proposed there are two identical worlds. In one, a butterfly flaps its wings in a certain position on the world. In the other world, the exact same butterfly does not. In the world where the butterfly flaps its wings, the weather conditions vary dramatically compared to in the world where the butterfly did not flap its wings. This is a representation of how a small change can bring about a major consequence.

We can also observe the starting points of the graphs, in relation to the butterfly effect. In the graphs below, the left hand side graphs differ against the graphs on the right hand side, by one part in a thousand of the x coordinate starting point. With the same values for the variables in the equations, it is clear that the graphs follow a different trajectory.

The difference in trajectories can only be seen in the values of ρ that give the unstable maps. For the stable maps, the paths track each other and no change can be seen. This can be seen more clearly in a colour representation. This is courtesy of Melos Campos, providing a Lorenz graph generator on his website (see reference in Bibliography section).

Here are two graphs, one with ρ=28 and one with ρ=23. There are two Lorenz graphs mapped on top of each other in each one, one indicated in blue and the other in yellow. These two trajectories differ only in their x-coordinate start point by 10-5 and it is clear that their paths are very different. This confirms the theory of the butterfly effect and the system is chaotic due to it's unpredictability.

In addition, the graphs can be viewed with respect to time. Here we have compared two maps, again each only differing slightly in the x-coordinate. As time increases initially, the system is stable and the manifold terminates at a fixed point. Here the two graphs are identical. Then as time increases and the torus knot starts to develop, the graphs diverge and their trajectories are different.

### Bibliography

M. Casco Associcates - The M. Casco Learning Centre (Order Course)

http://mcasco.com/ocourse.html

P Grassberger, H Kantz & U Moenig (Physics Department, University of Wuppertal) “On The Symbolic Dynamics of the Henon map”, pp. 1-3.

Images showing the fractal structure of Henon map ( 4)

http://hmf.enseeiht.fr/travaux/CD0203/travaux/optmfn/hi/03pa/hyd09/page2.htm

Lorenz Equations

http://planetmath.org/encyclopedia/LorenzEquation.html

M.Henon & Y. Pomeau, “Two Strange Attractors with a Simple Structure”, pp. 1-4, 36-40.

The Butterfly Effect:

http://www.zeuscat.com/andrew/chaos/lorenz.html

Lorenz graph generator

http://to-campos.planetaclix.pt/fractal/lorenz_eng.html