Choas Theory: Hénon Map (Part 3)

       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].

F_n = -x_{n+1}+a-x_n^2+bx_{n-1}, \qquad \qquad \qquad \qquad (5)

dx_n/dt = s_nF_n, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, \, \, \, (6)

where s_n = \pm1 and n=1,...,p. 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.

a8c2bb9303e1ad17b96b54a86bfc8428.epsUnstable periodic orbits of order n

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

henon_UPOsHénon 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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Chaos Theory: Hénon map (Part 2)

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

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

       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.

7efb335bcb61a3a363ecd14f7449fd7f.eps

Hénon map: Sensitivity to initial conditions

Using the classical Hénon map parameter values and comparing two different initial conditions (0.1,0.1) and (0.10001,0.10001) 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 a, with a fixed b=0.3. Like the logistic map, there’s a wide range of different behaviours dependent upon our choice of a. 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 x and y values.

Bifurcation diagrams for the classical Hénon map for 0 \leq a \leq 1.5

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 a=0.36, the second at a=0.91, the bifurcations keep occurring results in orbits of period 4,8,16,…,\infty. 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 a \approx 1.22 amongst the chaos, only to go through more period doublings and transition back into chaos.

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.

H$\acute{e}$non map

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.

51479eeaac8bda88aa302df90d8a0ca5.eps

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.

1

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.