Ambiguities in input-output behavior of driven nonlinear systems close to bifurcation

Open access

Abstract

Since the so-called Hopf-type amplifier has become an established element in the modeling of the mammalian hearing organ, it also gets attention in the design of nonlinear amplifiers for technical applications. Due to its pure sinusoidal response to a sinusoidal input signal, the amplifier based on the normal form of the Andronov-Hopf bifurcation is a peculiar exception of nonlinear amplifiers. This feature allows an exact mathematical formulation of the input-output characteristic and thus deeper insights of the nonlinear behavior. Aside from the Hopf-type amplifier we investigate an extension of the Hopf system with focus on ambiguities, especially the separation of solution sets, and double hysteresis behavior in the input-output characteristic. Our results are validated by a DSP implementation.

[1] Szalai R., Champneys A., Homer M., Maoileidigh D.O., Kennedy H., Cooper N., On the origins of the compressive cochlear nonlinearity, Working Paper (2011).

[2] Ruggero M.A., Responses to sound of the basilar membrane of the mammalian cochlea, Current Opinion in Neurobiology 2: 449-456 (1992).

[3] Eguíluz V.M., Ospeck M., Choe Y., Hudspeth A.J., Magnasco M.O., Essential nonlinearities in hearing, Physical Review Letters 84: 5232-5235 (2000).

[4] Stoop R., Jasa T., Uwate Y., Martignoli S., From Hearing to Listening: Design and Properties of an Actively Tunable Electronic Hearing Sensor, Sensors 7: 3287-3298 (2007).

[5] Reit M., Mathis W., Stoop R., Time-Discrete Nonlinear Cochlea Model Implemented on DSP for Auditory Studies, Proceedings of the 20th International Conference on Nonlinear Dynamics of Electronic Systems (NDES 2012).

[6] Camalet S., Duke T., Jülicher F., Prost J., Auditory sensitivity provided by self-tuned critical oscillations of hair cells, Proceedings of the National Academy of Sciences 97(7): 3183-3188 (2000).

[7] Montgomery K.A., Silber M., Solla S.A., Amplification in the auditory periphery: the effect of coupled tuning mechanisms, Physical Review E 75: 051924 (2007).

[8] Reit M., Stoop R., Mathis W., Analysis of Cascaded Canonical Dissipative Systems and LTI Filter Sections, Acta Technica 59(1): 79-96 (2014).

[9] Reit M., Mathis W., A Class of Sinusoidal Driven Nonlinear Input-Output Systems with Sinusoidal Response, International Symposium on Nonlinear Theory and its Applications (2014).

[10] Kuznetzov Y.A., Elements of Applied Bifurcation Theory, Springer (2004).

[11] Zhang Y., Golubitsky M., Periodically Forced Hopf Bifurcation, SIAM Journal on Applied Dynamical Systems 10(4): 1272-1306 (2011).

Archives of Electrical Engineering

The Journal of Polish Academy of Sciences

Journal Information


CiteScore 2016: 0.71

SCImago Journal Rank (SJR) 2016: 0.238
Source Normalized Impact per Paper (SNIP) 2016: 0.535

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