ARSAdvances in Radio ScienceARSAdv. Radio Sci.1684-9973Copernicus PublicationsGöttingen, Germany10.5194/ars-15-43-2017Discretization analysis of bifurcation based nonlinear amplifiersFeldkordSvenReitMarcoreit@tet.uni-hannover.deMathisWolfgangInstitute of Theoretical Electrical Engineering, Leibniz Universität Hannover,
Appelstraße 9A, 30167 Hannover, GermanyMarco Reit (reit@tet.uni-hannover.de)21September201715434716January20175March2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://ars.copernicus.org/articles/15/43/2017/ars-15-43-2017.htmlThe full text article is available as a PDF file from https://ars.copernicus.org/articles/15/43/2017/ars-15-43-2017.pdf
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.
Introduction
Early research has shown that the mammalian hearing process is
active and involves an active amplification within
the cochlea . This amplification
characteristic can be modeled by a system near the onset of an Andronov–Hopf
bifurcation . Regarding this bifurcation based
amplifier and after first analog realizations by
a discrete-time system was developed
and implemented on a digital signal processor (DSP) .
This implementation was latterly improved and extended to compare nonlinear
amplifiers based on Andronov–Hopf and Neimark–Sacker bifurcations
. The Neimark–Sacker bifurcation is also
called the Andronov–Hopf bifurcation for maps .
In this work we analyze the influence of explicit single-step integration
methods on the behavior of systems based on the normalform equation of the
Andronov–Hopf bifurcation. This is done exemplarily for the explicit Euler
method and extended to explicit Runge–Kutta methods. Lastly, a rescaled
normalform equation of the Neimark–Sacker bifurcation is derived that
approximates the characteristics of the rescaled Andronov–Hopf bifurcation
within the discrete-time domain while preserving the natural frequency and
the peak amplitude.
Fundamentals of the Andronov–Hopf and the Neimark–Sacker bifurcation
The analysis in this work focusses on the truncated normalform equation of
the Andronov–Hopf bifurcation dzdt=μ+iω0z+σ|z|2z,
where z∈C is the system variable, μ∈R the
bifurcation parameter, i the imaginary unit, σ∈C the first nonlinearity
coefficient that determines the type of the bifurcation and
ω0∈R the natural frequency. For ℜσ<0, the system is supercritical, branching a stable limit cycle for μ>0
from a stable fixed point (μ≤0) that loses stability. On the contrary
for ℜσ>0, the system exhibits an unstable limit cycle
for 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. A special characteristic of
Eq. () is the mapping of an additive sinusoidal
excitation signal a0eiωt to a
sinusoidal output signal z0eiωt+ϕ
without harmonic distortions
.
Similar to the normalform equation of the Andronov–Hopf bifurcation in the
continuous-time domain, the truncated normalform equation of the
Neimark–Sacker bifurcation in the discrete-time domain
,
z↦eiθz1+α+d|z|2,
is analyzed. As before, z is the complex system variable. The real-valued
bifurcation parameter is denoted by α. Similar to σ of the
Andronov–Hopf system, the type of bifurcation is dependent on the sign of
ℜ{d} while the bifurcation occurs at α=0. The Neimark–Sacker
bifurcation based system is embedded into a continuous-time environment by
iterating the map Eq. () with equidistant timesteps
h. From this iteration, the relation θ=h⋅ω0 results.
When using Andronov–Hopf and Neimark–Sacker bifurcation based systems as
nonlinear amplifiers, only the supercritical case with μ<0 is considered.
Adding an excitation signal for μ>0, the supercritical case results in
synchronization problems . The subcritical
case is unstable and therefore neglected. For modeling the cochlea, a
ω0-rescaled truncated normalform equation of the Andronov–Hopf
bifurcation with an added excitation term a(t),
dzdt=ω0μ+iz+ω0σ|z|2z+ω0a(t),
was introduced by . The special property of
Eq. () is that the amplitude response to a sinusoidal
excitation signal a(t)=a0eiωt is independent of the
absolute value of the natural frequency. Here, the fraction ω/ω0
determines the output amplitude for a given set of μ, σ and a0.
Recently, the rescaled truncated normalform equation of the Andronov–Hopf
bifurcation with excitation was implemented on a DSP platform
. In a comparative study of the amplitude responses, the
truncated normalform equation of the Neimark–Sacker bifurcation with the
excitation a(n),
zn+1=eiθzn1+α+d|zn|2+a(n),
was implemented and analyzed for a(n)=a0eiωhn. It can be shown, that the input-output
behavior of the Neimark–Sacker system is very similar to that of the not
rescaled Andronov–Hopf system Eq. (). Thus, to
approximate the behavior of the ω0-rescaled Andronov–Hopf system, an
appropriate normalform equation of the Neimark–Sacker bifurcation is
desirable which is introduced later in this work.
