Oscillators exhibiting an Andronov–Hopf bifurcation are candidates to mimic the functionality of the cochlea, since the transfer response of these oscillators is compressive and frequency selective. The former implies that small stimuli are amplified and strong stimuli are attenuated, while the latter means that the oscillator only reacts in a (small) frequency band. However, this implies that many oscillators are needed to cover a relevant frequency band. By introducing the notion of tunable characteristic frequencies, i.e., the characteristic frequency can be adjusted by a controllable input, the number of oscillators can be eventually reduced. Subsequently, the tunability enhancement of coupled oscillators is investigated by analyzing the local dynamics of a network of oscillators. For this, necessary conditions for the emergence of Andronov–Hopf bifurcations are determined for networks consisting of two groups, i.e., a group is a network of identical oscillators. By choosing the eigenvalues of the product of the cross-coupling matrix as bifurcation parameters and exploiting the structure of the transfer matrix of this network, the critical points and, thus, the characteristic frequency at this point can be derived. Tunability of the characteristic frequency is then enabled by controlling the asymmetry between the groups of oscillators.

In rapidly changing environments, technological speech procession is outperformed by mammalian hearing perception. This comes from the fact that hearing perception is superior in terms of energy and time consumption, which is induced by a nonlinear feedback loop in the cochlea. In particular, the dynamics of the feedback loop is similar to an Andronov–Hopf bifurcation. Thus, oscillators exhibiting this bifurcation can mimic the functionality of the cochlea so that the performance gap might be closed by implementing them into technology. However, many oscillators are needed since a system exhibiting an Andronov–Hopf bifurcation can only react to a (small) frequency range. Hence, methods to reduce this number are needed. This can be done, e.g., by enabling tunability of the characteristic frequency so that one oscillator can measure a frequency band. In this work, the influence of coupling oscillators on the dynamics is analyzed. In particular, it is shown that by coupling, the Andronov–Hopf bifurcation is preserved and the characteristic frequency becomes tunable.

## I. INTRODUCTION

Compared to technical solutions, the auditory perception of mammalians is superior. This performance difference comes from the different architectures of both systems: the technical solution has a feedforward structure consisting of three steps—a typical micro-electromechanical microphone with a passive, linear characteristics,^{1} nonlinear pre-processing,^{2} and an artificial neural network^{3,4}—while auditory perception has a feedback structure that induces an active process with a compressive nonlinearity.^{5,6} The latter implies that the dynamical range of the biological system is significantly extended compared to a feedforward architecture. Furthermore, auditory perception is frequency selective. With this, the biological system can detect both faint sound with low background noise and sound, which is heavily jammed, i.e., the latter effect is called the cocktail party effect.^{7,8} This high adaptability is also induced by the feedback loop, which is summarized subsequently.

The nonlinear dynamics of the auditory perception are located in the cochlea, where the pressure of an acoustic signal is transformed into an electrical signal. This is done by the interplay of the basilar membrane, the tectorial membrane, and the inner and outer hair cells. The inner hair cells are only connected to the basilar membrane, so these are hyperpolarized by bending them to one direction and are depolarized by bending them to the opposite direction. In contrast, the outer hair cells are connected both to the basilar membrane and the tectorial membrane and the stiffness of these hair cells can be changed. Thus, they can either block the inner hair cells from oscillating or amplify a signal by stimulating the inner hair cells by using the tectorial membrane. Moreover, the basilar membrane is frequency selective, so higher frequencies stimulate the front of the basilar membrane and lower frequencies stimulate its end.^{11,12} Note that the feedback mechanism of the inner and the outer hair cells couples to their neighbors since they are connected by the tectorial membrane. Hence, the cochlea can be viewed as oscillators coupled in a line topology.

Modeling of the cochlea commonly involves the ability to generate an Andronov–Hopf bifurcation as moving from the sub-critical to the super-critical regime (and vice versa) induces useful dynamical properties to mimic biological behavior. In the sub-critical regime, a system exhibiting an Andronov–Hopf bifurcation is asymptotically stable. After exceeding the critical point, the system can exhibit a stable limit cycle. Thus, by proving that this type of bifurcation exists, the existence of a limit cycle can be concluded.^{13} In addition, the synchronization problem can be addressed for coupled oscillators.^{14} Hence, the emergence of Andronov–Hopf bifurcations was analyzed for various systems arising, e.g., in engineering,^{15} biology,^{16–19} and physics.^{14,20,21} Moreover, systems exhibiting Andronov–Hopf bifurcations have remarkable properties in the sub-critical regime. Here, the response to an excitation can be either linear if the system is far away from the critical point or both frequency selective and compressive, if the system is close to the critical point. Thus, a system exhibiting an Andronov–Hopf bifurcation has similar dynamical properties as the cochlea.^{22–24} This implies in view of the realization of a bio-inspired sensor mimicking the functionality of the cochlea that Andronov–Hopf bifurcations must be enabled by design and/or suitable feedback control so that both the linear and the nonlinear properties in the sub-critical regime can be used to enhance sensor performance. Additionally, coupling these sensors may enhance frequency tunability^{20,25} or may enable broader bandwidth^{26} of the response in terms of external input. Thus, improving the performance of the system.

