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Chapter 8: Problem 5

(Plotting Lissajous figures) Using a computer, plot the curve whose parametric equations are \(x(t)=\sin t, y(t)=\sin \omega t\), for the following rational and irrational values of the parameter \(\omega\) : (a) \(\omega=3\) (b) \(\omega=\\}\) (c) \(\omega=\frac{5}{3}\) (d) \(\omega=\sqrt{2}\) (e) \(\omega=\pi\) (f) \(\omega=\frac{1}{2}(1+\sqrt{5})\). The resulting curves are called Lissajous figures. In the old days they were displayed on oscilloscopes by using two ac signals of different frequencies as inputs.

Short Answer

Expert verified
Plot Lissajous figures for given values of \(\omega\), using the parametric equations \(x(t) = \sin t\) and \(y(t) = \sin \omega t\). For example, when \(\omega = 3\), plot the curve with coordinates \(\sin t\) and \(\sin (3t)\). Use a graphing tool to visualize the curves, and observe how their shapes depend on the frequency ratios. Lissajous figures are used in oscilloscopes to display the relationship between two AC signals with different frequencies.

Step by step solution

01

1. Set up the coordinates and the parameter t

Define the parameter t within a certain range, for example \(t \in [0, 2\pi]\). Then, specify the functions \(x(t) = \sin t\) and \(y(t) = \sin \omega t\) for each selected value of \(\omega\).
02

2. Ploting the Lissajous figure for \(\omega = 3\)

Replace \(\omega\) with 3 in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (3t)\). Plot the resulting curve using a graphing tool.
03

3. Ploting the Lissajous figure for \(\omega = \frac{5}{3}\)

Replace \(\omega\) with \(\frac{5}{3}\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\frac{5}{3}t)\). Plot the resulting curve using a graphing tool.
04

4. Ploting the Lissajous figure for \(\omega = \sqrt{2}\)

Replace \(\omega\) with \(\sqrt{2}\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\sqrt{2}t)\). Plot the resulting curve using a graphing tool.
05

5. Ploting the Lissajous figure for \(\omega = \pi\)

Replace \(\omega\) with \(\pi\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\pi t)\). Plot the resulting curve using a graphing tool.
06

6. Ploting the Lissajous figure for \(\omega = \frac{1}{2}(1+\sqrt{5})\)

Replace \(\omega\) with \(\frac{1}{2}(1+\sqrt{5})\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\frac{1}{2}(1+\sqrt{5})t)\). Plot the resulting curve using a graphing tool.
07

7. Lissajous figures and oscilloscopes

Lissajous figures are used in oscilloscopes to visually display the relationship between two AC signals with different frequencies. By plotting these figures for various values of \(\omega\), we can observe how the shape of the curve depends on the ratio of frequencies.

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Key Concepts

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

Parametric Equations
Parametric equations are a way of representing mathematical functions where each axis is expressed as a function of one or more parameters, rather than a single variable. In the context of Lissajous Figures, parametric equations express the coordinates of the curve in terms of a parameter, usually denoted as \( t \). The equations for Lissajous figures are \( x(t) = \sin t \) and \( y(t) = \sin(\omega t) \). Here, \( t \) is the parameter that denotes time or a certain angle progression, and \( \omega \) is a variable influencing the curve's frequency.

These parametric equations allow us to create complex and often beautiful curves by changing the value of \( \omega \). These changes affect the trajectory traced out by the function in the \( xy \)-plane, leading to the unique figures recognized as Lissajous figures. Understanding parametric equations helps in visualizing the path taken by these figures over time by tracing out both the x and y components simultaneously. With Lissajous figures, different values of \( \omega \) yield varied patterns, capturing the essence of the relationship between x and y as they vary with \( t \).
Oscilloscopes
An oscilloscope is an electronic test instrument that graphically displays varying signal voltages. Historically, Lissajous figures were used with oscilloscopes to analyze the relationship between two periodic signals, usually for determining their frequency and phase differences.

By inputting two sinusoidal signals of different frequencies into the oscilloscope, each having its own channel, the display would intricately plot their interaction in real-time. The x-component of the signal would drive the horizontal axis, and the y-component would drive the vertical one. The resulting Lissajous figure represented the continuous interaction between the two signals.

