Chaotic behavior of some floor-squared maps

Posted on May 4, 2020 by mAnasa-taraMgiNI

Consider the one dimensional maps of the form:

x_{n+1}=\dfrac{\left(\lfloor x_n \rfloor \right)^2 + \{x_n\}^2 }{ax_n}, where \{x_n\}=x-\lfloor x \rfloor is the fractional part of x

What will be evolution of a x_0 under this map when a=2 or a=3? We can see that for x_0>0 it will tend converge. However, the behavior is far more interesting for x_0<0: It turns out that in these cases the trajectory of x_n exhibits chaotic behavior (Figure 1, 2, 3, 4).

floor_square01a2_evFigure 1

Here, a=2 and we use x_0=-1.464; of course all other x_0<0 show comparable behavior (but the choice of this for illustration x_0 will become clear below). The evolution is plotted after discarding the first few values of x_n. While the evolution of x_0 under the map is chaotic it is not entirely random. There are preferred zones and which x_n inhabits. This can be better visualized by plotting a histogram of all the values of x_n in the evolution under the map for 20000 iterations.

floor_square01a2_histFigure 2

We can see that for any value of x_0<0 the iterates will be quickly pushed below -1. Further, we can also see that once a value is -2.5 \le x \le -1 it will remain orbiting within these bounds. Thus, it settles into an attractor in this interval. However, we observe that there is are 2 zones of exclusion in this interval. Even though we initiate the mapping in the middle of the larger zone of exclusion (which is why we chose x_0=-1.464), we observe that x_n moves away from that zone and mostly keeps away from it. As it oscillates between -1 and -2.5 x_n repeatedly approaches the second zone of exclusion from either side but gets repelled by it. How can we precisely determine these repellors of the map? Those can be determined by solving the equation:

2x^2=\left(\lfloor x \rfloor \right)^2+\left(x-\lfloor x \rfloor \right)^2

We have to solve such equations piecemeal due to the discontinuity of the \lfloor x \rfloor function. Because of the interval within which the attractor lies we have to only consider its solutions where \lfloor x \rfloor=-2 and \lfloor x \rfloor=-3. By substituting these two values of the floors in the above floor equation we get the two quadratics x^2-4x-8=0 and x^2-6x-18=0, whose roots will yield the repellors. The solutions of the first are 2 \pm 2\sqrt{3}. Since only the negative root is within interval of for our attractor, we have one repellor as r_1=2 - 2\sqrt{3}. Similarly, from the second equation we get the second repellor to be r_2=3 - 3\sqrt{3}. These are shown as green lines in the above figures. One can see we initiated the map close to r_1 and saw how it was repelled. However, if x_0=2 - 2\sqrt{3}, x_0=3 - 3\sqrt{3} then x_n remains fixed. Thus, while r_1, r_2 repel x_n in their vicinity they are fixed points on the map (green points).

floor_square01a3_evFigure 3

Here, a=3 and we initiate the mapping with x_0=-0.618. We can see that in this case for all x_0 < 0 the iterates with be pushed below -\tfrac{1}{3}. Further, once -\tfrac{5}{3} \le x_n \le -\tfrac{1}{3} we can see that x_n will be trapped in an orbit within this interval. As in the above case, in the example illustrated in Figure 3, after briefly oscillating close to x_0, x_n gets repelled away from it and that region is an exclusion zone for x_0<0. However, unlike in the above case we do not have a clear second exclusion zone here (Figure 4).

floor_square01a3_histFigure 4

Instead, we see that there are repeated attempts to come close to a certain line followed by repulsion away from it to flanking bands. As a result we do not get the second exclusion zone but have a saddle-like distribution around the line that is repeatedly approached. As in the above case to identify the primary repellor and the secondary value that behaves like a pseudo-attractor and also a repellor we need to solve the floor equation:

3x^2=\left(\lfloor x \rfloor \right)^2+\left(x-\lfloor x \rfloor \right)^2

Again we solve it piecemeal. This time given the interval of our attractor [-\tfrac{5}{3}, -\tfrac{1}{3}] we have to consider only \lfloor x \rfloor= -1 and \lfloor x \rfloor= -2. By substituting the first floor value in the above equation we get the quadratic x^2-x-1=0 whose solutions are \phi, -\tfrac{1}{\phi}, where \phi is the Golden Ratio. Taking only the negative value that is in the interval which matters for us, we get the repellor of this map to be r_1=\tfrac{-1}{\phi}. With the next floor we get the quadratic x^2-2x-4=0 with roots 2 \phi, -\tfrac{2}{\phi}. The second of these give us r_2=-\tfrac{2}{\phi} the pseudo-attractor-repellor. This pseudo-attractor-repellor is like the superficially alluring woman who draws you but is a repellor you when you get too close. As in the above case r_1, r_2 are also fixed points of the map.

Finally, we can investigate the evolution of closely separated x_0 to see how closely they parallel each other (Figure 5).

floor_square01a2_evcompFigure 5

Here we compare the evolution of x_0=-0.1464 and x_0=-0.1465 under the first map. We observe that while they are statistically the same in behavior the actual trajectories rapidly diverge. This is a hall mark of chaotic behavior.

This entry was posted in Scientific ramblings and tagged , , , , , , , . Bookmark the permalink.