Chaos Theory: Topological conjugation

Sorry about the late posts all of a sudden, just started work and have little to no time most days!

In this post (on chaos theory) we’re going to be looking at the relationship between logistic map and tent map; both maps we have explored previously in this category. We’ll be proving that the maps are identical under iteration (topologically conjugate).

This occurs for the case \mu = 2 and r = 4 for the tent map and logistic map respectively, denoting that x_n = \dfrac{2}{\pi}\sin^{-1} \sqrt{y_n} then given the tent map (see below) we can procede to prove this relationship the two maps share.The tent map is given by

T(x)=    \begin{cases}    2 x_{n}, \qquad \qquad 0 \leq x \leq \frac{1}{2}\\    2 - 2 x_{n}, \qquad \frac{1}{2} \leq x \leq 1    \end{cases}

Case 1: x_{n+1} = 2 x_n

\dfrac{2}{\pi}\sin^{-1} \sqrt{y_{n+1}} = \dfrac{4}{\pi}\sin^{-1} \sqrt{y_n} \Longrightarrow \sqrt{y_{n+1}} = \sin(2sin^{-1} \sqrt{y_n}).

Using \theta = \sin^{-1} \sqrt{y_n} then it follows that \sin \theta = \sqrt{y_n} and cos \theta = \sqrt{1-y_n}.

Substituting gives \sqrt{y_{n+1}} = \sin 2 \theta = 2 \sin \theta \cos \theta = 2 y_n (\sqrt{1-y_n}).

Squaring the result gives the logistic map y_{n+1} = 4 y_n (1 - y_n).

Case 2: x_{n+1} = 2 - 2 x_n

\dfrac{2}{\pi}\sin^{-1} \sqrt{y_{n+1}} = 2 - \dfrac{4}{\pi}\sin^{-1} \sqrt{y_n} \Longrightarrow \sqrt{y_{n+1}} = \sin \pi - \sin(sin^{-1} \sqrt{y_n}).

We know that \sin \pi = 0. Similarly, we’ll use the same substitution \theta.

Substituting gives \sqrt{y_{n+1}} = - \sin 2 \theta = - 2 \sin \theta \cos \theta = - 2 y_n (\sqrt{1-y_n}).

Squaring the result gives the logistic map y_{n+1} = 4 y_n (1 - y_n).

With this result we can conclude that if the tent map has chaotic orbits then the logistic map must also have chaotic orbits.

P.S. I’m aware that sometimes the mathematics being explained here doesn’t view properly, just refresh the page and it should fix the problem. Thanks!

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Choas Theory: Tent Map (Part 3)

Following from the previous post, we’re going to explore the limitations of iterating the tent map for when \mu = 2. The graph below shows graphically what occurs given a rational number, say, x_0 = 0.4. For clarity, we’ve zoomed in on the point in which the system jumps straight to 0. This is the point in which the binary notation for x is all 0’s.

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Limitations when using \mu = 2

To see the behaviour of the tent map for all \mu across the range [0,1] of x we plot the bifurcation diagram (seen below).

Bifurcation diagram for the Tent map.

Bifurcation diagram for the Tent map.

Previously we stated that the behaviour of the tent map for \mu < 1 converges to 0 and so it isn’t of any interest to us, and so we’ve plotted the bifurcation for 1 \leq \mu \leq 2. Unlike the Logistic map, the tent map doesn’t follow the period-doubling route to chaos. In fact it can be shown that there are no period-doublings [1].

Tent map histograms: distribution of data

Tent map histograms: distribution of data

Similarly to the logistic map we can represent the distribution of x values and the range on a histogram. Note we cannot use \mu = 2 due to numerical limitations when computing the results as previously discussed. For \mu \leq 1 all the points converge to a fixed point and so nearly all the points are distributed to one value. For \mu > 1 the points converge to a number of fixed points as shown by the histogram for \mu = 1.2. As we can see from the graph above, the distribution for the tent map tends towards a uniform distribution as \mu tends to 2.