It was shown in Lenk *et al.*^{9} that this behavior can be realized in principle utilizing a thermally actuated, micro-electromechanical cantilevered (MEMS) sensor. Mathematically, this system can be modeled as a linear Euler–Bernoulli beam with a nonlinear actuation induced by the embedded heater. The deflection can be measured with an embedded piezoelectric sensor^{27} so that the velocity can be computed using a high pass filter. By considering a dominant mode approximation and by feeding back the velocity with a suitable gain, two different Andronov–Hopf bifurcations can be achieved as was shown experimentally in Refs. 9, 26, 28, and 29 and theoretically in Refs. 9 and 10. The principle bifurcation behavior is sketched in Fig. 1. Particularly, the system is locally asymptotically stable in the interval $[ k H , 2, k H , 2]$. After surpassing the critical value $ k H , 1$ or by falling below the critical value $ k H , 2$, an unstable limit cycle will emerge in the dominant mode model.^{10} In view of the conceptually similar dynamics, this device can also be called an artificial hair cell. However, different issues remain as for a single sensor the characteristic frequency of the model cannot be adjusted simultaneously in this way. Ideally, this can be changed by exploiting a controllable asymmetry of coupled MEMS sensors, such that the characteristic frequencies of the network can be changed by assigning the damping accordingly. For instance, a change in the characteristic frequency can be observed with two injectively coupled Andronov–Hopf oscillators in Gomez *et al.*^{20} and Rolf and Meurer.^{25}

In contrast, symmetry and symmetry-breaking is commonly investigated. Herein, the oscillators are assumed to be symmetric for some group, e.g., identical oscillator coupled in an undirected graph, such that the symmetry-breaking of equivariant ordinary differential equations (ODEs) can be shown by applying the equivariant branching lemma for the analysis of steady state bifurcations, e.g., saddle-node bifurcations, and the equivariant Hopf theorem for the analysis of the Andronov–Hopf bifurcation in these networks.^{30} In particular, multi-clustering and pattern formation appear after the symmetry breaks. Thus, symmetry and symmetry-breaking can be used to describe, for instance, animal gaits and central pattern generators^{31–33} and specification.^{34–37}

A different approach to simplify the analysis of coupled oscillators is the assumption that the coupling is weak. Then, each oscillator in the network can be described as an uncoupled oscillator with a small perturbation resembling the coupling between the oscillators. With this assumption, the models of the oscillators can be simplified to a phase model by applying perturbation methods.^{38} Note that different methods are needed to compute the phase models since the dynamics of the oscillators can be, for example, stiff or non-stiff.^{39–41} For instance, this simplification can be used to investigate chemical reactions^{42,43} and neuronal networks.^{44}

This paper aims to analyze the tunability enhancements of asymmetric coupled oscillators. This is done by analyzing the bifurcation behavior of two coupled groups of MEMS sensors with a focus on the emergence of Andronov–Hopf bifurcations and the qualitative behavior of the corresponding characteristic frequencies. Herein, the bifurcation parameters are assumed to be the eigenvalues of the product between the cross-coupling matrix and the internal feedback of the MEMS sensors is used to induce a controllable asymmetry in the network. It should be noted that the bifurcation analysis extends the results in Stan and Sepulchre,^{15} where the coupling of identical oscillators in symmetric networks is addressed. In particular, the necessary condition is shown by transforming the transfer function of the uncoupled system with the adjacency matrix into a weighted identity matrix. A rank drop satisfying the necessary condition of the Andronov–Hopf bifurcation can be easily shown for the network consisting of two groups of oscillators. The characteristic frequencies then follow from the bifurcation analysis and conditions for the tunability of the characteristic frequency as a function of the internal feedback are derived. In addition, the location of the critical points is analyzed by determining the critical point of a so-called Hopf–Hopf bifurcation.^{30,45,46} With this, the characteristic frequency in the sub-critical regime is characterized.

