9
$\begingroup$

The harmonically driven damped pendulum is often used as a simple example of a chaotic system, the equation is just $\ddot{\phi}+\frac1q\dot{\phi}+\sin\phi=A\cos(\omega t)$. As long as $A$ and $\omega$ are small it behaves like a driven harmonic oscillator, and asymptotically settles into regular oscillations with a fixed period. However, as $A$ (or $\omega$) are increased, with the rest of parameters fixed, the system undergoes a cascade of period doubling bifurcations leading to chaotic behavior, which then gives way to regular oscillations again when it is increased further. For example, when $q=2$ and $\omega=2/3$ the first period doubling ("symmetry breaking") occurs at $A\approx1.07$ and the first chaos at $A\approx1.08$.

Unfortunately, these results seem to be obtained by numerical simulations. I am actually interested in situations where chaos does not occur. Are there known rigorous conditions on $A,\omega$ and $q$ that put the system below the first period doubling?

The most common references are to Baker-Gollub's Chaotic Dynamics and Baker-Blackburn's The Pendulum that have a lot of phase diagrams with attractors, but no theorems. From what I understand there is Melnikov's method for detecting homoclinic bifurcations rigorously (e.g. Wiggins's Global Bifurcations and Chaos), but I could not find it used to obtain this kind of result for the pendulum. I searched for papers on driven damped pendulum on MathSciNet, but they seem to use physics-style approach and/or numerical simulations. A paper that gives more of an analytic insight by perturbation methods (treating $A$ as a small parameter) is Miles, Resonance and symmetry breaking for the pendulum, Physica D: Nonlinear Phenomena, Volume 31, Issue 2, June 1988, Pages 252-268.

$\endgroup$
4
  • $\begingroup$ Do you want the exact threshold or just not too weak sufficient condition that ensures the good behavior? $\endgroup$
    – fedja
    Aug 13, 2020 at 21:59
  • $\begingroup$ @fedja Just a lower estimate would be good. I am doing a control problem where one needs to select parameter values that induce stable regular oscillations, and it reduces to DDP. $\endgroup$
    – Conifold
    Aug 13, 2020 at 22:03
  • $\begingroup$ If you want to control a real system, why are simulations not good enough for you? You might also want to look into the shadowing lemma. $\endgroup$
    – Wrzlprmft
    Aug 14, 2020 at 8:22
  • 1
    $\begingroup$ @Wrzlprmft Because it is for a class of systems over a range of parameters. We want to induce stable oscillations with desired period and amplitude by specifying $A,\omega$ and $q$, and to know when it is possible we need to estimate the single period region of the parameter values and how they determine the induced values there. Thanks for the tip. $\endgroup$
    – Conifold
    Aug 15, 2020 at 8:18

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.