Adv. Radio Sci., 15, 43–47, 2017https://doi.org/10.5194/ars-15-43-2017© Author(s) 2017. This work is distributed underthe Creative Commons Attribution 3.0 License.

Discretization analysis of bifurcation based nonlinear amplifiersSven Feldkord, Marco Reit, and Wolfgang MathisInstitute of Theoretical Electrical Engineering, Leibniz Universität Hannover, Appelstraße 9A, 30167 Hannover, Germany

Correspondence to: Marco Reit (reit@tet.uni-hannover.de)

Received: 16 January 2017 – Accepted: 5 March 2017 – Published: 21 September 2017

Abstract. Recently, for modeling biological amplificationprocesses, nonlinear amplifiers based on the supercriticalAndronov–Hopf bifurcation have been widely analyzed ana-lytically. For technical realizations, digital systems have be-come the most relevant systems in signal processing appli-cations. The underlying continuous-time systems are trans-ferred to the discrete-time domain using numerical integra-tion methods. Within this contribution, effects on the qual-itative behavior of the Andronov–Hopf bifurcation basedsystems concerning numerical integration methods are an-alyzed. It is shown exemplarily that explicit Runge–Kuttamethods transform the truncated normalform equation of theAndronov–Hopf bifurcation into the normalform equation ofthe Neimark–Sacker bifurcation. Dependent on the order ofthe integration method, higher order terms are added duringthis transformation.

A rescaled normalform equation of the Neimark–Sackerbifurcation is introduced that allows a parametric design ofa discrete-time system which corresponds to the rescaledAndronov–Hopf system. This system approximates the char-acteristics of the rescaled Hopf-type amplifier for a largerange of parameters. The natural frequency and the peakamplitude are preserved for every set of parameters. TheNeimark–Sacker bifurcation based systems avoid large com-putational effort that would be caused by applying higher or-der integration methods to the continuous-time normalformequations.

1 Introduction

Early research has shown that the mammalian hearing pro-cess is active (Kemp, 1978) and involves an active ampli-fication within the cochlea (Zenner and Gitter, 1987). Thisamplification characteristic can be modeled by a system nearthe onset of an Andronov–Hopf bifurcation (Eguíluz et al.,

2000). Regarding this bifurcation based amplifier and af-ter first analog realizations by Stoop et al. (2005, 2007) adiscrete-time system was developed and implemented on adigital signal processor (DSP) (Reit et al., 2012). This im-plementation was latterly improved and extended to com-pare nonlinear amplifiers based on Andronov–Hopf andNeimark–Sacker bifurcations (Feldkord et al., 2016). TheNeimark–Sacker bifurcation is also called the Andronov–Hopf bifurcation for maps (Kuznetsov, 2013).

In this work we analyze the influence of explicit single-step integration methods on the behavior of systems based onthe normalform equation of the Andronov–Hopf bifurcation.This is done exemplarily for the explicit Euler method andextended to explicit Runge–Kutta methods. Lastly, a rescalednormalform equation of the Neimark–Sacker bifurcation isderived that approximates the characteristics of the rescaledAndronov–Hopf bifurcation within the discrete-time domainwhile preserving the natural frequency and the peak ampli-tude.

2 Fundamentals of the Andronov–Hopf and theNeimark–Sacker bifurcation

The analysis in this work focusses on the truncatednormalform equation of the Andronov–Hopf bifurcation(Kuznetsov, 2013)

dzdt= (µ+ iω0)z+ σ |z|2z, (1)

where z ∈ C is the system variable, µ ∈ R the bifurcationparameter, i the imaginary unit, σ ∈ C the first nonlinear-ity coefficient that determines the type of the bifurcation andω0 ∈ R the natural frequency. For 0 from astable fixed point (µ≤ 0) that loses stability. On the contraryfor 0, the system exhibits an unstable limit cycle for

Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.

44 S. Feldkord et al.: Discretization analysis of bifurcation based nonlinear amplifiers

µ < 0 that shrinks and disappears at the point of bifurcation(µ= 0) where the encircled stable fixed point loses stability.Here, for µ≥ 0, the system has an unstable fixed point. Aspecial characteristic of Eq. (1) is the mapping of an additivesinusoidal excitation signal a0eiωt to a sinusoidal output sig-nal z0ei(ωt+φ) without harmonic distortions (Eguíluz et al.,2000; Stoop et al., 2005; Reit et al., 2016).

