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English

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Seifert, Alexander

Bäuerle, Simon Andreas

Hetzler, Hartmut

Received: 1 July 2023 Accepted: 22 August 2023
DOI: 10.1002/pamm.202300235
RESEARCH ARTICLE
Numerical detection of synchronization phenomena in
quasi-periodic solutions
Alexander Seifert Simon Baeuerle Hartmut Hetzler
Engineering Dynamics Group, University
of Kassel, Kassel, Germany
Correspondence
Alexander Seifert, Engineering Dynamics
Group, University of Kassel, D 34125
Kassel, Germany.
Email: alexander.seifert@uni-kassel.de
Abstract
In science and technology, dynamical systems can show so-called quasi-periodic
solutions. These solutions are composed of two or more base frequencies. The
solution in the time domain can be represented by an invariant manifold. To
parametrize the invariant manifold, we choose the hyper-time parametrization.
If quasi-periodic solutions branches are continued by means of a path continua-
tion, the phenomenon of synchronization may occur. This is important, because
the hyper-time parametrization is only valid, as long as the number of base
frequencies remains unchanged. Therefore, it is essential to detect a parametriza-
tion to a synchronization point. Synchronization can happen in different types.
We address the mechanism of suppression, where one base frequency becomes
suppressed until its amplitude vanishes. This corresponds to the quasi-periodic
solution ending in a Neimark–Sacker bifurcation. We present a method to derive
a scalarmeasure from the quasi-periodic solution in the hyper-time parametriza-
tion, to detect an approach to aNeimark–Sacker bifurcationwhile continuing the
solution branch.
1 INTRODUCTION
In science and technology, the dynamic behavior of systems plays a key role. To ensure a safe operation it is essential to
understand themechanismswhich determine the behavior of solutions of dynamical systems. Therefore, themechanisms
which can excite stationary solutions require a thorough investigation.
In literature, there are many sources that describe the behavior of systems that are excited by one mechanism for exam-
ple, external excitation, parametric excitation, or self-excitation [1, p. 16]. The occurrence of more than one excitation
mechanism may lead to periodic, quasi-periodic, or chaotic solutions. Recently, quasi-periodic solutions have become
the center of interest in research. The occurrence of quasi-periodic solutions depends on the frequencies which excite
the system.
For stable-state and periodic solutions, there exists a bifurcation theory based on the stability of the solution [2,
p. 346]. However, for quasi-periodic solutions, there is no known systematic approach to detect bifurcations. Therefore,
this contribution will present an approach to detect Neimark–Sacker bifurcations in quasi-periodic solutions, which are
parametrized by hyper-time parametrization.
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium,
provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
© 2023 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.
Proc. Appl. Math. Mech. 2023;23:e202300235. wileyonlinelibrary.com/journal/pamm 1 of 8
https://doi.org/10.1002/pamm.202300235
SEIFERT et al. 2 of 8
TABLE 1 Parameter sets.
Parameter set 1 Parameter set 2
𝑠 (Excitation strength) 0.9 0.9
𝜀 (Nonlinearity) 0.5 0.5
Ω (Excitation frequency) 1.05 1.5
F IGURE 1 Periodic solution of the forced van-der-Pol equation in state-space, with the states 𝑧1 and 𝑧2 and in the frequency domain,
with the value of the Fourier transform (𝑧1) over the occuring frequencies ?̄?, for parameter set 1.
F IGURE 2 Quasi-periodic solution of the forced van-der-Pol equation in state-spacewith the states 𝑧1 and 𝑧2 and in the frequency
domain, with the value of the Fourier transform (𝑧1) over the occuring frequencies ?̄?, for parameter set 2.
2 QUASI-PERIODIC SOLUTIONS
Quasi-periodic solutions can occur in a dynamical system that experiences more than one mechanism of excitation. We,
therefore, demonstrated the occurrence of a quasi-periodic solution on the example of the forced van-der-Pol equation
𝑦′′ + 𝜀
(
𝑦2 − 1
)
𝑦′ + 𝑦 = 𝑠 cos(Ω𝑡). (2.1)
For the parameter sets in Table 1we canderive the state-space of the solution by time integration and theFourier transform,
which are depicted in Figures 1 and 2.
In Figures 1 and 2 we see that for both parameter-sets the spectrum remains discrete, however, the trajectory in state-
space appears completely different. While parameter set 1 leads to a periodic solution (i.e., a closed orbit in the state-
space), parameter set 2 leads to a solution that is no longer periodic, but, contrary to a chaotic solution, retains a discrete
Fourier spectrum.
