MATLAB Project 2
NJIT Math 222
Problem 1
Consider the differential equation
\[ \frac{dx}{dt}= f(x) = \sin{x}-a \sin{4x} \]This equation has period \(2\pi\), so we will consider it for \(-\pi\le x \le \pi. \) Things will be clearer if we plot instead on a slightly larger interval such as \(-4\le x \le 4\). When \(a=0\), this equation has two equilibria per period, at \(x=0\) and \(x=\pi\) (considering \(x=\pm\pi\) as the same point). When \(a\) is large (say \(a=4\)) then the graph of \(f(x)\) looks pretty much like \(-a\sin{4x}\) (mathematicians would say the second term dominates), and the system has 8 fixed points per period.
Starting with \(a=0\) plot the direction field for several increasing values of \(a\). You should find that at a certain value of \(a\) satisfying \(0.2<a<0.3\), two new equilibria appear and at a second critical value satisfying \(0.8<a<1\), four more equilibria appear.
Print out enough graphs to describe what happens in each case and describe what you observe in a sentence or two. Pay attention to the changes in the neighborhood of the equilibrium \(x=0\). Hint: it is useful to plot \(f(x)\) for various values of \(a\) as well. The website and apps from Desmos are easy to use and output beautiful graphics.
Problem 2
Graph the direction field and some solutions for the non-autonomous equation
\[ \frac{dx}{dt}=x - \frac{t x}{1+x^2}. \]It is up to you to pick the range over which you plot in \(x\) and \(t\) in order to see the behavior.
There is a finite number of things that can happen as \(t\to\infty\). Describe them.