Similar to the normalform equation of the Andronov–Hopfbifurcation in the continuous-time domain, the truncated nor-malform equation of the Neimark–Sacker bifurcation in thediscrete-time domain (Kuznetsov, 2013),

z 7−→ eiθz(

1+α+ d|z|2), (2)

is analyzed. As before, z is the complex system variable. Thereal-valued bifurcation parameter is denoted by α. Similar toσ of the Andronov–Hopf system, the type of bifurcation isdependent on the sign of 0, the super-critical case results in synchronization problems (Pikovskyet al., 2003). The subcritical case is unstable and thereforeneglected. For modeling the cochlea, a ω0-rescaled truncatednormalform equation of the Andronov–Hopf bifurcation withan added excitation term a(t),

dzdt= ω0 (µ+ i)z+ω0σ |z|2z+ω0a(t), (3)

was introduced by Stoop et al. (2007). The special propertyof Eq. (3) is that the amplitude response to a sinusoidal ex-citation signal a(t)= a0eiωt is independent of the absolutevalue of the natural frequency. Here, the fraction ω/ω0 de-termines the output amplitude for a given set of µ, σ anda0. Recently, the rescaled truncated normalform equation ofthe Andronov–Hopf bifurcation with excitation was imple-mented on a DSP platform (Reit et al., 2012). In a compara-tive study of the amplitude responses, the truncated normal-form equation of the Neimark–Sacker bifurcation with theexcitation a(n),

zn+1 = eiθzn

(1+α+ d|zn|2

)+ a(n), (4)

was implemented and analyzed for a(n)= a0eiωhn (Feldkordet al., 2016). It can be shown, that the input-output behaviorof the Neimark–Sacker system is very similar to that of thenot rescaled Andronov–Hopf system Eq. (1). Thus, to ap-proximate the behavior of the ω0-rescaled Andronov–Hopfsystem, an appropriate normalform equation of the Neimark–Sacker bifurcation is desirable which is introduced later inthis work.

Since we examine the purpose of digital processing appli-cations, where real-time processing and low latency are themain goals, only explicit integration methods are consideredto implement the Andronov–Hopf bifurcation based systems.The aforementioned DSP-Implementation uses the Runge–Kutta method of 4th order.

3 Analysis of the numerically integrated normalformequation of the Andronov–Hopf bifurcation

The following section focusses on the qualitative changesof the system behavior due to a discrete-time implementa-tion. We use the rescaled truncated normalform equation ofthe Andronov–Hopf bifurcation in Eq. (3). The intuitive ap-proach is to transform Eq. (3) into a system in the discrete-time domain using a numerical integration method. Then, theresulting difference equation can be analyzed. Due to thenonlinearity of the equation, mixed terms of the state vari-able and the excitation term can occur. This leads to compli-cated expressions that makes an analytical approach almostimpossible. Therefore, the following analysis focusses on theAndronov–Hopf system in Eq. (3) without the excitation.

To illustrate the transformation of the system parametersdue to an integration method, the explicit Euler method isapplied and the resulting iterative map

z 7−→ z+hω0

((µ+ i)z+ σ |z|2z

)(5)

is compared to Eq. (2). Equation (5) has the complex conju-gated eigenvalues λ1,2 = 1+hω0(µ± i). Since the eigenval-ues of the Neimark–Sacker normalform equation are knownas e±iθ (1+α), a comparison leads to

α =−1+√h2ω20 + (hω0µ+ 1)

2, (6)

θ = arg(1+hω0(µ± i)) . (7)

Moreover, the nonlinearity coefficient d can be calculated by

d = e−iθhω0σ. (8)

It can be concluded that the application of the explicit Eulermethod to Eq. (3), omitting the excitation, results in a sys-tem that is equivalent to the truncated normalform equationof the Neimark–Sacker bifurcation Eq. (2). An important as-pect of this mapping is the addition of a bifurcation offset.Thus, the bifurcation in the discrete-time domain does notoccur at µ= 0. The mapping also changes the natural fre-quency as well as the coefficient d of the nonlinear term. Thenonlinearity coefficient of the system is scaled by hω0 androtated by θ (cf. Eq. 8). Hence, the type of bifurcation canchange from supercritical to subcritical and vice versa whena change in the sign of

S. Feldkord et al.: Discretization analysis of bifurcation based nonlinear amplifiers 45

solution of very simple differential equations. To overcomethis issue, explicit Runge–Kutta methods are preferred. Theset of complex-valued terms that are equivalent to the righthand side of the not truncated normalform equation of theAndronov–Hopf bifurcation can be defined as

A=

{k∑n=0

ξn|z|2n· z |ξn ∈ C, k ∈ N\{0}

}. (9)

Every algorithmic step of an explicit Runge–Kutta method(Stoer and Bulirsch, 1978) applied to an Andronov–Hopfsystem can be generalized to either an insertion f (g(z)) oran addition b · f (z)+ c · g(z) with f , g ∈ A and b, c ∈ C.The addition b · f (z)+ c · g(z) is also an element of A. Theabove generalization is only possible under the conditionf (g(z)) ∈ A. Assuming f (g(z)) 6∈ A, any further operationsrelying on f (g(z)) would not result in elements of the set A.

