Chaos Theory: Hénon Map

Before we go onto looking at Lyapunov exponents, a statistic which helps us determine whether a system is chaotic or not, we will be looking at 2-D maps. In particular we will be examining the Hénon map, a discrete time dynamical system. First introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model, it has become on the of most studied examples of systems that exhibit chaotic behaviour. The Hénon map takes $(x_n,y_n)$ to a new point by the recurrence relation described by

$x_{n+1}=y_n+1-ax_n^2, \qquad \qquad y_{n+1}=bx_n \qquad \qquad \qquad \qquad \qquad (3).$

The map is dependent on two parameters a and b. We can see that if we have $b=0$ then the map reduces to a quadratic map. The classical Hénon map, which has $a=1.4$ and $b=0.3$. For these values the map is chaotic, and the system resembles a boomerang shape as seen below. Known as the Hénon attractor; it has become another icon of chaos theory alongside the bifurcation diagram and Lorenz attractor.

Hénon map for $a=1.4$, $b=0.3$ and initial conditions $x_0=0.5$ and $y_0=0.5$

The Hénon attractor is a strange attractor, this is because the dimension of the attractor is non-integer and is usually associated with systems that are chaotic.

Sources:

HILBORN, R.C. (2000). Chaos and Nonlinear Dynamics: An introduction for Scientists and Engineers. UK. Oxford University Press.

PEITGEN, H.O., JURGENS, H. & SAUPE, D. (2004). Chaos and Fractals: New Frontiers of Science Second Edition, New York, USA. Springer.

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!

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.

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.

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

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.

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

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

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.

Ajanta Caves

The Ajanta Caves are among the world’s most prestigious religious locations in the world, sacred to Buddhism. Located in India, these caves were carved thousands of years ago from the mountain side; displaying inhuman-like capabilities as the tools and technology we have today would not have been accessible or invented when the caves were formed. The caves include paintings, sculptures and signs of astronomical knowledge and have been subject to wide-spread tourism and speculation as to how they were built.

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The caves were discovered in 1819 by a British office on a hunting party. Hidden by dense forestation the caves were said to be lost for centuries after the downfall of Buddhism in India in the early millennium. The Ajanta caves were said to be a ancient monastery for Buddhists, including space for living, teaching and worship; this is reflected by the fact that most caves are connected by the exterior.

Another fascinating feature of these caves are their relationship with the summer and winter solstice’s. The sun shines directly through gaps in the mountain to light up and illuminate significant religious Stupa’s (a place for meditation) at the back of the caves. It’s amazing because precise calculations and tools would have been needed to be able to orient the caves to the solstices since the caves are carved inside the rock. This has led people to believe that alien activity or interaction was involved in the feat of constructing the caves. There’s plenty of information regarding these caves and documentaries surrounding them. Check out the video in the sources below!

Sources:

Chaos Theory: Periodic Windows

The bifurcation diagram is much more complex than just a simple division of regions of chaos and periodic behaviour [1]. As previously stated, in this post we will be looking at one of the most iconic features that comes from observing the bifurcation diagram; periodic windows. Amongst all the chaos, a stable period-3 orbit appears at r $\approx 3.83$ as shown below. These regions (periodic windows) exist for r > $r_{\infty}$, they are areas whereby the system isn’t chaotic.

Periodic window at approx. r=3.83

First we will explore how the period-3 orbit is created. Note, all periodic windows are created in this manner [2]. Using the logistic equation $x_{n+1}=rx_{n}(1-x_{n})$ then we can rewrite as $x_{n+1} = f(x_n)$, so $x_{n+2} = f(f(x_n)) = f^2(x_n)$, similarly $x_{n+3} = f^3(x_n)$. This mapping is vital in understanding the birth of the period-3 window. In the figure below we plot the third iterative mapping for $r = 3.84$ which has 8 fixed points such that $f^3(x) = x$.

Third iterative map of the logistic map for r=3.84

Two of the fixed points are actually solutions to $f(x)=x$, the other six are marked by dots coloured red and black. The red dots correspond to a stable period-3 orbit, alternatively, the black notes correspond to a unstable period-3 orbit. We can show this is true by looking at the gradient at each fixed point. If it’s negative then it’s stable, and if positive then it’s unstable. Decreasing r to $r = 3.8$ then the graph changes as indicated by the red line below.

Third iterative map for r=3.8, 3.8284, 3.84

Therefore we expect for some value between $r = 3.8$ and $r = 3.84$ the graph of $f^3(x)$ becomes tangent to the diagonal. At this value of r then the stable and unstable fixed points unite and annihilate in a tangent bifurcation [2], indicating the birth of the period-3 window. This value of r is approximately represented by the green line above, and is said to be $r = 3.8284...$ [3]. As previously stated we can show that each periodic window is created in this way. In addition, it can be shown that there are periodic windows in all chaotic regions and so is a infinite number of periodic windows [4].

Sources:

[1] GILMORE, R. & LEFRANC, M. (2002). The Topology of Chaos. New York, USA. John Wiley & Sons.

[2] STROGATZ, S.H. (1994). Nonlinear dynamics and chaos. New York, USA. Westview Press, Perseus.

[3] TAKAYASU, H. (1990). Fractals in the physical sciences. Manchester, UK. Manchester University Press.

[4] FELDMAN, D.P. (2012). Chaos and Fractals: An elementary introduction. Oxford, UK. Oxford University Press.