3 of 8
F IGURE 3 Quasi-periodic solution of the forced van-der-Pol equation in autonomous state-space for parameter set 2, showing the states
𝑧1 and 𝑧2 over the time Ω𝑡 modulo 2𝜋.
In the spectrum of parameter set 2, we can identify two base frequencies 𝝎 = (𝜔1, 𝜔2)⊤, from which all other occurring
frequencies Ω𝑖 can be composed by the linear combination ⟨𝒂𝑖, 𝝎⟩ = Ω𝑖 with a constant 𝒂𝑖 ∈ ℤ2. For the identified fre-
quencies 𝝎, the following statement holds. The elements of a set of frequencies 𝝎 ∈ ℝ𝑝 are called base frequencies, if the
equation
⟨𝒄, 𝝎⟩ = 𝟎, 𝒄 ∈ ℤ𝑝 (2.2)
is only satisfied for the constatnt 𝒄 = 𝟎 [3, p. 15]. The number 𝑝 ∈ ℕ being the number of base frequencies. We restrict
further investigations to 𝑝 = 2. In the case of the forced van-der-Pol equation, we can identify the two base frequencies
as 𝝎 = (Ω, 𝜔)⊤, where Ω represents the external excitation frequency and 𝜔 the self-excitation frequency. From the time
solution in Figure 2 we cannot extract key information about the solution because we cannot know, for example, if the
solution has yet passed the maximum amplitude or velocity.
We therefore transform the solution to parameter set 2 into autonomous state-space and introduce a periodicity cor-
responding to the external (known) excitation frequency Ω in the time axis by 𝑧3 = mod(Ω𝑡, 2𝜋). If we transform the
solution into an autonomous state-space we derive the graphic shown in Figure 3. We can identify an invariant object,
which is represented by the surface of an opened torus. The invariant object will be smooth because the number of base
frequencies is finite [4, p. 56], therefore the object satisfies the requirements for a manifold. The quasi-periodic solution
will fill thismanifold densely if 𝑡 approaches infinity [3, p. 17]. Given the ergodic theorem [5, p. 146–147] we can identify the
quasi-periodic time solutionwith the geometric invariantmanifold as the carrier of the solution. To calculate the invariant
manifold, we need a parametrization, which represents the invariant manifold.
3 HYPER-TIME PARAMETRIZATION
If we want to describe the invariant manifold, we need a parametrization. Such a parametrization is given by the hyper-
time parametrization. Given a quasi-periodic solution with the frequency base 𝝎 = (Ω1,Ω2, … ,Ω𝑝)⊤, the hyper-time
parametrization introduces a multiple time axis 𝜽 = (𝜃1, 𝜃2, … , 𝜃𝑝)⊤, the so-called hyper-time coordinates. The connec-
tion between the physical time and the hyper-time is given by 𝜃𝑖 = mod(Ω𝑖𝑡, 2𝜋). Here, the discrete nature of the Fourier
spectrum is used to introduce periodicities in the hyper-time axis [2, p. 364]. The quasi-periodic solution in the hyper-time
parametrization represents the stationary solution in the state space [3, p. 53].
The quasi-periodic solution in the hyper-time parametrization for parameter set 2 is depicted in Figure 4. For each state
variable, there exists one≫surface≪. To determine the solution in the hyper-time parametrization, we need an equation.
Due to the fact that the solution domain is no longer scalar but p-dimensional, the equation will be a partial differential
equation (PDE). The solution in physical time is described by the system of ordinary differential equations
𝒛′ = 𝒇(𝑡, 𝒛), (3.1)
with𝒇 ∶ ℝ × ℝ𝑛 → ℝ𝑛.We changed the domain of themapping; therefore, we have to replace the derivative operatorwith
a partial differential operator [3, p. 54]. Keeping the restriction to 𝑝 = 2we introduce the torus function 𝒁 ∶ 𝕋2 → ℝ𝑛 and
SEIFERT et al. 4 of 8
F IGURE 4 Quasi-periodic solution of the forced van-der-Pol equation in hyper-time parametrization for parameter set 2, showing the
values of the torus function 𝑍𝑖 over the hyper-time coodinates 𝜃𝑘 .
