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Chapter 12: Problem 51

Consider the Mandelbrot sequence with seed \(s=-0.75\). Show that this Mandelbrot sequence is attracted to the value -0.5 . (Hint: Consider the quadratic equation \(x^{2}-0.75=x\), and consider why solving this equation helps.)

Short Answer

Expert verified
The Mandelbrot sequence with the seed s=-0.75 is attracted to the value -0.5.

Step by step solution

01

Solve the Quadratic Equation

Rearrange the quadratic equation \(x^{2}-0.75=x\) to standard form by moving all terms to one side: \(x^{2}-x-0.75=0\). Solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) where \(a=1\), \(b=-1\), and \(c=-0.75\).
02

Calculate the Discriminant

Calculate the discriminant \(b^{2}-4ac = (-1)^{2}-4*1*-0.75 = 4\) which is greater than zero. This means that the quadratic equation has two different real roots.
03

Find the roots

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula to find the roots: \(x_{1,2} = \frac{-(-1) \pm \sqrt{4}}{2*1} = 0.5 \pm 1\). So the roots are \(x_{1}=1.5\) and \(x_{2}=-0.5\).
04

Relate to the Mandelbrot sequence

The roots \(x_1\) and \(x_2\) represent the values to which the Mandelbrot sequence could be attracted to. But because the seed is supposed to be \(s=-0.75\), which is closer to \(-0.5\) it can be safely assumed that the Mandelbrot sequence is attracted to the value \(-0.5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
A quadratic equation is a polynomial equation of degree two. It takes the general form:
  • \[ ax^{2} + bx + c = 0 \]
Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable to solve for. Quadratic equations are essential in many areas of mathematics, especially in algebra.

To solve a quadratic equation like the one in our problem, \( x^{2} - x - 0.75 = 0 \), we can use several methods. The most common is the quadratic formula:
  • \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
This formula provides a way to find the roots, which are the solutions for \( x \) that satisfy the equation, by using the coefficients \( a \), \( b \), and \( c \) from our quadratic equation.

For our specific equation, \( a = 1 \), \( b = -1 \), and \( c = -0.75 \), which we use in the quadratic formula to find the roots.
Discriminant
The discriminant is a crucial component of solving a quadratic equation when using the quadratic formula. It is found under the square root part of the formula:
  • \[ b^{2} - 4ac \]
This discriminant helps determine the nature of the roots:
  • If the discriminant is positive, there are two distinct real roots.
  • If it is zero, there is exactly one real root, or a repeated root.
  • If negative, the roots are complex or imaginary.
In this exercise, the discriminant calculation was:
  • \[ (-1)^{2} - 4 \times 1 \times (-0.75) = 4 \]
Since 4 is positive, it confirms that our quadratic equation has two different real roots. This separation of roots is important to determine potential values to which a Mandelbrot sequence might be attracted.
Real Roots
Real roots are the solutions of a quadratic equation that are real numbers. They indicate the points at which the graph of the quadratic equation intersects the x-axis in a Cartesian plane.

From the quadratic formula, once we calculate the discriminant and determine it to be positive, we can compute the real roots. In our example:
  • \[ x_{1,2} = \frac{-(-1) \pm \sqrt{4}}{2 \times 1} \]
This simplifies to:
  • \[ x_{1} = 1.5 \]
  • \[ x_{2} = -0.5 \]
The two real roots, \( 1.5 \) and \( -0.5 \), can each be a potential attractor for a Mandelbrot sequence. However, given the seed \( s = -0.75 \), it is reasonable to assume the sequence will gravitate towards the closer value, \( -0.5 \). This understanding helps conclude why the Mandelbrot sequence aligns with one particular root over another.

