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Frequency locking in forced oscillatory systems typically occurs in'V'-shaped domains in the plane spanned by the forcing frequency and amplitude,the so-called Arnol'd tongues. Here, we show that if the medium is spatiallyextended and monotonically heterogeneous, e.g., through spatially-dependentnatural frequency, the resonance tongues can also display 'U' and 'W' shapes;to the latter, we refer as "inverse camel" shape. We study the generic forcedcomplex Ginzburg-Landau equation for damped oscillations under parametricforcing and, using linear stability analysis and numerical simulations, uncoverthe mechanisms that lead to these distinct shapes. Additionally, we study theeffects of discretization, by exploring frequency locking of oscillatorschains. Since we study a normal-form equation, the results aremodel-independent near the onset of oscillations, and, therefore, applicable toinherently heterogeneous systems in general, such as the cochlea. The resultsare also applicable to controlling technological performances in variouscontexts, such as arrays of mechanical resonators, catalytic surface reactions,and nonlinear optics.

Spatial heterogeneity may form an inverse camel shape Arnol'd tongue in parametrically forced oscillations