derive the invariance equation
Ω
𝜕𝒁
𝜕𝜃1
+ 𝜔
𝜕𝒁
𝜕𝜃2
= 𝒇(𝜽, 𝒁), (3.2)
which describes the quasi-periodic solution in the hyper-time parametrization. By the introduction of the hyper-time coor-
dinates, we achieve a 2𝜋-periodicity 𝒁(𝜃1, 𝜃2) = 𝒁(𝜃1 + 2𝜋, 𝜃2) and 𝒁(𝜃1, 𝜃2) = 𝒁(𝜃1, 𝜃2 + 2𝜋). To solve Equation (3.2)
any numerical method for solving PDE’s is suitable, but given the periodicity the quasi-periodic shooting algorithm and
the quasi-periodic Fourier–Galerkin method are particularly suitable [3, p. 65]. The major drawback of the hyper-time
parametrization, however, is that this parametrization is only valid as long as the number of base frequencies remains
unchanged (cf. [3, p. 53–54]). This is particularly critical if a quasi-periodic solution branch is continued and a synchroniza-
tion point is approached. We need a mechanism to detect an approach to such bifurcation points based on the properties
of the quasi-periodic solution.
4 SYNCHRONIZATION
If a dynamical system produces different types of solutions, such as periodic or quasi-periodic solutions, the phenomenon
of synchronization may occur. Here, a dynamical system may change its solution type from quasi-periodic to periodic
as a parameter changes. This behavior is called synchronization. Restricting to the case of the synchronization of quasi-
periodic solutions with 𝑝 = 2 and combined forced and self-excited systems (i.e., the forced van-der-Pol equation), we can
find two major mechanisms of synchronization [6, p. 56–62].
The first mechanism is called entrainment or frequency locking. Here, the dynamical system adjusts the base frequencies
of its solution so that they are rationally dependent and therefore the number of base frequencies reduces to 𝑝 = 1. The
quasi-periodic solution branch ends in a saddle-node bifurcation [6, p. 33] [7, p. 194]. The second mechanism is called
suppression of the natural dynamic or short suppression. Here, the dynamical system suppresses one base frequency of its
solution until its amplitude in the Fourier spectrum vanishes and only one base frequency remains. The quasi-periodic
solution branch ends in a Neimark–Sacker bifurcation [8, p. 330]. In the following, we will focus on the mechanism
of suppression.
5 NEIMARK–SACKER BIFURCATION
Given the forced van-der-Pol equation (2.1), we can derive the bifurcation diagram shown in Figure 5 for the parameters
𝑠 = 0.9, 𝜀 = 0.5.
The periodic solution loses its stability due to a pair of complex conjugated Floquet multipliers leaving the unit circle [8,
p. 329–330]. The unstable periodic solution continues to exist [8, p. 330]. From the bifurcation point branches off a quasi-
5 of 8
F IGURE 5 Bifurcation diagram of the forced van-der-Pol equation with Neimark–Sacker bifurcation (NSB), showing the maximal value
of the solutions𝑚𝑎𝑥 (|𝒛|2) over the excitation frequency Ω.
F IGURE 6 Comparison between a quasi-periodic solution far from the synchronizationΩ = 1.5 (left), and close to the synchronization
point Ω = 1.244 (right), showing the states 𝑧1 and 𝑧2 over the time Ω𝑡 modulo 2𝜋.
F IGURE 7 Quasi-periodic solution of the forced van-der-Pol equation in hyper-time parametrization forΩ ≈ Ωsync close to
synchronization, showing the values of the torus function 𝑍𝑖 over the hyper-time coodinates 𝜃𝑘 .
periodic solution branch. To detect Neimark–Sacker bifurcations while continuing the periodic branch is easily doable,
because the solution continues to exist after the bifurcation. Thus, it is easy to iterate the bifurcation point.
The major difference is that the quasi-periodic solution starts/ends in the Neimark–Sacker bifurcation and does not
continue beyond the bifurcation point. Furthermore, there exists no test function based merely on the quasi-periodic
solution. Therefore, it is not possible to simply detect a sign change. For a quasi-periodic solution far from the synchro-
nization point, we find that the solution in Figure 6 fills an invariant manifold. If we now vary the parameter Ω and
approach the bifurcation, we find that the torus shrinks until only a line remains.
This corresponds to the suppression of the self-excitation frequency. If we visualize the approach to theNeimark–Sacker
bifurcation in the hyper-time parametrization, we can see that the hyper-time manifold becomes constant in 𝜃2 which
likewise corresponds to the suppression of the self-excitation frequency (Figure 7).
SEIFERT et al. 6 of 8
F IGURE 8 Comparison of the Poincaré sections between a quasi-periodic solution far from the synchronizationΩ = 1.5 (left), and close
to the synchronization point Ω = 1.244 (right), showing the cross-section of the state-space with the states 𝑧1 and 𝑧2.
