# Plotting the orbit of an ODE

I'm looking to see how I can plot the following ODE orbit:

$$\phi(t) =\begin{pmatrix} \sin(2t) \\ \cos(t) \end{pmatrix}$$

The author of my ODE textbook has recommended that I use a vector plot. The textbook has an example for

VectorPlot[{y, −Sin[x]}, {x, −2π, 2π}, {y, −5, 5}]


To plot the above orbit, would I do a vector plot similar to

VectorPlot[{Sin[2t], Cos[t]}, {x, −2π, 2π}, {y, −5, 5}]


This doesn't appear to work when I test it in Wolfram|Alpha.

• Any reference to the book? – zhk Nov 10 '18 at 4:21
• I included the book as a hyperlink. The book is ODE by Teschl. He goes through the VectorPlot in chapter 6.2. – Axion004 Nov 10 '18 at 4:23
• Notice the example used functions of x and y given ranges of x and y. Then you want to plot functions of t given ranges of x and y. Can you see the disconnect? This VectorPlot[{Sin[2 x], Cos[y]}, {x, -2 Pi, 2 Pi}, {y, -5, 5}] will plot using WolframAlpha, but I cannot be certain that is exactly what you want. Can you turn what you want to functions of x and y? Or turn everything into a function of t? – Bill Nov 10 '18 at 5:14
• Yes, set $x(t)=sin(2t)$ and $y(t)=cos(t)$. This would produce a two dimensional system for $\frac{dx}{dt} = f(x(t),y(t))$ and $\frac{dy}{dt} = g(x(t),y(t))$. The phase plot of the system could be drawn by ParametricPlot[{Sin[2 t], Cos[t]}, {t, -50, 50}]. – Axion004 Nov 10 '18 at 18:56