The paper is structured as follows: In Sec. II, the dominant mode model of a coupled MEMS sensor is introduced and the preliminaries for the local analysis and the challenges in analyzing the bifurcations are discussed. Afterward, the notion of tunability is defined in Sec. III. In addition, it is shown that a single MEMS sensor is not tunable to motivate the bifurcation analysis for a network of MEMS sensors. Then, two groups of identical and nonidentical MEMS sensors are analyzed in Sec. IV with respect to the emergence of Andronov–Hopf bifurcations. Herein, the controllable asymmetry between two groups of non-identical is used to achieve tunability of the system. The results are then numerically verified in Sec. V. The focus of the simulation studies is to provide insight if the Andronov–Hopf bifurcation of the coupled, MEMS sensors is tunable and to compare the reaction of the bio-inspired system with the cochlea. Finally, some remarks conclude this paper.

## II. NETWORKS OF MEMS SENSORS

^{27,47}In this sense, the mathematical model of the $ i$th MEMS sensor in a network of $ N\u2208 N$ MEMS sensors is described by

^{13,45}In particular, the eigenvalues of the adjacency matrix $ K$ are of interest for the bifurcation analysis and to compute the characteristic frequency as is elaborated in Secs. IV and V.

^{48,49}so that the necessary conditions of an Andronov–Hopf bifurcation cannot be investigated directly. To circumvent this issue, the structure of the system matrix $ A$ is exploited subsequently by considering the transfer function of an uncoupled MEMS sensor,

## III. PROBLEM STATEMENT

In addition to determining the dynamics of the coupled MEMS sensors, the number of oscillators in an acoustic sensor has to be reduced and the design has to be robust. This can be done, e.g., by asserting that the characteristic frequency is adjustable by external feedback. This property of an oscillator or a network of oscillators is subsequently called (frequency) tunability.

### (Frequency tunability)

An oscillator or a network of oscillators is called tunable, if its characteristic frequency (or synchronization frequency, respectively) can be changed by a controllable input. In addition, the parameter controlling the (frequency) tunability of the oscillator or network of oscillators is called tunability parameter.

It should be noted that Definition 1 is not satisfied in general for an oscillator and that this property can be induced in a MEMS sensor by changing the geometry of the MEMS sensor,^{47,50–56} so that geometric nonlinearities arise for small pre-deflections of the MEMS sensor. For instance, these nonlinearities can then be modeled by a Duffing oscillator.^{47} In contrast to this, subsequently frequency tunability is investigated by exploiting asymmetries between coupled oscillators so that the frequency can be tuned without adapting the geometry of the MEMS sensor.

### A. Example 1: Tunability of two coupled Kuramoto oscillators

^{57}

### B. Example 2: Tunability of two injectively coupled Andronov–Hopf oscillators

^{25}

### C. Example 3: Tunability of a single MEMS sensor

#### (Rolf and Meurer10)

**(Rolf and Meurer**

^{10})In particular, the characteristic frequencies are determined by the geometry and the material constants of the sensor so that the characteristic frequency cannot be controlled by the feedback strength $ k$ and the DC-voltage $ u DC$. This comes from the fact that the considered system represents a cantilever, e.g., see Example 6.7 in Reedy.^{58} Hence, the characteristic frequency of system (10) is not tunable.

### D. Focus of the work

Based on the results from the examples, the aim of this work is to investigate, if frequency tunability can be achieved in a network of MEMS sensors by controlling the network’s asymmetry. Herein, the asymmetry is adjusted by the effective Q-factor of each MEMS sensor. Particularly, these Q-factors are controlled by the feedback strengths $ k i i$ for the oscillators $ i=1,\u2026, N$. This conclusion follows from analyzing the necessary conditions of the Andronov–Hopf bifurcations of the coupled MEMS sensors given by (1), i.e., the eigenvalues of the system matrix $ A$ have two complex conjugated eigenvalues on the imaginary axis. It should be noted that the remaining conditions, i.e., the eigenvalue crossing condition and the stability of the limit cycle, are not analyzed in detail. In particular, it is trivial to show that the limit cycle is unstable. This comes from the fact that the nonlinearity in (1) is quadratic so that by employing the center manifold theorem^{13,45} and inserting the critical point a center manifold with a quadratic nonlinearity in the neighborhood of the equilibrium arises.