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Most popular questions from this chapter

Refer to a variation of the chaos game. In this game you start with a square \(A B C D\) with sides of length 27 as shown in Fig. \(12-41\) and a fair die that you will roll many times. When you roll a 1 , choose vertex \(A\); when you roll a 2, choose vertex \(B\); when you roll a 3 , choose vertex \(C\); and when you roll a 4 choose vertex \(D .\) (When you roll a 5 or \(a\) 6, disregard the roll and roll again.) A sequence of rolls will generate a sequence of points \(P_{1}, P_{2}, P_{3}, \ldots\) inside or on the boundary of the square according to the following rules. Start. Roll the die. Mark the chosen vertex and call it \(P_{1}\). Step 1. Roll the die again. From \(P_{1}\) move two-thirds of the way toward the new chosen vertex. Mark this point and call it \(P_{2}\) Steps \(2,3,\) etc. Each time you roll the die, mark the point two-thirds of the way between the previous point and the chosen vertex. Using a rectangular coordinate system with \(A\) at \((0,0), B\) at \((27,0), C\) at \((27,27),\) and \(D\) at \((0,27),\) find the sequence of rolls that would produce the given sequence of marked points. (a) \(P_{1}:(27,0), P_{2}:(27,18), P_{3}:(9,24), P_{4}:(3,8)\) (b) \(P_{1}:(0,27), P_{2}:(18,9), P_{3}:(24,3), P_{4}:(8,19)\) (c) \(P_{1}:(27,27), P_{2}:(9,9), P_{3}:(21,3), P_{4}:(7,19)\)

Are a review of complex number arithmetic. Recall that (1) to add two complex numbers you simply add the real parts and the imaginary parts: e.g., \((2+3 i)+(5+2 i)=\) \(7+5 i ;\) (2) to multiply two complex numbers you multiply them as if they were polynomials and use the fact that \(i^{2}=-1:\) e.g., \((2+3 i)(5+2 i)=10+4 i+15 i+6 i^{2}=4+19 i .\) Finally, if you know how to multiply two complex numbers then you also know how to square them, since \((a+b i)^{2}=(a+b i)(a+b i)\). Simplify each expression. (Give your answers rounded to three significant digits.) (a) \((-0.25+0.125 i)^{2}+(-0.25+0.125 i)\) (b) \((-0.2+0.8 i)^{2}+(-0.2+0.8 i)\)

Refer to the chaos game as described in Section 12.2. You should use graph paper for these exercises. Start with an isosceles right triangle \(A B C\) with \(A B=A C=32,\) as shown in Fig. \(12-40 .\) Choose vertex \(A\) for a roll of 1 or \(2,\) vertex \(B\) for a roll of 3 or \(4,\) and vertex \(C\) for \(a\) roll of 5 or \(6 .\) Using a rectangular coordinate system with \(A\) at \((0,0), B\) at \((32,0),\) and \(C\) at \((0,32),\) complete Table \(12-21\) $$ \begin{array}{c|c|c} \text { Roll } & \text { Point } & \text { Coordinates } \\ \hline 3 & P_{1} & (32,0) \\ \hline 1 & P_{2} & (16,0) \\ \hline 2 & P_{3} & \\ \hline 3 & P_{4} & \\ \hline 5 & P_{5} & \\ \hline 5 & P_{6} & \\ \hline \end{array} $$

Consider the construction of a Koch snowflake starting with a seed triangle having area \(A=729 .\) Let \(R\) denote the number of triangles added at a particular step, \(S\) denote the area of each added triangle, \(T\) the total new area added, and \(Q\) the area of the "snowflake" obtained at a particular step of the construction. Complete the missing entries in Table \(12-4 .\) $$ \begin{array}{l|c|c|c|c} & R & S & T & Q \\ \hline \text { Start } & 0 & 0 & 0 & 729 \\ \hline \text { Step 1 } & 3 & 81 & 243 & 972 \\ \hline \text { Step 2 } & 12 & 9 & 108 & 1080 \\ \hline \text { Step 3 } & & & & \\ \hline \text { Step 4 } & & & & \\ \hline \text { Step 5 } & & & & \end{array} $$

Consider the Mandelbrot sequence with seed \(s=-1.25 .\) Is this Mandelbrot sequence escaping, periodic, or attracted? If attracted, to what number?
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