Frequency locking in extended systems: The impact of a turing mode

PACS. 47.20.Ky – Nonlinearity (including bifurcation theory). PACS. 47.54.+r – Pattern selection; pattern formation. Abstract. – A Turing mode in an extended periodically forced oscillatory system can change the classical resonance boundaries of a single forced oscillator. Using the normal form equation for forced oscillations, we identify a Hopf-Turing bifurcation point around which we perform a weak nonlinear analysis. We show that resonant standing waves can exist outside the 2: 1 resonance region of uniform oscillations, and non-resonant mixed-mode oscillations may prevail inside the resonance region. An oscillator, periodically forced in time, can adjust its oscillation frequency to make it a rational fraction of the forcing frequency [1]. At very low forcing amplitudes, the forced oscilla-tor exhibits quasi-periodic oscillations consisting of the forcing frequency and the frequency of the unforced oscillator. Frequency adjustment, or frequency locking, occurs when the forcing amplitude exceeds a threshold value that depends on the forcing frequency. In the parameter plane spanned by the forcing amplitude and forcing frequency, these thresholds form tongue-shaped regions, the so-called Arnol’d tongues. Each tongue corresponds to a different rationa