Bugs #11

This is a zoom into yesterday’s image, getting a closer look at the tricorn. The parameter is still a = 2.5. The image center is at $a*\pi*i$.

Recall the definition of the parameter version of bug: $bug(x+y i) = x + a * sin(y/a) i$. Then, a little algebra,

$\begin{matrix} bug(z+a \pi i) & = & bug( x + (y+a \pi)i) \\ & = & x + a *sin(y/a+\pi)i \\& = & x - a * sin(y/a) i \\ & = & \overline{bug(z)} \end{matrix}$

If you are a little rusty the definition of complex conjugate is $\overline{x+yi} = x-yi$, and a well known trig identity is $sin(\theta+\pi) = -sin(\theta)$.

Also recall that near z=0 for large a it is almost true that bug(z) = z. More formally, for any $\epsilon, R > 0$ if a is large enough then $|bug(z)-z|< \epsilon \text{ whenever } z < R$. Now we can also say $|bug(z+\pi i)- \overline{z}|< \epsilon \text{ whenever } z < R$

There is a little more to be done, I will skip the details. If you track what is going on in the iteration of $bug(z)^2+c$ near $c = a * \pi i$ you will see that it behaves like the tricorn, $\overline z^2+c$ near $c = 0$.

Given the periodicity of the sin function, the same is true for c near $c = n * a * \pi i$ for any integer n. Even n looks like a Mandelbrot set, odd n looks like the Tricorn.

Bugs #10

Here is a zoom out by a factor of two from yesterday’s picture. The triangular shape at the top is called a Tricorn or Mandelbar. The parameter a is 2.5. (See Bugs #8 for a description of the formula.) The image is centered at $(a * \pi /2) * i$.

The Tricorn set is generated by $\bar{z}^2+c$. The standard formula is modified to use the complex conjugate of z. Here is a non-paywall academic paper on Tricorns

Bugs #9

Zoom out by a factor of 2 from yesterday’s image, we see the rope/web supports extend above and below the main body.

The rest of this post is for the more mathematically inclined readers. Yesterday I stated that the bug function is close to the identity. The real part is already there, so let’s focus on the imaginary part, and treat just the imaginary part as a real valued function.

Here is a table of values for three function and their first few derivatives evaluated at 0.

 f f(0) f'(0) f”(0) f”'(0) f””(0) f””'(0) x 0 1 0 0 0 0 sin(x) 0 1 0 -1 0 1 a*sin(x/a) 0 1 0 $-1/a^2$ 0 $1/a^4$

Notice that for the parameterized sin function, a*sin(x/a), with a large value of a, all of the derivatives are close to the derivatives of the identity function, x. UsingĀ  Taylor series, the difference can be made arbitrarily small of an arbitrary large neighborhood of the origin.

For example, $\text{ for } \epsilon, R \text{ where }0 < e < 1 < R, \text{ set }a = R^2/\epsilon$ and use Taylor series to show that

$|x-a*sin(x/a)| < \epsilon \text{ whenever } |x| < R$

The same result carries over to the complex bug function.

Bugs #8

Today’s image uses the parameterized version of the bug formula with parameter a = 2.5.

The parameterized version of the bug formula is $bug_a(x+iy) = x + i*a*sin(y/a)$. When a = 1.0, this is the same as the non-parameterized version.

For large values of a, the bug formula gets close to the identity function, and the images get closer to the Mandelbrot set. By adjusting the parameter a, you can control the degree of distortion / similarity when compared to $z^2+c$.

In this view, with a = 2.5, the non-cyclic, non-escaping chaotic region in the upper right and lower right of Bugs #3 and Bugs #4 are gone. The body looks pretty normal. Surrounding the body is a bifurcating webbing that is characteristic of the bug formula.