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Register a new userSpatial heterogeneity may form an inverse camel shape Arnold tongue in parametrically forced oscillations
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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 Arnold 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.
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Spatially localized oscillations in periodically forced systems are intriguing phenomena. They may occur in spatially homogeneous media (oscillons), but quite often emerge in heterogeneous media, such as the auditory system, where localized oscillations are believed to play an important role in frequency discrimination of incoming sound waves. In this paper, we use an amplitude-equation approach to study the spatial profile of the oscillations and the factors that affect it. More specifically, we use a variant of the forced complex Ginzburg-Landau (FCGL) equation to describes an oscillatory system below the Hopf bifurcation with space-dependent Hopf frequency, subject to both parametric and additive forcing. We show that spatial heterogeneity, combined with bistability of system states, results in spatial asymmetry of the localized oscillations. We further identify parameters that control that asymmetry, and characterize the spatial profile of the oscillations in terms of maximum amplitude, location, width and asymmetry. Our results bear qualitative similarities to empirical observation trends that have found in the auditory system.
Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially-coupled oscillators: the complex Ginzburg-Landau equation (CGLE) and a generalization of the CGLE in which oscillators are coupled through an external medium (emCGLE). We focus on external control drives that vary in both space and time. We find that the spatial distribution of the drive signal controls the frequency ranges over which oscillators synchronize to the drive and that boundary conditions strongly influence synchronization to external drives for the CGLE. Our calculations also show that the emCGLE has a low density regime in which a broad range of frequencies can be synchronized for low drive amplitudes. We study the bifurcation structure of these models and find that they are very similar to results for the driven Kuramoto model, a system with no spatial structure. We conclude by discussing the implications of our results for controlling coupled oscillator systems such as the social amoebae emph{Dictyostelium} and populations of BZ catalytic particles using spatially structured external drives.
We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as `strangely dispersed. Along the way, we generalise some well-known results about circle endomorphisms to the uniquely ergodically forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically forced circle endomorphism are realised on minimal sets, and if the rotation interval has non-empty interior then the topological entropy is strictly positive. The results apply in particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm example.
The emergence of localised vibrations in cyclic and symmetric rotating structures, such as bladed disks of aircraft engines, has challenged engineers in the past few decades. In the linear regime, localised states may arise due to a lack of symmetry, as for example induced by inhomogeneities. However, when structures deviate from the linear behaviour, e.g. due to material nonlinearities, geometric nonlinearities like large deformations, or other nonlinear elements like joints or friction interfaces, localised states may arise even in perfectly symmetric structures. In this paper, a system consisting of coupled Duffing oscillators with linear viscous damping is subjected to external travelling wave forcing. The system may be considered a minimal model for bladed disks in turbomachinery operating in the nonlinear regime, where such excitation may arise due to imbalance or aerodynamic excitation. We demonstrate that near the resonance, in this non-conservative regime, localised vibration states bifurcate from the travelling waves. Complex bifurcation diagrams result, comprising stable and unstable dissipative solitons. The localised solutions can also be continued numerically to a conservative limit, where solitons bifurcate from the backbone curves of the travelling waves at finite amplitudes.
We address the nonlinear inverse source problem of identifying a time-dependent source occurring in one node of a network governed by a wave equation. We prove that time records of the associated state taken at a strategic set of two nodes yield uniqueness of the two unknown elements: the source position and the emitted signal. We develop a non-iterative identification method that localizes the source node by solving a set of well posed linear systems. Once the source node is localized, we identify the emitted signal using a deconvolution problem or a Fourier expansion. Numerical experiments on a $5$ node graph confirm the effectiveness of the approach.
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