Sources:

[1] LAM, L. (1998). Non-Linear Physics for Beginners: Fractals, Chaos, Pattern Formation, Solutions, Cellular Automata and Complex Systems. London, UK. World Scientific Publishing.

Chaos Theory: Tent Map (Part 2)

Continuing from the previous post; the graph below shows the tent map for a range of \mu values, here we’ll use initial condition 0.4.

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Tent map for a range of values of \mu

For \mu < 1 the system converges to 0 for all initial conditions. If \mu = 1 then all initial conditions less than or equal to \frac{1}{2} are fixed points of the system, otherwise for initial conditions x_0 > \frac{1}{2} they converge to the fixed point 1 - x_0. For example, for x_0 = 0.93 then it converges to the fixed point 0.07 as seen by the red points in the top right plot for \mu = 1 in the figure above.

For \mu > 1 the system has fixed points at 0 and the other at \frac{\mu}{\mu + 1}, we can show this mathematically using the tent map. For when 0 \leq x \leq \frac{1}{2} then the fixed point is when \mu x = x which implies that x = 0 is a fixed point. For when \frac{1}{2} \leq x \leq 1 then the fixed point is when \mu - \mu x = x. Rearranging gives \mu = x + \mu x = (\mu + 1) x. It follows that the fixed point is at \frac{\mu}{\mu + 1}. However, both fixed points are unstable, we can show this mathematically by looking at the gradient of the fixed point. If the gradient is less than one then its said to be stable, if its greater than then its unstable.

Let T(x^*) denote a fixed point of the tent map, then T'(x^*) is the gradient. Given the fixed point 0, then T'(x^*) = T'(0) = \mu. For the fixed point \frac{\mu}{\mu + 1}, then T'(x^*) = T'(\frac{\mu}{\mu + 1}) = \frac{1}{(\mu + 1)^2}. Since \mu > 1 in both cases then both fixed points are unstable. In other words, for a value x near the given fixed point, it will diverge away from it rather than converge. For \mu = 2, the system maps the interval [0,1] onto itself, becoming chaotic.

The dynamics will be aperiodic for initial conditions that are irrational, and periodic when rational. One effect of this is that we cannot run long term simulations of the tent map on a computer because all numbers in a computer are rational. Since x_n is expressed in binary notation, each successive iteration of the tent map will eventually hit 0 since the leftmost bit will always be removed (Ex 1). For irrational numbers there’s an infinite binary expansion and so will never go to 0 (Ex 2) [1].

(Ex 1) \quad x_0 = 0.5322265625 = \frac{1}{2} + \frac{1}{32} + \frac{1}{1024} = 0.1000100001.

(Ex 2) \quad x_0 = \frac{\pi}{10}.

In the next post we will (again) be exploring further the tent map, looking at computational limitations regarding the calculation of the tent map, bifurcation diagrams and histogram.

Sources:

[1] GLEICK, J. (1998). Chaos: the amazing science of the unpredictable. London, UK. Vintage.

Chaos Theory: Tent Map (Part 1)

In previous posts regarding chaos theory we have investigated the logistic map. In this post we’re going to be looking at a similar system known as the tent map; it is also commonly referred to as the triangle map.

The tent map is a recurrence relation, written as:

x_{n+1}=    \begin{cases}    \mu x_{n}, \qquad \qquad 0 \leq x \leq \frac{1}{2}\\    \mu - \mu x_{n}, \qquad \frac{1}{2} \leq x \leq 1 \qquad \qquad \qquad \qquad \qquad \qquad (2)    \end{cases}

for  0 \leq \mu \leq 2 and 0 \leq x \leq 1.

As shown below, the graph of (2) has maximum value \frac{\mu}{2} at x = \frac{1}{2}. The tent map is piecewise linear, because of this characteristic this makes the tent map easier to analyse than the logistic map. However, although the form of the tent map is simple and the equations are linear, for certain parameter values the map can yield complex and chaotic behaviour [1].