## IV. BIFURCATION ANALYSIS OF TWO GROUPS OF MEMS SENSORS

Theoretical results are presented to determine the critical point of an interconnection of two different groups of oscillators. It is assumed that the uncoupled oscillators exhibit at least a single Andronov–Hopf bifurcation in terms of a bifurcation parameter. For instance, (10) satisfies this assumption in terms of the feedback strength $ k$. Based on this, the bifurcations of MEMS sensors in the coupled case are analyzed. It turns out that the necessary conditions for the emergence of an Andronov–Hopf bifurcation of two coupled groups of MEMS sensors are satisfied in three cases. Finally, the three critical points are examined in view of the emergence of a Hopf–Hopf bifurcation. The latter arises if two critical points are identical for a given parameter configuration.^{30,45,46}

### A. Theoretical results

Before discussing the main results of this paper, the definition of a group of oscillators is given.

A group is a network of identical oscillators. The coupling inside a group is called self-coupling, the coupling between groups is called cross coupling, and the feedback of each oscillator onto itself is called self-feedback.

To determine the critical points of two coupled groups, the structure of the linearization of (1) is exploited. Note that this structure can be induced in the following way:

For a network composed of oscillators, the following assumptions are imposed:

The equilibria of the decoupled oscillators are invariant to coupling, i.e., they remain the sole equilibria of the coupled nonlinear system, and the system matrix is influenced by the adjacency matrix.

For each uncoupled oscillator, there exists a critical value $ k H , i i$ of the self-feedback strength so that it undergoes at least one Andronov–Hopf bifurcation.

As a consequence of (A1), the transfer matrix of the coupled system is obtained by coupling the transfer functions of the uncoupled oscillators. Assumption (A2) implies that the Jacobian $ A i i$ has a pair of complex conjugated eigenvalues on the imaginary axis for $ k i i= k H , i i$. For instance, the critical feedback strength of a single MEMS sensor is given in Theorem 1. The conditions for the critical point of the two coupled groups of oscillators are elaborated as follows.

Equation (11) must be interpreted in the sense that at least one eigenvalue $ \lambda k$ of the matrix $ K 21 K 12$ must take the value $ \lambda H$. Note that this also enables us to deduce information about the topology.

To prove the claim, it has to be shown that the transfer matrix of the network has a complex conjugated pair of eigenvalues on the imaginary axis. This is done in three steps: First, the transfer function from output to input of the network is derived depending on the coupling in terms of the adjacency matrix $ K$. Second, the adjacency matrix $ K$ is shifted by a diagonal matrix $ Q\u2208 C N \xd7 N$ so that $ K ~= K+ Q$ becomes singular. Third, this shift introduces a linear pseudo-feedback into each transfer function. This is then used to show that the transfer function of the network has a complex conjugated pair of eigenvalues on the imaginary axis in terms of the eigenvalues $ \lambda H$ of the matrix $ K 12 K 21$. According to (A2), the linearization of the oscillators at their equilibrium yields the transfer functions $ g 1( s)$ and $ g 2( s)$ given by (4).

^{59}with the characteristic frequency $ \omega C>0$. For this, the Jordan decomposition of $ K ~= W \u2217 J W$ is used with the Jordan matrix $ J\u2208 R N \xd7 N$ and a transformation matrix $ W\u2208 R N \xd7 N$.

^{60}For the output-to-input transfer function of the network, this implies

Theorem 2 is also satisfied for two coupled groups of general oscillators if Assumptions (A1) and (A2) are fulfilled and if in addition, no pole-zero cancellation occurs in the elements of the transfer matrix.

The condition imposed on the real-valued eigenvalues of the product of the cross-coupling matrices $ K 12 K 21$ and the self-coupling matrices $ K 12$ and $ K 22$ might seem difficult to satisfy. However, this condition is immediately fulfilled, e.g., if

the network is undirected so that the adjacency matrix becomes symmetric, i.e., $ K= K T$, and

the size of one group is $1$.

^{15}In particular, the critical point is also shifted by the eigenvalues of the self-coupling matrix in asymmetric networks. This is summarized subsequently.

Consider a single group consisting of $ N\u2208 N$ oscillators and denote the self-coupling matrix by $ K\u2208 R N \xd7 N$. In addition, assume that Assumptions (A1) and (A2) are satisfied and that the eigenvalues of the self-coupling matrix $ \lambda i\u2208\sigma ( K)$ are real-valued for the oscillators $ i=1,\u2026, N$. Then the critical point $ k H$ of an Andronov–Hopf bifurcation of each oscillator are given by $ k H= k+ \lambda i$.

