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In the course of doing some physics I've encountered serveral systems of the form $$a''= -f(b,t) b'\\ b''=f(b,t) a'$$ where prime denotes derivative in $t$ and $f$ is maybe a polynomial or e.g. $\cos (a-t)$. I know that it's a non-linear system and I shouldn't expect exact soultions, but I would like to know what sort of methods I might use to obtain analytic approximations (I've successfully used Picard iteration in the linear case $f(t)$) and maybe some indication as to why they are hard to solve. I think but haven't proven that one gets chaotic dynamics even for simple $f$ like $\cos (b)$.

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  • $\begingroup$ Do you think it's possible to reduce the order of the system with a substitution like $x=a'$ and $y=b'$? This is one way of working with higher-order systems of ODEs. $\endgroup$ – user170231 Oct 2 '14 at 17:19
  • $\begingroup$ First of all you immediately have constraint : $a'^2+b'^2=const$. So probably it is more natural to make change of variables:$a'(t)=r \cos (w(t) t), b'(t)=r \sin(w(t) t)$ $\endgroup$ – Alexander Vigodner Oct 2 '14 at 18:04
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You can decouple them simply by noticing that by integration by parts, $$ \int_0^t s b^{ \prime \prime } (s) ds = tb^{\prime}(t) - \int_0^t b^{\prime}(s)ds = t b^\prime (t) - b(t) + b(0), $$ so that, using the equation for $( a^{\prime\prime }, b^{\prime \prime })$ it follows that $$ b(t) = b(0) + tb^\prime (t) - \int_0^t s f( b(s),s ) a^\prime(s)ds, \text{ and } a^\prime (t) = a^\prime (0 ) - \int_0^t f(b(s),s) b^\prime (s) ds. $$ So putting these two together, $$ b(t) = b(0) + tb^\prime(t) - a^\prime(0)\int_0^tsf(b(s),s)ds+ \int_0^t\int_0^ssf(b(s),s)f(b(r),r)b^\prime(r)drds, $$ which, after taking the derivative implies that for every $t$, $$ 0 = tb^{\prime \prime }(t) - a^\prime(0)tf(b(t),t) + tf(b(t),t)\int_0^tf(b(s),s)b^\prime(s)ds. $$ Since $t$ was arbitrary, it follows that $$ b^{\prime \prime} (t) = a^\prime (0 ) f( b(t), t ) + f( b(t), t ) \int_0^tf(b(s),s)b^\prime(s)ds. $$ At this point I don't see any general tricks and it might depend on the function $f$. I am curious to see what folks come up with.

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