Since we examine the purpose of digital processing applications, where
real-time processing and low latency are the main goals, only explicit
integration methods are considered to implement the Andronov–Hopf
bifurcation based systems. The aforementioned DSP-Implementation uses the
Runge–Kutta method of 4th order.
Analysis of the numerically integrated normalform equation of the Andronov–Hopf bifurcation
The following section focusses on the qualitative changes of the system
behavior due to a discrete-time implementation. We use the rescaled truncated
normalform equation of the Andronov–Hopf bifurcation in
Eq. (). The intuitive approach is to transform
Eq. () into a system in the discrete-time domain using a
numerical integration method. Then, the resulting difference equation can be
analyzed. Due to the nonlinearity of the equation, mixed terms of the state
variable and the excitation term can occur. This leads to complicated
expressions that makes an analytical approach almost impossible. Therefore,
the following analysis focusses on the Andronov–Hopf system in
Eq. () without the excitation.
To illustrate the transformation of the system parameters due to an
integration method, the explicit Euler method is applied and the resulting
iterative map
z↦z+hω0μ+iz+σ|z|2z
is compared to Eq. (). Equation () has the
complex conjugated eigenvalues λ1,2=1+hω0(μ±i). Since the eigenvalues of the Neimark–Sacker normalform
equation are known as e±iθ(1+α), a comparison leads
to
α=-1+h2ω02+(hω0μ+1)2,θ=arg1+hω0(μ±i).
Moreover, the nonlinearity coefficient d can be calculated by
d=e-iθhω0σ.
It can be concluded that the application of the explicit Euler method to
Eq. (), omitting the excitation, results in a system that
is equivalent to the truncated normalform equation of the Neimark–Sacker
bifurcation Eq. (). An important aspect of this mapping
is the addition of a bifurcation offset. Thus, the bifurcation in the
discrete-time domain does not occur at μ=0. The mapping also changes the
natural frequency as well as the coefficient d of the nonlinear term. The
nonlinearity coefficient of the system is scaled by hω0 and rotated
by θ (cf. Eq. ). Hence, the type of bifurcation can
change from supercritical to subcritical and vice versa when a change in the
sign of ℜ{d} depending on θ occurs. Regarding the relation
Eq. (), the nonlinearity coefficient is dependent on μ
besides hω0. The explicit Euler method often requires very small
timesteps to satisfactorily approximate the solution of very simple
differential equations. To overcome this issue, explicit Runge–Kutta methods
are preferred. The set of complex-valued terms that are equivalent to the
right hand side of the not truncated normalform equation of the
Andronov–Hopf bifurcation can be defined as
A=∑n=0kξn|z|2n⋅z|ξn∈C,k∈N\{0}.
Every algorithmic step of an explicit Runge–Kutta method
applied to an Andronov–Hopf system can be
generalized to either an insertion f(g(z)) or an 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. The above generalization
is only possible under the condition f(g(z))∈A. Assuming f(g(z))∉A, any further operations relying on f(g(z)) would not result in elements
of the set A.
Dependent on the order of the Runge–Kutta method, the insertion f(g(z)) is
an operation that can occur multiple times through the calculation of one
timestep. It can be shown 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 are elements of the set
A. Thus, they are also terms of the right hand side of the not truncated
normalform equation of the Neimark–Sacker bifurcation. It can be concluded
that any explicit Runge–Kutta method maps any kind of normalform equation of
the Andronov–Hopf bifurcation to a normalform equation of the
Neimark–Sacker bifurcation. The value of k for the highest term by using
the Runge–Kutta method of 4th order applied to Eq. ()
without excitation can be calculated by the scheme above to k=40.
The normalform equation is given completely by the coefficients ξn. The
truncated normalform Eq. () is obtained by omitting all
higher order terms with n>1. This can be used for an approximation that is
easy to analyze. However, the higher order nonlinearities might change the
qualitative behavior of the bifurcation . Using
the system as amplifier, for small values of the excitation amplitude and
near the point of bifurcation, the higher order terms cannot be neglected due
to large nonlinearity coefficients, e.g. ℜ{ξ2}≈0.42 for
h=1/48kHz-1, ω0=2π⋅5kHz,
μ=-0.1 and σ=-1. Thus, any analysis of the truncated
normalform equation must be taken with care and a higher order term
approximation should be considered.
Derivation of the Rescaled Neimark–Sacker Bifurcation
From Sect. , we know that explicit Runge–Kutta methods map
an Andronov–Hopf system to a Neimark–Sacker system. Moreover, the
steady-state response of the driven Neimark–Sacker bifurcation resembles the
driven Andronov–Hopf system for a wide range of parameters.