Dependent on the order of the Runge–Kutta method, theinsertion f (g(z)) is an operation that can occur multipletimes through the calculation of one timestep. It can beshown that one single insertion f (g(z)) with k =K for f (z)and k =M for g(z), which calculates the highest exponent,results in an element of A with k = 2 ·M+2 ·K+2 ·M ·K .The resulting terms of any explicit Runge–Kutta method areelements of the set A. Thus, they are also terms of the righthand side of the not truncated normalform equation of theNeimark–Sacker bifurcation. It can be concluded that anyexplicit Runge–Kutta method maps any kind of normalformequation of the Andronov–Hopf bifurcation to a normalformequation of the Neimark–Sacker bifurcation. The value of kfor the highest term by using the Runge–Kutta method of 4thorder applied to Eq. (3) without excitation can be calculatedby the scheme above to k = 40.

The normalform equation is given completely by the coef-ficients ξn. The truncated normalform Eq. (2) is obtained byomitting all higher order terms with n > 1. This can be usedfor an approximation that is easy to analyze. However, thehigher order nonlinearities might change the qualitative be-havior of the bifurcation (Reit et al., 2016). Using the systemas amplifier, for small values of the excitation amplitude andnear the point of bifurcation, the higher order terms cannot beneglected due to large nonlinearity coefficients, e.g. θ > 0 (Kuznetsov, 2013), thebifurcation point at α = 0 and the sign of the nonlinear-ity coefficient remain unchanged. The parametric designof the rescaled Neimark–Sacker bifurcation to approximatethe rescaled Andronov–Hopf bifurcation is done by settingα = µ, θ = hω0 and d = σ . By adding the excitation terma0e

ihωn, we can compute algebraic amplitude responses forthe Neimark Sacker system Eq. (4), the rescaled Neimark–Sacker system Eq. (11) and the Runge–Kutta 4 method ap-plied to the rescaled Andronov–Hopf bifurcation (Feldkordet al., 2016). In the excitation term, n is the iteration countand h ·ω the scaled excitation frequency. An algebraic equa-tion for the input-output relation can also be given for theAndronov–Hopf bifurcation in the continuous-time domain(Eguíluz et al., 2000; Reit et al., 2016). Applying the Runge–Kutta 4 method analytically to the differential equation withexcitation, mixed terms of the state variable and the excita-tion occur. Thus, the differential equation does not map tothe normalform equation of the Neimark–Sacker bifurcation.In this case, an algebraic input-output relation is difficult orimpossible to compute.

The amplitude responses of the rescaled Neimark–Sackerbifurcation compared to the amplitude response of therescaled Andronov–Hopf bifurcation, the Neimark–Sackerbifurcation and the difference equation resulting from theRunge–Kutta 4 method applied to the rescaled Andronov–Hopf bifurcation are shown in Fig. 1a). Besides the inevitable

www.adv-radio-sci.net/15/43/2017/ Adv. Radio Sci., 15, 43–47, 2017

46 S. Feldkord et al.: Discretization analysis of bifurcation based nonlinear amplifiers

-1.0 -0.5 0.0 0.5 1.0-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

Neimark-SackerAndronov-HopfRunge-Kutta 4Rescaled Neimark-Sacker

a)

-1.0 -0.5 0.5 1.04.0

3.5

3.0

2.5

2.0

1.5

1.0

-

-

-

-

-

-

-

-0.5

Andronov-HopfRunge-Kutta 4Rescaled Neimark-Sacker

0.0

b)

)

)

Figure 1. Amplitude responses with h= 1/fS = 1/48 kHz−1, σ =d =−1, µ= α =−0.1 and ω0 = 2π · 3 kHz (a) respectively ω0 =2π · 15 kHz (b). The input amplitudes are 0.1, 0.01 and 0.001.

periodicity of the discrete-time systems in the frequency, therescaled Andronov–Hopf and the rescaled Neimark–Sackersystems match very well. On the contrary, the Neimark–Sacker system shows a wider bandwidth and is only equalto the Andronov–Hopf system in the peak amplitude and thenatural frequency. The rescaled Neimark–Sacker bifurcationshows only small differences in amplitude apart from the nat-ural frequency (Fig. 1b). Using the Runge–Kutta 4 method,the amplitude response deviates for higher natural frequen-

cies in amplitude and peak frequency from the solutions ofthe Andronov–Hopf system (Fig. 1b).

