Articles | Volume 15
21 Sep 2017
 | 21 Sep 2017

Discretization analysis of bifurcation based nonlinear amplifiers

Sven Feldkord, Marco Reit, and Wolfgang Mathis

Abstract. Recently, for modeling biological amplification processes, nonlinear amplifiers based on the supercritical Andronov–Hopf bifurcation have been widely analyzed analytically. For technical realizations, digital systems have become the most relevant systems in signal processing applications. The underlying continuous-time systems are transferred to the discrete-time domain using numerical integration methods. Within this contribution, effects on the qualitative behavior of the Andronov–Hopf bifurcation based systems concerning numerical integration methods are analyzed. It is shown exemplarily that explicit Runge–Kutta methods transform the truncated normalform equation of the Andronov–Hopf bifurcation into the normalform equation of the Neimark–Sacker bifurcation. Dependent on the order of the integration method, higher order terms are added during this transformation.

A rescaled normalform equation of the Neimark–Sacker bifurcation is introduced that allows a parametric design of a discrete-time system which corresponds to the rescaled Andronov–Hopf system. This system approximates the characteristics of the rescaled Hopf-type amplifier for a large range of parameters. The natural frequency and the peak amplitude are preserved for every set of parameters. The Neimark–Sacker bifurcation based systems avoid large computational effort that would be caused by applying higher order integration methods to the continuous-time normalform equations.

Short summary
Recently, for modeling biological amplification processes, nonlinear amplifiers based on the Andronov–Hopf bifurcation have become a focus of attention. In this contribution, we analyze discrete-time implementations of this type of amplifiers. The effects of the time-discretization by explicit integration methods are discussed. A novel discrete-time system based on the Neimark–Sacker bifurcation is introduced, that outstandingly approximates the behavior of a particular Hopf-type amplifier.