Tent map

Tent map

The logistic and tent map are topologically conjugate for r = 4 and \mu = 2 respectively. In other words, the behaviour of the tent map for \mu = 2 is the same as that of the logistic map when r = 4. The proof of this is will be looked at in future posts. First lets look to see if the tent map is sensitive to initial conditions. Here we’ll use similar initial conditions to that found in previous posts for the logistic map; x_{0} = 0.4 and x_{0}* = 0.40001 and \mu = 1.9999

Tent map: Sensitivity to Initial Conditions

Tent map: Sensitivity to Initial Conditions

Similarly to the logistic map, we can see that the tent map is also sensitive to initial conditions. In addition to being sensitive to initial conditions, the tent map is also dependent on its parameter value \mu, ranging from predictable to chaotic behaviour. We will explore the tent map further in the next post.

Sources:

[1] LYNCH, S. (2007). Dynamical Systems with Applications Using Mathematica (eBook). Birkhäuser Basel, 1st edition.

Chaos Theory: Logistic Map Introduction

This chapter looks at dynamical systems where by time is discrete. These systems are also known as iterated maps or difference equations. In particular, we will be focusing on 1-D dynamical systems, when plotted the orbit is just a sequence of points in the real numbers (\mathbb{R}). The two systems we will be looking at (as previously stated) are the logistic map and the tent map, these maps illustrate how chaotic behaviour arises from simple systems.

The logistic map was first proposed to describe the dynamics of insect populations by biologist Robert May in 1976. Often speculated as the archetypal example of how chaotic behaviour arises from simple dynamical equations, it is a recurrence relation, written as

x_{n+1}=rx_{n}(1-x_{n}), \qquad 0 \leq x \leq 1\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)

where x_{n} represents the ratio of the existing populations to a possible maximum population at year n such that 0 \leq x_n \leq 1. For example, x_{0} represents the initial ratio at year 0. The parameter r is a positive number, representing the rate of reproduction and starvation such that 0 \leq r \leq 4.

logistic mapThe graph of (1) is a parabola with maximum value of r/4 at x=\frac{1}{2} as shown above. Increasing the value of r=0 to r=4 make the map undergo a series of period-doubling bifurcations, something we will be exploring later on. The logistic map has fixed points at x=0 and x=1, regardless of r. However, the stability of the fixed points depend upon r. Again, something we will be exploring in the future.

P.S. All graphical output produced on MATLAB.

Chaos Theory: Dynamical Systems

Light_Rain_ShowersChaos is typically understood as a mathematical property of a dynamical system. A dynamical system can be considered to be a model that evolves through time, where time is either continuous or discrete.  Together with chaos theory they explore the evolution of qualitative behaviour. Examples of dynamical systems include the weather or even the number of cars going through a junction.

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We will look at investigating non-linear dynamical systems that are defined at discrete times. A discrete-time system uses current states as input and generates new states as output. Moreover, we will be exploring questions such as why chaotic behaviour arises. We will focus on looking at sensitivity to initial conditions, and, explore the different ways to represent the behaviour of dynamical systems.

Two maps that we will be looking at are the quadratic map (logistic map) and the tent map, both of which are the simplest examples of how models illustrate chaotic behaviour.

Chaos Theory

Ever wondered why we can’t predict the weather in the long term? Ever wondered what the meaning behind the ‘butterfly effect’ is? Well, in this section of my blog I’m going to be exploring the most intriguing and probably the least understood part of science there is, Chaos Theory.

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Chaos Theory is one of the newest branches of science there is, becoming popular in the 1960’s thanks to the work of Edward Lorenz, an American mathematician and meteorologist. In the future, I will be updating daily or weekly more information and interesting mathematics that spring from Chaos Theory. Starting with the history behind it, moving on to discrete dynamical systems – looking at basic models such as the logistic map and tent map, then moving on to look at fractals.

Just remember, chaos theory arises and exists everywhere. Check out this first post on this forum, got told about this from a friend, it’s how chaos theory is related to Star Wars, I’m sure there’s a few “geeks” out there that would be interested in knowing the relationship!

Star Wars and Chaos Theory.

So if you’re interested in the weird and wonderful or just want to learn something new, watch this space!