/*! 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 (Flow on a torus; cf. \([\mathrm... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 2

(Flow on a torus; cf. \([\mathrm{H} / \mathrm{S}]\), p. 241\()\). Identify \(\mathbf{R}^{4}\) with \(\mathbf{C}^{2}\) having two complex coordinates \((w, z)\) and consider the linear system $$ \begin{aligned} \dot{w} &=2 \pi i w \\ \dot{z} &=2 \pi \alpha i z \end{aligned} $$ where \(\alpha\) is an irrational real number. (a) Set \(a=e^{2 \text { *ai }}\) and show that the set \(\left\\{a^{n} \in \mathbf{C} \mid n=1,2, \ldots\right\\}\) is dense in the unit circle \(C=\\{z \in \mathbf{C}|| z \mid=1\\}\). (b) Let \(\phi_{t}\) be the flow of this system. Show that for any integer \(n\), $$ \phi_{n}(w, z)=\left(w, a^{n} z\right) $$ (c) Let \(x_{0}=\left(w_{0}, z_{0}\right)\) belong to the torus \(T^{2}=C \times C \subset C^{2} .\) Use (a) and (b) to show that \(\omega\left(\Gamma_{\mathrm{x}_{0}}\right)=T^{2}\).

Short Answer

Expert verified
The set \(\{a^n\}\) is dense in \(C\); \(\phi_n(w, z) = (w, a^n z)\); \(\omega(\Gamma_{x_0}) = T^2\).

Step by step solution

01

Understand the Given System of Equations

The system is given by \(\begin{aligned}\dot{w} &=2 \pi i w \dot{z} &=2 \pi \alpha i z\end{aligned}\)where \(w\) and \(z\) are complex numbers, and \(\alpha\) is an irrational real number.
02

Define and Analyze the Exponential Term

Set \(a = e^{2 \pi \alpha i}\). We need to show that the set \(\left\{a^{n} \in \mathbf{C} \mid n=1,2, \ldots\right\}\) is dense in the unit circle \(C=\{z \in \mathbf{C} \mid |z|=1\}\).
03

Show Density in the Unit Circle

Since \(\alpha\) is irrational, the argument \(2 \pi \alpha\) is an irrational multiple of \(\pi\). The powers of \(e^{2 \pi \alpha i}\) form a dense set on the unit circle because for any point on the unit circle, we can find an integer \(n\) such that \(\left(a^n \right)\) is arbitrarily close to it. This is a property of irrational rotations on the circle.
04

Define the Flow \(\phi_{t}\)

Let \(\phi_{t}\) be the flow of the system. Consider the solution of the differential equations given by: \(w(t) = w(0)e^{2\pi it}\), and \(z(t) = z(0)e^{2\pi \alpha it}\).
05

Find \(\phi_{n}\)

Evaluate the flow at integer time \(t = n\). For integer \(n\), we have: \(w(n) = w(0)e^{2 \pi i n} = w(0)\), and \(z(n) = z(0)e^{2 \pi \alpha i n} = z(0)a^n\). Thus, \(\phi_{n}(w, z) = (w, a^n z)\).
06

Use Results to Show \(\omega(\Gamma_{x_{0}})=T^{2}\)

Given \(x_{0}=(w_0, z_0)\) in \(T^2\), the set of points visited by \((w_0, z(t))\) as \(t\) varies is dense in \(T^2\), because the powers \(a^n\) are dense in the unit circle. Therefore, the omega-limit set of any trajectory passing through \(x_{0}\) is the entire torus \(T^2\).

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Key Concepts

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

Complex Dynamical Systems
The given exercise tackles concepts from the realm of complex dynamical systems. Complex dynamical systems refer to systems that evolve over time according to complex-valued functions. These systems are described using differential equations or iterative mappings. The main goals typically involve understanding how the system's state changes over time and predicting its long-term behavior.
In this context, the system involves complex coordinates \(w\) and \(z\), and their evolution is governed by linear differential equations. This system can exhibit complex behaviors such as periodicity, quasi-periodicity, or chaos, depending on the properties of the coefficients and initial conditions.
Linear systems are simpler but still display interesting dynamics when complex numbers are involved. Complex dynamical systems are essential for studying various phenomena in physics, engineering, and even biology, where growth and oscillatory behaviors can be modeled effectively.
Linear Differential Equations
Linear differential equations play a crucial role in understanding the flow on our torus. The equations given are:

\begin{aligned} \ \ \dot{w} &=2\pi iw\ \ \dot{z} &=2\pi\alpha iz \ \ \ \end{aligned}
These are first-order linear differential equations with complex coefficients, representing how the complex variables \(w\) and \(z\) evolve over time. Here, \(2\pi i\) and \(2\pi\alpha i\) essentially act as the growth rates, determining the rate and nature of oscillation.
Solving these equations involves integrating with respect to time. For example, if we solve for \(w(t)\), we get:
\[ w(t) = w(0)e^{2\pi it}\]
Similarly for \(z(t)\):
\[ z(t) = z(0)e^{2\pi\alpha it}\]
The solutions reveal that \(w\) oscillates with a period of \(((2\pi)/1)\), and \(z\) oscillates with a period of \(((2\pi)/\alpha)\). Given \alpha\ is irrational, \(z\)'s oscillations don't repeat, leading to dense sets on the unit circle.
Unit Circle Density
The density of the set \(\{a^n \ffer\ n=1,2, \ldots\}\) in the unit circle is pivotal for comprehending the flow on the torus. To understand this, remember that \(a = e^{2\pi\alpha i}\)\, where \alpha\ is an irrational number. Because \(\alpha \) is irrational, \(2\pi\alpha i\) is also irrational, causing the exponentiation to cover the unit circle densely.
This implies for any point on the unit circle\( C = \{z \in\mathbf{C}|\mid z\mid = 1\} \), there exists some integer \(n\) such that \(a^n\) gets arbitrarily close to that point. Ultimately, this ensures that the sequence cycles through values that come infinitely close to any complex number with magnitude 1. Therefore, powers of \(a\) densely populate the unit circle, ensuring that trajectories on a system such as the given torus exhaustively explore all possible points on \(T^2\).
Understanding this concept allows us to see why the omega-limit set is the whole torus.