By assuming that the oscillators are passive, i.e., the unforced oscillator possesses a stable limit cycle and the feedback system satisfies the dissipation inequality $ S \u02d9\u2264( k\u2212 k H) y 2\u2212 y h( y)+ y\Delta v$ with a nonlinearity $ h( y)\u2208 R$, the critical point of the network becomes unique, i.e., only the minimal eigenvalue changes the critical point. For more details, see Stan and Sepulchre.^{15}

In summary, the bifurcation analysis for a network of two groups of oscillators is simplified by employing Theorem 2 and Lemma 1 in two different ways: First, the conditions of Theorem 2 reduce the degree of the characteristic polynomial. Second, the bifurcation parameter is identified easier, since it is possible to reduce the degree of the characteristic polynomial with Lemma 1 to $min{ N 1, N 2}$. Thus, it even might become possible to compute the bifurcation parameter analytically. This is addressed subsequently for two groups motivated by the MEMS-based oscillators (1). In particular, by analyzing the bifurcations of two coupled groups of these oscillators, the characteristic frequency directly follows. Hence, the frequency tunability of two coupled groups can be investigated in this way.

### B. Andronov–Hopf bifurcation

#### 1. Identical MEMS sensors

#### 2. Non-identical MEMS sensors

In the case of two non-identical groups of MEMS sensors, (19) has to be solved generally. This is done in the following way: (19a)–(19c) are solved for the constants $ q 11$, $ q 12$, $ q 22$, and the characteristic frequency $ \omega C$. After substituting the results into (19d), a polynomial of third degree arises. Particularly, the emergence of three different critical points can be explained in two different ways. First is by considering the simpler bifurcation behavior of $ N$ injectively coupled Andronov–Hopf oscillators. Herein, the eigenvalues of the adjacency matrix are assumed to be the bifurcation parameters. Then, this network has at maximum $ N\u22121$ different Andronov–Hopf bifurcations.^{61} For the simplest case, two coupled Andronov–Hopf oscillators have one Andronov–Hopf bifurcation.^{25} Second is by considering the results on two coupled groups of identical oscillators, which also have three real-valued critical points.

With these analogies, it follows that a network consisting of two different coupled groups of MEMS sensors has three real-valued bifurcation points, since Theorem 1 implies that one MEMS sensor can described by two Andronov–Hopf oscillators. Thus, it is reasonable to assume that the cubic polynomial has only real solutions.^{62} The resulting values for the constants $ q 11$, $ q 12$, $ q 21$, $ q 22$ and the characteristic frequency $ \omega C$ are given in Appendix C.

#### 3. Frequency tunability

In addition to the emergence of the three Andronov–Hopf bifurcations, the three respective characteristic frequencies become tunable by changing the damping of the bifurcation.^{25} This is achieved by means of the self-feedback strengths $ k 11$ and $ k 22$ as these values in the physical setup change the sensitivity by heating the MEMS sensor, such that the asymmetry in the network can be changed. The limits of the characteristic frequencies can be, e.g., derived by employing first Proposition 1 and then Theorem 2. Proposition 1 implies that the critical points and characteristic frequencies of one group will be given by the individual oscillators if the two groups are not coupled. Thus, if one group is in bifurcation, Theorem 2 implies that there is one critical point equal to zero in the case of two coupled groups. In this case, the characteristic frequency of the two groups is then given by the characteristic frequency of the individual oscillators. In particular, these characteristic frequencies will be the limits for the individual Andronov–Hopf bifurcation, since the characteristic frequency of the system will move toward the characteristic frequency of the other group by increasing the bifurcation parameter of the system. In view of a practical realization, it is desirable that the tunability of the resulting Andronov–Hopf bifurcations is constrained by the neighboring characteristic frequencies so that the closure of the three intervals is empty. This ideal situation is sketched in Fig. 5.

### C. Hopf–Hopf bifurcation

^{30,45,46}Subsequently, the coupled groups of MEMS sensors are investigated for the emergence of the Hopf–Hopf bifurcation with respect to the frequency difference between the natural frequencies $ \omega 1$ and $ \omega 2$ and the feedback strength $ k 11$ and $ k 22$ for two different scenarios: First, two coupled groups of identical MEMS sensors are investigated. This is done by imposing the additional constraint,

#### 1. Identical MEMS sensors

#### 2. Non-identical MEMS sensors

## V. SIMULATION RESULTS AND NUMERICAL EVALUATION

Subsequently, the analytical results are verified numerically. First, Theorem 2 is verified for two groups consisting of 6 different MEMS sensors. Herein, it is assumed that $ N 1=4$ and $ N 2 = 2$. Second, the Andronov–Hopf bifurcations are analyzed for an arbitrary topology of the coupled groups. This is done in the following way: First, two coupled, identical groups of MEMS sensors are simulated. Here, the setup is focused on the occurrence of the Hopf–Hopf bifurcation. Afterward, the characteristic frequencies of two coupled non-identical groups of MEMS sensors are investigated. Thereby, it is compared, which pair of eigenvalues passes the imaginary axis first. From now on, this bifurcation will be called dominant since this bifurcation is observed first in the network. Finally, the Hopf–Hopf bifurcation is investigated in terms of the differences between the natural frequencies $ \omega 1$ and $ \omega 2$ and the feedback strengths $ k 11$ and $ k 22$, respectively.

