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Chapter 10: Problem 5

(Crudest possible estimate of \(\alpha\) ) Let \(f(x, r)=r-x^{2}\). a) Write down explicit expressions for \(f\left(x, R_{0}\right)\) and \(\alpha f^{2}\left(x / \alpha, R_{1}\right)\). b) The two functions in (a) are supposed to resemble each other near the origin, if \(\alpha\) is chosen correctly. (That's the idea behind Figure \(10.7 .3\) ) Show the \(O\left(x^{2}\right)\) coefficients of the two functions agree if \(\alpha=-2\).

Short Answer

Expert verified
a) To find the explicit expressions, replace r with R鈧� and R鈧� respectively in the given function f(x, r) = r - x虏: - \(f(x, R_{0}) = R_{0} - x^{2}\) - \(\alpha f^{2}\left(\frac{x}{\alpha}, R_{1}\right) = \alpha\left(R_{1} - \left(\frac{x}{\alpha}\right)^{2}\right)\) b) To compare the O(x虏) coefficients of the functions, note that when 伪 = -2, the following holds: - In \(f(x, R_{0}) = R_{0} - x^{2}\), the coefficient of \(x^{2}\) is -1. - In \(-2\left(R_{1} + \frac{x^2}{4}\right)\), the coefficient of \(x^{2}\) is -1/2 (-2). Thus, the coefficient of the \(x^{2}\) terms are the same for both functions when 伪 = -2.

Step by step solution

01

Find f(x, R鈧�)

To find the expression for f(x, R鈧�), we just need to replace r with R鈧� in the given function f(x, r) = r - x虏. So, we have: \[f(x, R_{0}) = R_{0} - x^{2}\] #Step 2: Write down the explicit expressions for 伪f虏(x/伪, R鈧�)#
02

Find 伪f虏(x/伪, R鈧�)

To find the explicit expression for 伪f虏(x/伪, R鈧�), we need to plug in x/伪 for x and R鈧� for r in the function f(x, r) = r - x虏, and then multiply the result by 伪. We have: \[伪f^{2}\left(\frac{x}{伪}, R_{1}\right) = 伪\left(R_{1} - \left(\frac{x}{伪}\right)^{2}\right)\] #Step 3: Find the O(x虏) coefficients of f(x, R鈧�) and 伪f虏(x/伪, R鈧�) when 伪=-2#
03

Compare O(x虏) coefficients

We first rewrite the expression for 伪f虏(x/伪, R鈧�) by simplifying the square term and multiplying by 伪: \[\alpha\left(R_{1} - \left(\frac{x}{\alpha}\right)^{2}\right) = -2\left(R_{1} + \frac{x^2}{4}\right)\] Now we look at the coefficients of the \(x^{2}\) terms in both functions: - In \(f(x, R_{0}) = R_{0} - x^{2}\), the coefficient of \(x^{2}\) is -1. - In \(-2\left(R_{1} + \frac{x^2}{4}\right)\), the coefficient of \(x^{2}\) is -1/2 (-2). Notice that when 伪 is -2, the coefficients of the \(x^{2}\) terms are the same for both functions, i.e., they agree on the order of \(O\left(x^{2}\right)\).

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Key Concepts

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

Chaos theory
Chaos theory explores systems that seem random but are actually deterministic, meaning they follow specific laws or principles. In chaotic systems, small differences in initial conditions can lead to dramatically different outcomes, a phenomenon often referred to as the butterfly effect.
This theory applies to a wide range of systems, from weather patterns to financial markets. Here are some key aspects of chaos theory:
  • **Sensitivity to Initial Conditions:** Even tiny changes at the start can cause large differences later.
  • **Determinism:** Although outcomes appear random, they follow specific rules or equations.
  • **Non-linear Equations:** These often model chaotic systems, where outputs are not directly proportional to inputs.
Understanding chaos theory helps us make sense of complex and unpredictable systems. Instead of attempting to predict exact outcomes, scientists look for probabilities and patterns that help explain these chaotic behaviors.
Dynamical systems
Dynamical systems are mathematical models used to describe how a point in a geometric space evolves over time according to specific rules. These systems can be continuous (changing smoothly over time) or discrete (changing at specific intervals).
They are widely used in physics, engineering, biology, and more. Key characteristics of dynamical systems include:
  • **State Space:** The geometric space that represents all possible states of the system.
  • **Evolution Rule:** Governs how the system changes over time.
  • **Trajectory:** The path that a state follows through the state space over time.
By analyzing dynamical systems, we gain insights into the behavior and stability of various phenomena. This includes predicting long-term behavior based on initial states, which can be crucial for understanding systems ranging from celestial mechanics to population dynamics.
Bifurcation theory
Bifurcation theory studies changes in the qualitative or topological structure of a given family of dynamical systems. As parameters within the system are varied, the system can undergo sudden changes in behavior, known as bifurcations.
These changes can signal the appearance or disappearance of equilibrium states or periodic orbits. Here are some important points about bifurcation theory:
  • **Bifurcation Points:** The parameter values at which a qualitative change occurs in the system's behavior.
  • **Types of Bifurcations:** Include saddle-node, transcritical, pitchfork, and Hopf bifurcations, each describing different ways that solutions can change.
  • **Applications:** Used to understand critical transitions in various fields like ecology, economics, and engineering.
By exploring how small changes can precipitate major system shifts, bifurcation theory provides essential insights into the stability and transition phenomena. This understanding is crucial for fields requiring prevention or exploitation of sudden system changes.