5 Conclusions

It is shown that applying an explicit Runge–Kutta methodto the truncated normalform of the Andronov–Hopf bifurca-tion leads to a discrete-time system represented by an itera-tive map which is equal to the normalform equation of theNeimark–Sacker bifurcation with higher order terms.

For the use as amplifier where the input-output character-istic is approximately independent of the natural frequency,a rescaled truncated normalform equation of the Neimark–Sacker bifurcation is introduced. A comparison of the im-plemented systems is given. It reveals the advantage ofdesigning the Neimark–Sacker bifurcation based system apriori over using an integration method to implement thecontinuous-time Andronov–Hopf bifurcation based systemon a digital platform. It is a much better approximation tothe behavior of the rescaled Andronov–Hopf bifurcation. Itpreserves the peak amplitude and the natural frequency.

Competing interests. The authors declare that they have no conflictof interest.

The publication of this article was funded by the open-accessfund of Leibniz Universität Hannover.

Edited by: J. AndersReviewed by: B. Schlecker and one anonymous referee

References

Eguíluz, V. M., Ospeck, M., Choe, Y., Hudspeth, A., and Magnasco,M. O.: Essential Nonlinearities in Hearing, Phys. Rev. Lett., 84,5232, https://doi.org/10.1103/PhysRevLett.84.5232, 2000.

Feldkord, S., Reit, M., and Mathis, W.: Implementation of a digitalevaluation platform to analyze bifurcation based nonlinear am-plifiers, Adv. Radio Sci., 14, 47–50, https://doi.org/10.5194/ars-14-47-2016, 2016.

Kemp, D. T.: Stimulated Acoustic Emissions from Within the Hu-man Auditory System, J. Acoust. Soc. Am., 64, 1386–1391,https://doi.org/0.1121/1.382104, 1978.

Kuznetsov, Y. A.: Elements of Applied Bifurcation Theory, in: Ap-plied Mathematical sciences, Vol. 112, edited by: Antman, S. S.,Marsden, J. E., and Sirovich, L., Springer, New York, 2004

Pikovsky, A., Rosenblum, M., and Kurths, J.: Synchronization: auniversal concept in nonlinear sciences, in: Cambridge NonlinearScience Series (Book 12), Cambridge University Press, 2003.

Reit, M., Mathis, W., and Stoop, R.: Time-Discrete NonlinearCochlea Model Implemented on DSP for Auditory Studies,in: Nonlinear Dynamics of Electronic Systems, Proceedings ofNDES 2012, 11–13 July 2012, Wolfenbüttel, Germany, 1–4,2012.

Adv. Radio Sci., 15, 43–47, 2017 www.adv-radio-sci.net/15/43/2017/

https://doi.org/10.1103/PhysRevLett.84.5232https://doi.org/10.5194/ars-14-47-2016https://doi.org/10.5194/ars-14-47-2016https://doi.org/0.1121/1.382104

S. Feldkord et al.: Discretization analysis of bifurcation based nonlinear amplifiers 47

Reit, M., Berens, M., and Mathis, W.: Ambiguities in input-output behavior of driven nonlinear systems close to bifurcation,Archives of Electrical Engineering, 65, 337–347, 2016.

Stoer, J. and Bulirsch, R.: Einführung in die Numerische Math-ematik II, https://doi.org/10.1007/978-3-662-06866-3, Springer,Berlin, 1978.

Stoop, R., Kern, A., and van der Vyver, J. J.: How Ears Go Elec-tronic, in: 2005 International Symposium on Nonlinear The-ory and its Applications (NOLTA2005), 18–21 October 2005,Bruges, Belgium, 533–536, 2005.

Stoop, R., Jasa, T., Uwate, Y., and Martignoli, S.: From Hear-ing to Listening: Design and Properties of an ActivelyTunable Electronic Hearing Sensor, Sensors, 7, 3287–3298,https://doi.org/10.3390/s7123287, 2007.

Zenner, H.-P. and Gitter, A. H.: Die Schallverarbeitung des Ohres– ein Bericht über aufregende Experimente mit mikrochirurgischisolierten Haarzellen, Physik in unserer Zeit, 18, 97–105, 1987.

www.adv-radio-sci.net/15/43/2017/ Adv. Radio Sci., 15, 43–47, 2017

https://doi.org/10.1007/978-3-662-06866-3https://doi.org/10.3390/s7123287

AbstractIntroductionFundamentals of the Andronov--Hopf and the Neimark--Sacker bifurcationAnalysis of the numerically integrated normalform equation of the Andronov--Hopf bifurcationDerivation of the Rescaled Neimark--Sacker BifurcationConclusionsCompeting interestsReferences