Parameters for the MEMS sensors are given in Table I and the parameters for the numerical methods are given in Table II, respectively. It should be noted that the parameters are related to actual MEMS sensors; see, e.g., Lenk *et al.*^{9} and Rolf and Meurer.^{10} For the assumed parameter set, the two uncoupled MEMS sensors have the possible critical points of the Andronov–Hopf bifurcations $ k H , 11=0.109$, $ k H , 12=\u22120.296$, $ k H , 21=0.163$, and $ k H , 22=\u22120.387$. Furthermore, the characteristic frequencies at the critical point are given by $ \omega C , 11=2\pi \xd73.505$ $ 1 s$, $ \omega C , 12=2\pi \xd7159.44$ $ 1 s$, $ \omega C , 21=2\pi \xd74.0053 1 s$, and $ \omega C , 22=2\pi \xd7159.47$ $ 1 s$. Note that the equations for the critical point and the characteristic frequency are given in Theorem 1, respectively.

Parameter . | . | Values . |
---|---|---|

Natural frequency | ω_{1} | 2π × 3500 $ 1 s$ |

ω_{2} | 2π × 3750 $ 1 s$ | |

Q-factor | $ Q$ | 30, |

Offset voltage | $ u DC$ | −0.2 V |

Transfer factor | α | 19.2 $ m Ks$ |

Time constant | β | 1006.6 $ 1 s$ |

Transfer factor | ζ | 4.2588 × 10^{5} |

Time constant | τ | 10^{−3} $ 1 s$ |

Calibration factor | κ | 10^{6} $ V m$ |

Height | $ h$ | 1.45 × 10^{−6} m |

Density | ρ | 2, 329 $ kg m 3$ |

Parameter . | . | Values . |
---|---|---|

Natural frequency | ω_{1} | 2π × 3500 $ 1 s$ |

ω_{2} | 2π × 3750 $ 1 s$ | |

Q-factor | $ Q$ | 30, |

Offset voltage | $ u DC$ | −0.2 V |

Transfer factor | α | 19.2 $ m Ks$ |

Time constant | β | 1006.6 $ 1 s$ |

Transfer factor | ζ | 4.2588 × 10^{5} |

Time constant | τ | 10^{−3} $ 1 s$ |

Calibration factor | κ | 10^{6} $ V m$ |

Height | $ h$ | 1.45 × 10^{−6} m |

Density | ρ | 2, 329 $ kg m 3$ |

Parameter . | . | Frequency range (Hz) . | Values . |
---|---|---|---|

Step size | $ \eta k$ | … | 10^{−1} |

η_{ω} | [0, 1 000] | 10^{6} | |

[1 000, 7 500] | 10^{3} | ||

[7 500, 10 000] | 10^{−1} | ||

Threshold | $ e _ k$ | … | 10^{−12} |

e_{ω} | [0, 1 000] | 10^{−12} | |

[1 000, 7 500] | 10^{−9} | ||

[7 500, 10 000] | 10^{−7} | ||

[10 000, 20 000] | 10^{−5} |

Parameter . | . | Frequency range (Hz) . | Values . |
---|---|---|---|

Step size | $ \eta k$ | … | 10^{−1} |

η_{ω} | [0, 1 000] | 10^{6} | |

[1 000, 7 500] | 10^{3} | ||

[7 500, 10 000] | 10^{−1} | ||

Threshold | $ e _ k$ | … | 10^{−12} |

e_{ω} | [0, 1 000] | 10^{−12} | |

[1 000, 7 500] | 10^{−9} | ||

[7 500, 10 000] | 10^{−7} | ||

[10 000, 20 000] | 10^{−5} |

### A. Critical points

In the following, the effects of the network topology on the bifurcations are verified and the location of critical points in terms of the feedback strength is investigated numerically.