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Most popular questions from this chapter

Let \(F\) satisfy the hypotheses of Lienard's Theorem. Show that $$ \ddot{z}+F(\dot{z})+z=0 $$ has a unique, asymptotically stable, periodic solution.

Consider the nonlinear system $$ \begin{aligned} &\dot{x}=-2 y+a x\left(4-4 x^{2}-y^{2}+z\right) \\ &\dot{y}=8 x+a y\left(4-4 x^{2}-y^{2}+z\right) \\ &\dot{z}=z\left(x-a^{2}\right) \end{aligned} $$ where \(a\) is a parameter. Show that \(\gamma(t)=(\cos 4 t, 2 \sin 4 t, 0)^{T}\) is a periodic solution of this system of period \(T=\pi / 2\). Determine the linearization of this system about the periodic orbit \(\gamma(t)\), $$ \dot{\mathbf{x}}=A(t) \mathbf{x} $$ and show that is the fundamental matrix for this nonautonomous linear system which satisfies \(\Phi(0)=I\), where \(\alpha(t)\) and \(\beta(t)\) are \(\pi / 2\)-periodic functions which satisfy the nonhomogeneous linear system $$ \begin{gathered} \left(\begin{array}{l} \dot{\alpha} \\ \dot{\beta} \end{array}\right)=\left[\begin{array}{cc} a(a-4)-\cos 4 t-4 a \cos 8 t & -2-2 a \sin 8 t \\ 8-8 a \sin 8 t & a(a-4)-\cos 4 t+4 a \cos 8 t \end{array}\right] \\ \cdot\left(\begin{array}{c} \alpha \\ \beta \end{array}\right)+\left(\begin{array}{c} a \cos 4 t \\ 2 a \sin 4 t \end{array}\right) \end{gathered} $$ and the initial conditions \(\alpha(0)=\beta(0)=0\). (This latter system can be solved using Theorem 1 and Remark 1 in Section \(1.10\) of Chapter 1 and the result of Problem 4 in Section \(2.2\) of Chapter \(2 ;\) however, this is not necessary for our purposes.) Write \(\Phi(t)\) in the form of equation (3), show that the characteristic exponents \(\lambda_{1}=-8 a\) and \(\lambda_{2}=-a^{2}\), determine the characteristic multipliers of \(\gamma(t)\), and determine the dimensions of the stable and unstable manifolds of \(\gamma(t)\) for \(a>0\) and for \(a<0\)

Consider the system $$ \begin{aligned} &\dot{x}=-y+x\left(1-x^{2}-y^{2}\right)\left(4-x^{2}-y^{2}\right) \\ &\dot{y}=x+y\left(1-x^{2}-y^{2}\right)\left(4-x^{2}-y^{2}\right) \\ &\dot{z}=z \end{aligned} $$ Show that there are two periodic orbits \(\Gamma_{1}\) and \(\Gamma_{2}\) in the \(x, y\) plane represented by \(\gamma_{1}(t)=(\cos t, \sin t)^{T}\) and \(\gamma_{2}(t)=(2 \cos t, 2 \sin t)^{T}\), and determine their stability. Show that there are two invariant cylinders for this system given by \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\); and describe the invariant manifolds \(W^{\prime}\left(\Gamma_{j}\right)\) and \(W^{u}\left(\Gamma_{j}\right)\) for \(j=1,2\).

Sketch the phase portrait and show that the interval \([-1,1]\) on the \(x\)-axis is an attracting set for the system $$ \begin{aligned} &\dot{x}=x-x^{3} \\ &\dot{y}=-y \end{aligned} $$ Is the interval \([-1,1]\) an attractor? Are either of the intervals \((0,1]\) or \([1, \infty)\) attractors? Are any of the infinite intervals \((0, \infty),[0, \infty)\), \((-1, \infty),[-1, \infty)\) or \((-\infty, \infty)\) on the \(x\)-axis attracting sets for this system?

Show that the nonlinear system $$ \begin{aligned} &\dot{x}=-y+x z^{2} \\ &\dot{y}=x+y z^{2} \\ &\dot{z}=-z\left(x^{2}+y^{2}\right) \end{aligned} $$ has a periodic orbit \(\gamma(t)=(\cos t, \sin t, 0)^{T}\). Find the linearization of this system about \(\gamma(t)\), the fundamental matrix \(\Phi(t)\) for this (autonomous) linear system which satisfies \(\Phi(0)=I\), and the characteristic exponents and multipliers of \(\gamma(t)\). What are the dimensions of the stable, unstable and center manifolds of \(\gamma(t)\) ?
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