F IGURE 9 Derivation of the Poincaré sections from the quasi-periodic solution of the forced van-der-Pol equation in the hyper-time
parametrization for Ω = 1.5, showing the values of the torus functions 𝑍𝑖 over the hyper-time coordinates 𝜃𝑘 .
6 SYNTHESIZING POINCARÉ SECTION FROMA SOLUTION IN THE HYPER-TIME
PARAMETRIZATION
Poincaré sections Σ represent a sub-manifold of the state-space. In the analysis of dynamical systems, these sections are
usually executed as stroboscopic sections, [2, p. 321] which means that for every integer multiple of a predetermined time
𝑇, the current state is represented in the section. Therefore, the stroboscopic Poincaré section of a periodic orbit with
period 𝑇, is represented by a single point. The stroboscopic Poincaré section of a quasi-periodic solution, with the period
of one base-frequency being 𝑇, is represented by a densely filled curve.
The closed curve shrinks to eventually become a point once the Neimark–Sacker bifurcation is reached (c.f. Figure 8).
These Poincaré sections are derived from a time solution. If we continue the quasi-periodic solution in the hyper-time
parametrization we have to derive the Poincaré sections from this representation of the solution.
For the example of the forced van-der-Pol equation, we know that the amplitude along 𝜃2 corresponds to the self-
excitation frequency and that the mechanism of synchronization is the suppression of the self-excitation frequency. We
therefore derive sections for constant 𝜃1 (c.f. Figure 10). If we combine the ≫lines≪ derived in Figure 9 we get the object
depicted in Figure 10. We define the Poincaré section for 𝜃𝓁 =
2𝜋
𝑘
as Σ𝓁
𝑘
(𝒁).
7 DETECTING NEIMARK-SACKER BIFURCATIONS
As shown before, the Poincaré sections Σ1
𝑘
(𝒁) shrink to a point when approaching a Neimark–Sacker bifurcation. We can
utilize the area which is enclosed by the Poincaré as a scalar measure to determine an approach to a Neimark–Sacker
bifurcation. To get a measure for the whole torus, we take the area of the Poincaré section𝐴𝑘 = 𝐴[Σ1𝑘(𝒁)] and average this
7 of 8
F IGURE 10 Poincaré sections derived from the quasi-periodic solution in the hyper-time parametrization, showing the values of the
torus function 𝑍𝑖 over one torus dimension 𝜃1.
F IGURE 11 Averaged area of Poincaré sections ?̄?, derived from the quasi-periodic solution in the hyper-time parametrization, over the
excitation frequencyΩ.
area for all sections (𝑚 being the number of Poincaré sections) along 𝜃1 by
?̄? =
1
𝑚
𝑚∑
𝑘=1
𝐴𝑘. (7.1)
If we plot the averaged area over the external excitation frequency, we get the diagram shown in Figure 11.
The averaged area approaches zero. Terminating the continuation if 𝐴 < tol = 10−3 we can determine an approach to
the bifurcation point. Comparing the value forΩend = 1.2442138 as the approximateΩ for which synchronization occurs
with an iterated bifurcation point based on a periodic continuationΩNSB = 1.2442044, we get a relative error
𝜀 =
(
Ωend −ΩNSB
ΩNSB
)
× 100% = 7.56 × 10−4%. (7.2)
8 CONCLUSION
In this contribution, we present a method to detect Neimark–Sacker bifurcations (synchronization points), while contin-
uing quasi-periodic solution branches with two base frequencies. Themethod uses the averaged area of Poincaré sections,
which are derived from the quasi-periodic solution in the hyper-time parametrization. Here we can show for the forced
van-der-Pol equation that the averaged area of the Poincaré sections tends to zero as the Neimark–Sacker bifurcation is
approached.We can show that the detected value of the bifurcation parameter for which the bifurcation is detected differs
from the actual value by less than 10−3%.
ACKNOWLEDGMENTS
Open access funding enabled and organized by Projekt DEAL.
SEIFERT et al. 8 of 8
ORC ID
Alexander Seifert https://orcid.org/0000-0001-8806-0451
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How to cite this article: Seifert, A., Baeuerle, S., & Hetzler, H. (2023). Numerical detection of synchronization
phenomena in quasi-periodic solutions. Proceedings in Applied Mathematics and Mechanics, 23, e202300235.
https://doi.org/10.1002/pamm.202300235


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