Came across this and was just amazed, having studied mathematics at university it’s hard to believe that the following statement could be true
However, check out this “proof” to show that it is.
Now, can be written as
So we can deduce that . So it follows that .
Now lets take .
It follows that .
Rearranging gives .
Therefore, . Right?
I found this result absolutely amazing – but this result is used in string theory and quantum mechanics!
Unstable periodic orbits
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.
[BW89] WENZEL, B. (1989). PHYSICAL REVIEW LETTERS. Volume 63, Number 8, p.819-822. Ohio State University, USA.
Continuing on from our introduction of the Hénon map, we will look at its dependence on initial conditions and take a look at the bifurcation diagram which is created by varying our variable a, such that
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.
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.
Located in the middle of the Karakum Desert in Turkmenistan lies a burning crater with a diameter of 70 metres and depth of 20 metres: The Door to Hell – noted for its natural gas fire.
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’.
I first got introduced to this amazing memorial by my brother – apparently there have been several people exploring Google maps and what appears to be in the African desert is a black spot. When zoomed in we’re met with the following picture (alternatively click here to see for yourself on Google maps).
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.
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 to a new point by the recurrence relation described by
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.
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.
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 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
Using then it follows that and .
Substituting gives .
Squaring the result gives the logistic map .
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!