This technique provided a practical visual method for measuring frequency ratios and assessing phase differences between signals without needing complicated equipment or calculations beyond the simple plotting on the oscilloscope screen. The shapes helped technicians and scientists easily interpret what was happening with the signal properties, making it a valuable tool in both educational and professional settings.
Frequency Ratio
The frequency ratio in Lissajous figures is a crucial element in determining the resulting pattern of the figure. This ratio is defined by the frequencies of the two sinusoidal inputs. For the parametric equations \( x(t) = \sin t \) and \( y(t) = \sin(\omega t) \), the frequency ratio is \( 1:\omega \).

If \( \omega \) is a rational number (such as \( \omega = 3 \)), the resulting figure will close and repeat after a complete cycle. This closure indicates a stable, repeated pattern like a simple loop or figure-eight shape. However, when the frequency ratio becomes irrational (values such as \( \omega = \sqrt{2} \)), the Lissajous figure does not repeat perfectly, resulting in what appears as a more chaotic or densely filled pattern.

By adjusting this frequency ratio, different Lissajous figures represent the harmonious or discordant interaction of the two signals. In the context of real-world applications, understanding frequency ratios aids in tasks such as tuning musical instruments or studying waveforms in physics and engineering.

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Most popular questions from this chapter

(Bacterial respiration) Fair茅n and Velarde (1979) considered a model for respiration in a bacterial culture. The equations are $$ \dot{x}=B-x-\frac{x y}{1+q x^{2}}, \quad \dot{y}=A-\frac{x y}{1+q x^{2}} $$ where \(x\) and \(y\) are the levels of nutrient and oxygen, respectively, and \(A, B, q>0\) are parameters. Investigate the dynamics of this model. As a start, find all the fixed points and classify them. Then consider the nullclines and try to construct a trapping region. Can you find conditions on \(A, B, q\) under which the system has a stable limit cycle? Use numerical integration, the Poincar茅-Bendixson theorem, results about Hopf bifurcations, or whatever else seems useful. (This question is deliber- ately open-ended and could serve as a class project; see how far you can go.)

Consider the system \(\dot{x}=y-2 x, \dot{y}=\mu+x^{2}-y\). a) Sketch the nullclines. b) Find and classify the bifurcations that occur as \(\mu\) varies. c) Sketch the phase portrait as a function of \(\mu\).

(Logistic equation with periodically varying carrying capacity) Consider the logistic equation \(\dot{N}=r N(1-N / K(t))\), where the carrying capacity is positive, smooth, and \(T\)-periodic in \(t .\) a) Using a Poincar茅 map argument like that in the text, show that the system has at least one stable limit cycle of period \(T\), contained in the strip \(K_{\min } \leq N \leq K_{\max ^{\prime}}\) b) Is the cycle necessarily unique?

By calculating the linearization at the origin, show that the system \(\dot{x}=-y+\mu x+x y^{2}, \dot{y}=x+\mu y-x^{2}\) has pure imaginary eigenvalues when \(\mu=0\).

(Budworms vs. the forest) Ludwig et al. (1978) proposed a model for the effects of spruce budworm on the balsam fir forest. In Section 3.7, we considered the dynamics of the budworm population; now we turn to the dynamics of the forest. The condition of the forest is assumed to be characterized by \(S(t)\), the average size of the trees, and \(E(t)\), the "energy reserve" (a generalized measure of the forest's health). In the presence of a constant budworm population \(B\), the forest dynamics are given by $$ \dot{S}=r_{5} S\left(1-\frac{S}{K_{5}} \frac{K_{E}}{E}\right), \quad \bar{E}=r_{E} E\left(1-\frac{E}{K_{\bar{E}}}\right)-P \frac{B}{S} $$ where \(r_{5}, r_{E}, K_{5}, K_{E}, P>0\) are parameters. a) Interpret the terms in the model biologically. b) Nondimensionalize the system. c) Sketch the nullclines. Show that there are two fixed points if \(B\) is small, and none if \(B\) is large. What type of bifurcation occurs at the critical value of \(B\) ? d) Sketch the phase portrait for both large and small values of \(B\).
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