In this section, we introduce a rescaled version of the Neimark–Sacker
bifurcation based amplifier. To understand the process of reparametrization,
some insights into the mapping of the Andronov–Hopf bifurcation to the
Neimark–Sacker bifurcation are required. One of the conditions for the
Andronov–Hopf bifurcation is the crossing of the imaginary axis by a complex
conjugated pair of eigenvalues. For the Neimark–Sacker bifurcation, the
eigenvalues cross the unit circle. The most simple mapping is done by
applying the explicit Euler method to the differential
Eq. (). The iterative map Eq. () results,
that resembles the normalform equation of the Neimark–Sacker bifurcation of
the same degree since the cubic nonlinearity is the highest order term. When
the normalform equation of the Andronov–Hopf bifurcation is rescaled and
integrated by the explicit Euler method, the terms hμ and hσ are
rescaled by ω0,
z↦z1+hω0μ+ihω0+hω0σ|z|2.
In order to transfer this rescaling to the normalform equation of the
Neimark–Sacker bifurcation, it is considered that in the discrete-time
domain not the natural frequency itself but θ=hω0 is the
parameter of interest. Modifying the parameters α and d in
Eq. () by multiplication with θ, the
θ-rescaled truncated normalform equation of the Neimark–Sacker
bifurcation
z↦eiθz1+θα+θd|z|2
results. Since θ∈R and π>θ>0, the bifurcation point at α=0 and the
sign of the nonlinearity coefficient remain unchanged. The parametric design
of the rescaled Neimark–Sacker bifurcation to approximate the rescaled
Andronov–Hopf bifurcation is done by setting α=μ, θ=hω0 and d=σ. By adding the excitation term
a0eihωn, we can compute algebraic amplitude responses
for the Neimark Sacker system Eq. (), the rescaled
Neimark–Sacker system Eq. () and the Runge–Kutta 4
method applied to the rescaled Andronov–Hopf bifurcation
. In the excitation term, n is the
iteration count and h⋅ω the scaled excitation frequency. An
algebraic equation for the input-output relation can also be given for the
Andronov–Hopf bifurcation in the continuous-time domain
. Applying the Runge–Kutta
4 method analytically to the differential equation with excitation, mixed
terms of the state variable and the excitation occur. Thus, the differential
equation does not map to the normalform equation of the Neimark–Sacker
bifurcation. In this case, an algebraic input-output relation is difficult or
impossible to compute.
Amplitude responses with h=1/fS=1/48kHz-1,
σ=d=-1, μ=α=-0.1 and ω0=2π⋅3kHz(a) respectively ω0=2π⋅15kHz(b). The input amplitudes are 0.1, 0.01 and
0.001.
The amplitude responses of the rescaled Neimark–Sacker bifurcation compared
to the amplitude response of the rescaled Andronov–Hopf bifurcation, the
Neimark–Sacker bifurcation and the difference equation resulting from the
Runge–Kutta 4 method applied to the rescaled Andronov–Hopf bifurcation are
shown in Fig. a). Besides the inevitable periodicity of the
discrete-time systems in the frequency, the rescaled Andronov–Hopf and the
rescaled Neimark–Sacker systems match very well. On the contrary, the
Neimark–Sacker system shows a wider bandwidth and is only equal to the
Andronov–Hopf system in the peak amplitude and the natural frequency. The
rescaled Neimark–Sacker bifurcation shows only small differences in
amplitude apart from the natural frequency (Fig. b). Using
the Runge–Kutta 4 method, the amplitude response deviates for higher natural
frequencies in amplitude and peak frequency from the solutions of the
Andronov–Hopf system (Fig. b).
Conclusions
It is shown that applying an explicit Runge–Kutta method to the
truncated normalform of the Andronov–Hopf bifurcation leads to a
discrete-time system represented by an iterative map which is equal to the
normalform equation of the Neimark–Sacker bifurcation with higher order
terms.
For the use as amplifier where the input-output characteristic is
approximately independent of the natural frequency, a rescaled truncated
normalform equation of the Neimark–Sacker bifurcation is introduced. A
comparison of the implemented systems is given. It reveals the advantage of
designing the Neimark–Sacker bifurcation based system a priori over using an
integration method to implement the continuous-time Andronov–Hopf
bifurcation based system on a digital platform. It is a much better
approximation to the behavior of the rescaled Andronov–Hopf bifurcation. It
preserves the peak amplitude and the natural frequency.
The authors declare that they have no conflict of
interest. The publication of this article was funded by the open-access fund of Leibniz Universität Hannover.
Edited by: J. Anders
Reviewed by: B. Schlecker and one anonymous referee
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