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

TL;DR

This paper investigates how spatial heterogeneity in oscillatory systems can lead to novel resonance tongue shapes, including inverse camel shapes, using the complex Ginzburg-Landau equation and numerical simulations.

Contribution

It reveals that spatial heterogeneity causes Arnol'd tongues to form 'U' and 'W' shapes, expanding understanding of frequency locking in heterogeneous oscillatory media.

Findings

Heterogeneity induces inverse camel-shaped Arnol'd tongues.

Linear stability analysis explains the formation of these shapes.

Results are applicable to various physical and biological systems.

Abstract

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 spatially extended and monotonically heterogeneous, e.g., through spatially-dependent natural frequency, the resonance tongues can also display 'U' and 'W' shapes; to the latter, we refer as "inverse camel" shape. We study the generic forced complex Ginzburg-Landau equation for damped oscillations under parametric forcing and, using linear stability analysis and numerical simulations, uncover the mechanisms that lead to these distinct shapes. Additionally, we study the effects of discretization, by exploring frequency locking of oscillators chains. Since we study a normal-form equation, the results are model-independent near the onset of oscillations, and, therefore, applicable to inherently heterogeneous systems in general, such as the cochlea. The results are also applicable to controlling technological performances in various contexts, such as arrays of mechanical resonators, catalytic surface reactions, and nonlinear optics.