#### 1. Critical points of two groups of MEMS sensors

#### 2. Dominant critical point

Subsequently, consider two coupled identical groups of MEMS sensors for the case $ k= k 11= k 22$ with a natural frequency $ \omega 1=2\pi \xd73500$ $ 1 s$. This system has in total nine possible Hopf–Hopf bifurcations since there are two polynomials of degree 4 and one polynomial of degree 1. The solution of these polynomials is given by

$ k HH , 1 ( 1 , 2 )=\u22120.3$, $ k HH , 2 ( 1 , 2 )=\u22120.3$, $ k HH , 3 ( 1 , 2 )=0.11$, and $ k HH , 4 ( 1 , 2 )=0.11$ between the bifurcations $1$ and $2$,

$ k HH , 1 ( 1 , 3 )=\u22120.3$, $ k HH , 2 ( 1 , 3 )=\u22120.3$, $ k HH , 3 ( 1 , 3 )=0.11+0.05 i$, and $ k HH , 4 ( 1 , 3 )=0.11\u22120.05 i$ between the bifurcations $1$ and $3$, and

$ k HH ( 2 , 3 )=\u22120.093$ between the bifurcations $2$ and $3$.

### B. Tunability

The tunability of two groups of coupled MEMS sensors is evaluated numerically. This is done by verifying the predictions and investigating the connection between the dominant critical point and the tunability in the sub-critical regime.

#### 1. Characteristic frequency

In the following, the characteristic frequencies $ \omega C , 1$, $ \omega C , 2$, and $ \omega C , 3$ are investigated when the network of two coupled groups is at their corresponding critical points. Furthermore, the variation of the characteristic frequencies in the sub-critical regime is of interest, such that the simulation is performed in terms of the feedback strengths $ k 11\u2208[ k H , 12, k H , 11]$ and $ k 22\u2208[ k H , 22, k H , 21]$. Moreover, the Hopf–Hopf bifurcation between the first and third Andronov–Hopf bifurcation is computed numerically with (25a), since the numerical values of these two Andronov–Hopf bifurcations have the same sign when choosing appropriate parameters. The variations of the characteristic frequencies $ \omega C , 1$, $ \omega C , 2$, and $ \omega C , 3$ in terms of the feedback strengths $ k 11$ and $ k 22$ are shown in Fig. 9. Similar to the two coupled Andronov–Hopf oscillators, the characteristic frequency can be assigned by choosing the feedback strengths $ k 11$ and $ k 22$, accordingly. Interestingly, the simulations lead to the conclusion that the resulting characteristic frequencies can be tuned by controlling the asymmetry of the corresponding network. Moreover, the Hopf–Hopf bifurcation of the bifurcations corresponding to characteristic frequencies $ \omega C , 1$ and $ \omega C , 3$ is depicted in Figs. 9(a) and 9(b) by a blue line. Particularly, the regime above the blue line is interesting for a tunable characteristic frequency, since then the bifurcation corresponding to the characteristic frequency $ \omega C , 1$ is dominant in this regime. Thus, this bifurcation is called tunable from now on. Moreover, this observation leads to the conclusion that the feedback strengths $ k 11$ and $ k 22$ influence, which bifurcation becomes dominant. This comes from the fact that these feedback strengths also influence the eigenvalues, which correspond to the bifurcation.

#### 2. Maximal frequency difference

To evaluate which MEMS sensors can be coupled so that the tunable bifurcation becomes dominant, the maximal difference between the natural frequencies $ \omega 1$ and $ \omega 2$ of two MEMS sensors is evaluated numerically for different Q-factors, i.e., $ Q i\u2208{30,50}$ for all $ i\u2208{1,2}$. For this, denote the normalized frequency for a given frequency $\omega >0$ by $ f=\omega /2\pi $. Then, the analysis is done in two steps:

First, the frequency interval $ [ f _ , f \xaf ]$ in terms of the feedback strengths $ k 11$ and $ k 22$ is investigated. For this, the results obtained for two coupled groups of identical MEMS sensors are used, i.e., the relationship between the critical coupling strengths $ k H , 1$ and $ k H , 2$. This is done by evaluating the intersection of the bifurcation points $ k H , 1$ and $ k H , 2$ numerically.^{63} Second, the frequency difference $\Delta f= | f 1\u2212 f 2 |$ is computed with (25b) for the quantified regions numerically. Note that the maximal interval for this investigation is the hearing range of humans, i.e., the interval $ R=[20,20000]$ Hz.^{12}

The critical feedback strengths of a single MEMS sensor $ k H , 1$ and $ k H , 2$ for the frequency interval $ R$ are depicted in Figs. 10(a) and 10(c) for two different situations. The solid line shows the critical feedback strengths of a MEMS sensor, which is not shifted by feedback. In this situation, the critical feedback strength $ k H , 1$ is smaller than the other critical feedback strength $ k H , 2$ in an interval from approximately $165$ to $9550$ Hz for $ Q i=30$ and from approximately $165$ to $15170$ Hz for $ Q i=50$, respectively. Note that this interval is marked by the gray region in Fig. 10. In particular, the bifurcation of the characteristic frequency $ \omega C , 1$ has a smaller magnitude in these intervals rendering it the dominant bifurcation when coupling two identical MEMS sensors. This can be seen from the fact that the critical point of the two coupled identical MEMS sensors is given by the square of the critical feedback strength of a MEMS sensor. For comparison, the network is moved closer to the critical point of the tunable bifurcation. This is done by setting the self-feedback strength critical of one MEMS sensor to $ k(\omega )=0.8\xd7 k H , 1(\omega )$. The results are depicted by the dashed lines. Interestingly, the interval, in which the tunable bifurcation is dominant, is extended to the whole frequency interval $ R$.

