Nonlinear Dynamics

Question

Good afternoon,

I am working on Nonlinear Dynamics and Chaos by Strogatz $2$nd Edition.

I am having trouble with problem $2.14$ on pg $36$.

The question is about the analytic solution of $\frac{dx}{dt} = \sin x$. We are given the $x(t)$ as follows:

 x(t) = 2arctan(e^t / 1+2^1/2 )

We are asked to derive this from

                      t = ln abs[ csc(x_0) + cot(x_0) / cscx + cotx ]

For $x_0 = \frac{pi}{4}$

My problem is how to deal with this problem at $x_0$ = integer multiples of $\pi$

Also, how do I get rid of the absolute value sign when attempting to solve for $x$ in terms of $t$.

Lastly, how would you do the same problem for arbitrary initial value $x_0$.

Thank you!

Answer 1

• You're missing a logarithm from your expression for $\ t\ $, which should be $$ t=\ln\left|\frac{\csc x_0+\cot x_0}{\csc x + \cot x}\right| $$

• Your first equation $$ x=2\arctan\left(\frac{e^t}{1+\sqrt{2}}\right) $$ is the solution for $\ x_0=\frac{\pi}{4}\ $. You're not asked to derive that from the expression for $\ t\ $ with arbitrary $\ x_0\ $, but merely to show that it is the solution when $\ x_0=\frac{\pi}{4}\ $.

• Since $\ e^t>0\ $, then if $\ x\ $ satisfies the equation $$ e^t= \frac{\csc x_0+\cot x_0}{\csc x + \cot x} $$ it will automatically satisfy the equation $$ e^t=\left|\frac{\csc x_0+\cot x_0}{\csc x + \cot x}\right|\ . $$ So, provided any solution you propose satisfies the first of these equations, you can effectively ignore the absolute value signs.

• Part (b) of the question does ask you to find an "analytic solution" for $\ x(t)\ $ given an arbitrary initial value $\ x_0\ $. If $\ x_0\ $ is an integer multiple of $\ \pi\ $, then $\ \sin x_0=0\ $, and the solution of the differential equation is $\ x(t)=0\ $, for all $\ t\ $. After you've derived an analytic expression for $\ x(t)\ $ for $\ x_0\ $ not an integer multiple of $\ \pi\ $, you should be able to write it in a form such that when you substitute $\ x_0=2n\pi\ $ in that expression, where $\ n\ $ is an integer, it will give you $\ x(t)=0\ $ anyway, so the expression will in fact then be valid for those cases as well.

  However, since I think the expression must contain a term of the form $\ \tan\frac{x_0}{2}\ $ (or equivalent), which is undefined when $\ x_0\ $ is an odd multiple of $\ \pi\ $, the general formula will not be valid for those cases.