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.