With these considerations, the frequency difference $\Delta f$ is computed numerically. This is done for a system without feedback and a system with feedback relatively close to the critical point of the first MEMS sensor, i.e., $ k 11(\omega )=0.8\xd7 k H , 1(\omega )$ and $ k 22=0$. The results are shown in Figs. 10(b) and 10(d). In the situation without the shifted feedback, the numerical method converges in an interval from approximately $165$ to $9550$ Hz for $ Q i=30$ and from approximately $165$ to $15170$ Hz for $ Q i=50$, respectively. This is depicted by the solid line. Note that in this situation, the dominant bifurcation outside of this interval is given by the characteristic frequency $ \omega C , 3$. This can be changed by assigning a feedback closer to the critical point $ k H , 1$ of the first MEMS sensor. Then, the frequency difference has both a larger interval and a larger magnitude. In addition, it can be concluded that an increased Q-factor increased both the intervals, in which the oscillators without shifted feedback, can be coupled.

With the previous simulation, it is shown that the tunability of the characteristic frequencies is achieved in three independent intervals restricted by the characteristic frequencies of the uncoupled MEMS sensors. Herein, the emergence and its respective characteristic frequency of the dominant bifurcation are controlled by the asymmetry of the network, such that the asymmetries can either improve or deteriorate the tunability of two coupled groups. This comes from the fact that the asymmetry between two coupled groups can change their consensus for a limit cycle, such that the tunable critical point might be dominant.

### C. Reaction of the system

^{64,65}

## VI. CONCLUSIONS

The tunability of the resonance frequency of two coupled groups of oscillators undergoing Andronov–Hopf bifurcations is investigated. For this, the respective critical points in the network are derived. Herein, the eigenvalues of the product of the adjacency matrices between these two groups are the bifurcation parameters. In particular, injectively coupled mathematical models of MEMS sensors described by a dominant mode model are considered and the critical points of the arising Andronov–Hopf bifurcations are computed analytically. It turns out that the two coupled groups can exhibit three Andronov–Hopf bifurcations for each eigenvalue of the product between the cross-coupling matrices. In addition, the resonance frequencies of these Andronov–Hopf bifurcations become tunable within physical limits by adjusting the asymmetry of the network. Moreover, the emergence of Hopf–Hopf bifurcations in these networks is studied numerically. This is done by finding a critical feedback strength or a critical natural frequency such that two bifurcation points are equal, such that regions for different consensus are identified. In view of a practical realization, this analysis yields design rules on how two coupled artificial hair cells to achieve high tunability.

## ACKNOWLEDGMENTS

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 434434223—SFB 1461. The authors would like to thank Dr. Petro Feketa, Kalpan Ved, and Professor Dr. Claudia Lenk for the enlightening discussions and advice.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**H. F. J. Rolf:** Conceptualization (lead); Formal analysis (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **T. Meurer:** Formal analysis (supporting); Funding acquisition (lead); Supervision (supporting); Visualization (supporting); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: DETERMINANT OF A BLOCK MATRIX

#### (Ref. 66)

**(Ref. 66)**

### APPENDIX B: ENVELOPE MODEL OF INJECTIVELY COUPLED MEMS SENSOR

^{64,65}The theoretical values of the compression can be obtained by evaluating the equilibrium of the so-called envelope model,

^{67–69}which can be determined by writing the state vector $ x$ and the external input $ p$ as a time-dependent Fourier series,

^{10}

^{70–73}

### APPENDIX C: CRITICAL POINT OF THE ANDRONOV–HOPF BIFURCATIONS

The equations for the matrix $ Q$ and the resonance frequency are summarized. For this, let $ i=1,2,3$. Then, the bifurcation point of a network consisting of two different MEMS sensor is given by (C1) and (C2). In addition, the nonlinearity $ f q$ of the envelope model for $ N=2$ is given by (C3).

#### 1. Critical point and resonance frequency

#### 2. Auxiliary constants

#### 3. Nonlinearity of the envelope model

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