/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Calculate the Liapunov exponent ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 2

Calculate the Liapunov exponent for the decimal shift map \(x_{u+1}=10 x_{n}(\bmod 1)\)

Short Answer

Expert verified
The Liapunov exponent for the decimal shift map \(x_{n+1} = 10x_n \pmod{1}\) is calculated through the following steps: finding the Jacobian matrix which is the derivative with respect to \(x_n\) (equals to 10), calculating the average Jacobian at each point in the orbit (also 10), and finding the logarithm of the magnitude of the eigenvalue (which is \(\log(10)\)). Therefore, the Liapunov exponent for this map is \(\log(10)\).

Step by step solution

01

Define the map

We are given the map as: \(x_{n+1} = 10x_n \pmod{1}\).
02

Find the Jacobian matrix

Here we have a one-dimensional map, so the Jacobian is simply the derivative with respect to \(x_n\). The derivative of the map with respect to \(x_n\) is: \(\frac{dx_{n+1}}{dx_n} = 10\).
03

Calculate the average Jacobian at each point in the orbit

Since the Jacobian is constant and equals to 10, the average Jacobian at each point in the orbit is also 10.
04

Find the logarithm of the magnitude of the eigenvalue

In the one-dimensional case, the Jacobian itself is the eigenvalue. So, we need to find the logarithm of the magnitude of the average Jacobian matrix, which is: \(\log(|10|) = \log(10)\).
05

Calculate the Liapunov exponent

The Liapunov exponent is the average of the logarithm of the magnitudes of the eigenvalues of the average Jacobian matrix. In this case, it's just \(\log(10)\). Therefore, the Liapunov exponent for the decimal shift map \(x_{n+1} = 10x_n \pmod{1}\) is \(\log(10)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Decimal Shift Map
The decimal shift map is a type of mathematical transformation. It is defined by the function \( x_{n+1} = 10x_n \pmod{1} \). This map effectively shifts the decimal point one place to the right at every iteration. This operation is closely related to decimal expansion, meaning it involves moving the decimal place and discarding the integer part, if any.

Why is it called a 'mapping'? In mathematics, a 'map' is often a synonym for a function. It takes an input and assigns it to an output based on some defined rule. Here, the rule involves multiplication by 10 and taking the modulus with 1.

This map is interesting in the study of chaos and dynamical systems. Because it consistently expands the number, it can show growth behavior, leading to chaotic dynamics.
Jacobian Matrix
The Jacobian matrix is a crucial concept in understanding how functions behave. When you extend the idea of derivatives from single-variable functions to functions of several variables, you use the Jacobian matrix. It provides a linear approximation of a multi-dimensional function near a point.

However, in our exercise, the decimal shift map is a one-dimensional function. The Jacobian matrix simplifies to just the derivative in this case. So, the derivative for our map, as a function of \( x_n \), is simply \( 10 \).

This concept captures the rate of change: how sensitive the output is to changes in the input. The constant derivative means the map's behavior is uniform across its domain in our one-dimensional scenario.
Eigenvalue
The eigenvalue is a central concept when dealing with matrices and linear transformations. An eigenvalue is a scalar which gives you information on how a linear transformation changes a vector along a given direction.

For a one-dimensional map like the one in this exercise, the Jacobian is constant and effectively becomes the eigenvalue. This simplifies the problem significantly. The eigenvalue here is simply \( 10 \), the same as the derivative.

In broader terms, eigenvalues are essential for stability analysis in systems of differential equations and dynamical systems. They help in understanding how systems evolve over time and can indicate stability or chaos.
Logarithm
Logarithms are a mathematical tool for solving problems involving exponential growth or decay. They are inverses of exponentiation. In simple terms, understanding logarithms can be thought of as asking the question, "What power should we raise a base number to produce another number?"

In the context of the decimal shift map exercise, we calculate the Liapunov exponent using \( \log(10) \). The Liapunov exponent itself gives a measure of the rate of separation of infinitesimally close trajectories. It effectively tells us whether the system is stable (negative exponent) or chaotic (positive exponent).

Here, the positive outcome \( \log(10) \) indicates exponential divergence, emphasizing the chaotic nature of the decimal shift map.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given a map \(y_{n+1}=f\left(y_{n}\right)\), rewrite the map in terms of a rescaled variable \(x_{n}=\alpha y_{n} .\) Use this to show that rescaling and inversion converts \(f^{2}\left(x, R_{1}\right)\) into \(\alpha f^{2}\left(x / \alpha, R_{1}\right)\), as claimed in the text.

(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 \(\alpha=-2\).

(Binary shift map) Show that the binary shift map \(x_{n+1}=2 x_{n}(\bmod 1)\) has sensitive dependence on initial conditions, infinitely many periodic and aperiodic orbits, and a dense orbit. (Hint: Redo Exercises \(10.3 .7\) and \(10.3 .8\), but write \(x_{n}\) as a binary number, not a decimal.)

(Renormalization approach to intermittency: algebraic version) Consider the map \(x_{n+1}=f\left(x_{n}, r\right)\), where \(f\left(x_{n}, r\right)=-r+x-x^{2} .\) This is the normal form for any map close to a tangent bifurcation. a) Show that the map undergoes a tangent bifurcation at the origin when \(r=0\). b) Suppose \(r\) is small and positive. By drawing a cobweb, show that a typical orbit takes many iterations to pass through the botleneck at the origin. c) Let \(N(r)\) denote the typical number of iterations of \(f\) required for an orbit to get through the bottleneck. Our goal is to see how \(N(r)\) scales with \(r\) as \(r \rightarrow 0\). We use a renormalization idea: Near the origin, \(f^{2}\) looks like a rescaled version of \(f\), and hence it too has a bottleneck there. Show that it takes approximately \(\frac{1}{2} N(r)\) iterations for orbits of \(f^{2}\) to pass through the bottleneck. d) Expand \(f^{2}(x, r)\) and keep only the terms through \(O\left(x^{2}\right)\), Rescale \(x\) and \(r\) to put this new map into the desired normal form \(F(X, R) \approx-R+X-X^{2}\). Show that this renormalization implies the recursive relation e) Show that the equation in \((\mathrm{d})\) has solutions \(N(r)=a r^{b}\) and solve for \(b\).

(Quadratic map) Consider the quadratic map \(x_{n+1}=x_{n}^{2}+c\). a) Find and classify all the fixed points as a function of \(c .\) b) Find the values of \(c\) at which the fixed points bifurcate, and classify those bifurcations. c) For which values of \(c\) is there a stable 2 -cycle? When is it superstable? d) Plot a partial bifurcation diagram for the map. Indicate the fixed points, the 2cycles, and their stability.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.