.

However, check out this “proof” to show that it is.

Lets take

.

.

.

Now, can be written as

1-2+3-4+5-6+…

+ 1-2+3-4+5-6+…

= 1-1+1-1+1-1+1-…

= .

So we can deduce that . So it follows that .

Now lets take .

It follows that .

Rearranging gives .

Giving .

Now, .

Therefore, . Right?

I found this result absolutely amazing – but this result is used in string theory and quantum mechanics!

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In this section we’ll be introducing a method to calculate the unstable periodic orbits (UPO’s) of the Hénon map, showing that by plotting them we can create the Hénon attractor. By using equation (5) and solving the differential (6) then we can find the UPO’s and plot them [BW89].

where and . Using MATLAB we can solve this differential and find the UPO’s of order n for the Hénon map. Since we’re introducing the method we will look up to period 6. The Figure below shows the UPO’s for n=1, n=2, n=4 and n=6, note that when n=3 or n=5 the orbits are non-existent.

Unstable periodic orbits of order n

By using MATLAB coding we can compute the fixed points that lie on the periodic orbits.

Hénon attractor with points of UPO’s denoted by colours black, red, yellow and green

The figure above shows that for the first 6 periodic orbits, the fixed points lie on the attractor, if we continue this process up to period n and plotted the orbits we will be able to form the Hénon attractor. The black point denotes the period-1 UPO, red period-2, yellow period-4 and green period-6. Note that there exists more periodic orbits and fixed points up to period-6, i.e. there are two period-6 orbits.

**Sources:**

[BW89] WENZEL, B. (1989). PHYSICAL REVIEW LETTERS. Volume 63, Number 8, p.819-822. Ohio State University, USA.

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.

** Dependence on initial conditions**

For the 1-D maps we have explored in the previous chapters, we have shown that they are sensitive to initial conditions, we can show this is true for the Hénon map as well.

Hénon map: Sensitivity to initial conditions

Using the classical Hénon map parameter values and comparing two different initial conditions and represented by the coloured lines blue and red respectively then we can observe that the map is sensitive to initial conditions (Figure above). Note that the orbits overlap or are close for the first 26 iterations then appear to become unidentical to one another.

** Bifurcation diagram**

In this section we’ll be looking at the Hénon map for different values of parameter , with a fixed . Like the logistic map, there’s a wide range of different behaviours dependent upon our choice of . We can show this in a bifurcation diagram. Note that since the Hénon map is a 2-D map then we have bifurcation diagrams for our and values.

Bifurcation diagrams for the classical Hénon map for

The diagrams show that the Hénon map shares the same route to chaos as the Logistic map, i.e. period doubling route to chaos. The first bifurcation occurs approximately at , the second at , the bifurcations keep occurring results in orbits of period 4,8,16,…,. It can be shown that the rate at which these bifurcations occur converge to the Feigenbaum constant (see previous posts). It’s interesting to observe that despite the diagrams having different y-axis ranges, the diagram is identical. In addition, its interesting to note that the bifurcation diagram has periodic windows, with a large period-7 orbit appearing at amongst the chaos, only to go through more period doublings and transition back into chaos.

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The history behind this crater starts in 1971, whereby Russian scientists thought it was a great oil field site. Pleased with finding gas resources they started to store the gas. However, the ground collapsed; creating the crater seen above – releasing methane gas. Fearing the release of poisonous gases the scientists decided to burn it off. Expectations were that the gas would burn off in a couple of days, however its still burning after four decades.

The name of the crater isn’t too hard to understand its origins, it really does look like the opening of hell. A man-made one at that – and a wonder of the world in my eyes.

Here are a few more pictures to ‘tickle your fancy’.

**Sources:**

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So what is it? It’s a memorial for the UTA Flight 772, which unfortunately succumb to an explosion – scattering and breaking up over the Sahara Desert, killing 155 passengers and 15 crew members. The flight was scheduled to depart at Brazzaville, Republic of Congo and arrive at Paris, France.

18 years later, families of the victims gathered at the crash site to build a memorial. One of epic proportions and meaning. Even then, pieces of wreckage were found at the crash site – due to the remoteness of the location. With help of local inhabitants the memorial was built mostly by hand; dark stones were used to create a 200ft diameter circle, depicting the outline of a compass. 170 broken mirrors were placed around the circumference – used to represent the victims. Among the things being used to create the memorial, a wing from the aircraft (which was rescued 10 miles away from the site) was used to display the names of those who had died.

The finished memorial was completed a few months later, depicting a compass with an aeroplane in the centre – as said before, the memorial is so vast that it can be seen from Google maps and Google earth.

If you’d like to see pictures of its construction please see the link in the sources section below.

Sources:

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The map is dependent on two parameters *a* and *b*. We can see that if we have then the map reduces to a quadratic map. The classical Hénon map, which has and . 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.

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.

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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 and for the tent map and logistic map respectively, denoting that = then given the tent map (see below) we can procede to prove this relationship the two maps share.The tent map is given by

* Case 1*:

.

Using then it follows that and .

Substituting gives .

Squaring the result gives the logistic map .

* Case 2*:

.

We know that . Similarly, we’ll use the same substitution .

Substituting gives .

Squaring the result gives the logistic map .

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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To see the behaviour of the tent map for all across the range of *x* we plot the bifurcation diagram (seen below).

Previously we stated that the behaviour of the tent map for converges to 0 and so it isn’t of any interest to us, and so we’ve plotted the bifurcation for . 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].

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

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For the system converges to 0 for all initial conditions. If then all initial conditions less than or equal to are fixed points of the system, otherwise for initial conditions they converge to the fixed point . For example, for then it converges to the fixed point 0.07 as seen by the red points in the top right plot for in the figure above.

For the system has fixed points at 0 and the other at , we can show this mathematically using the tent map. For when then the fixed point is when which implies that is a fixed point. For when then the fixed point is when . Rearranging gives . It follows that the fixed point is at . 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 denote a fixed point of the tent map, then is the gradient. Given the fixed point 0, then . For the fixed point , then . Since 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 , 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 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) **.

**(Ex 2)** .

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.

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The tent map is a recurrence relation, written as:

for and .

As shown below, the graph of has maximum value at . 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].

The logistic and tent map are *topologically conjugate* for and respectively. In other words, the behaviour of the tent map for is the same as that of the logistic map when . 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; and and

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 , 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.

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