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Most popular questions from this chapter

(A trick for locating superstable cycles) Hao and Zheng (1989) give an amusing algorithm for finding a superstable cycle with a specified iteration pattern. The idea works for any unimodal map, but for convenience, consider the map \(x_{n+1}=r-x_{n}^{2}\), for \(0 \leq r \leq 2 .\) Define two functions \(R(y)=\sqrt{r-y}, L(y)=-\sqrt{r-y} .\) These are the right and left branches of the inverse map. a) For instance, suppose we want to find the \(r\) corresponding to the superstable 5 cycle with pattern \(R L L R\). Then Hao and Zheng show that this amounts to solving the equation \(r=R L L R(0)\). Show that when this equation is written out explicitly, it becomes $$ r=\sqrt{r+\sqrt{r+\sqrt{r-\sqrt{r}}}} $$ b) Solve this equation numerically by the iterating the map $$ r_{n+1}=\sqrt{r_{n}+\sqrt{r_{n}+\sqrt{r_{n}-\sqrt{r_{n}}}}} $$ starting from any reasonable guess, e.g., \(r_{0}=2\). Show numerically that \(r_{n}\) converges rapidly to \(1.860782522 \ldots\) c) Verify that the answer to (b) yields a cycle with the desired pattern.

(Conjugacy) Show that the logistic map \(x_{i n 1}=r x_{n}\left(1-x_{n}\right)\) can be transformed into the quadratic map \(y_{u+1}=y_{n}^{2}+c\) by a linear change of variables, \(x_{n}=a y_{11}+b\), where \(a, b\) are to be determined. (One says that the logistic and quadratic maps are "conjugate." More generally, a conjugacy is a change of variables that transforms one map into another. If two maps are conjugate, they are equivalent as far as their dynamics are concerned; you just have to translate from one set of variables to the other. Strictly speaking, the transformation should be a homeomorphism, so that all topological features are preserved.)

(Exact solutions for the logistic map with \(r=4\) ) The previous exercise shows that the orbits of the binary shift map can be wild. Now we are going to see that this same wildness occurs in the logistic map when \(r=4\). a) Let \(\left\\{\theta_{n}\right\\}\) be an orbit of the binary shift map \(\theta_{n+1}=2 \theta_{n}(\bmod 1)\), and define a new sequence \(\left\\{x_{n}\right\\}\) by \(x_{n}=\sin ^{2}\left(\pi \theta_{n}\right)\). Show that \(x_{n+1}=4 x_{n}\left(1-x_{n}\right)\), no matter what \(\theta_{0}\) we started with. Hence any such orbit is an exact solution of the logistic map with \(r=41\) b) Graph the time series \(x_{n}\) vs. \(n\), for various choices of \(\theta_{0}\).

(Superstable 2 -cycle) Let \(p\) and \(q\) be points in a 2 -cycle for the logistic map. a) Show that if the cycle is superstable, then either \(p=\frac{1}{2}\) or \(q=\frac{1}{2}\). (In other words, the point where the map takes on its maximum must be one of the points in the 2 -cycle.) b) Find the value of \(r\) at which the logistic map has a superstable 2 -cycle.

(Dense orbit for the decimal shift map) Consider a map of the unit interval into itself. An orbit \(\left\\{x_{n}\right\\}\) is said to be "dense" if it eventually gets arbitrarily close to every point in the interval. Such an orbit has to hop around rather crazily! More precisely, given any \(\varepsilon>0\) and any point \(p \in[0,1]\), the orbit \(\left\\{x_{n}\right\\}\) is dense if there is some finite \(n\) such that \(\left|x_{n}-p\right|<\varepsilon\). Explicitly construct a dense orbit for the decimal shift map \(x_{n+1}=10 x_{n}(\bmod 1)\).
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