This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

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1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

2 Physica D 24 (211) Contents lists available at ScienceDirect Physica D journal homepage: Frequency down-conversion using cascading arrays of coupled nonlinear oscillators Visarath In a,, Patrick Longhini a, Andy Kho a, Norman Liu a, Suketu Naik a, Antonio Palacios b, Joseph D. Neff a a Space and Naval Warfare Systems Center, Code Hull Street, San Diego, CA , USA b Nonlinear Dynamical Systems Group, Department of Mathematics, San Diego State University, San Diego, CA 92182, USA a r t i c l e i n f o a b s t r a c t Article history: Received 28 January 21 Received in revised form 3 November 21 Accepted 5 December 21 Available online 6 January 211 Communicated by A. Pikovsky Keywords: Nonlinear oscillators Bistable oscillators Multifrequency Frequency down-conversion Coupled arrays Coupled nonlinear oscillators A novel coupling scheme using M 2 arrays of coupled nonlinear elements arranged in a specific configuration can produce multifrequency patterns or a frequency down-converting effect on an external (input) signal. In such a configuration, each array contains N 3 nonlinear elements with similar dynamics and each element is coupled unidirectionally within the array. The subsequent arrays in the cascade are coupled in a similar fashion except that the coupling direction is arranged in the opposite direction with respect to that of the preceding array. Previous theoretical work and numerical results have already been reported in [P. Longhini, A. Palacios, V. In, J. Neff, A. Kho, A. Bulsara, Exploiting dynamical symmetry in coupled nonlinear elements for efficient frequency down-conversion, Phys. Rev. E 76 (27) 2621]. This paper is focused on results of experiments implemented on two distinct systems: the first system is fabricated using discrete component circuits to approximate an overdamped bistable Duffing oscillator described by a quartic potential system, and the second system is built in a microcircuit, where the nonlinearity is described by a hyperbolic tangent function, with the option of applying an external signal to investigate resonant effects. In particular, the circuit implementations for each case use M = 2 arrays, but their voltage oscillations already demonstrate that the frequency relations between each of the successive arrays decrease by a rational factor, conforming to earlier theoretical and numerical results for the general case containing M arrays. This behavior is important for efficient frequency down-converting applications which are essential in many communication systems where heterodyning is typically used and it involves multi-step processes with complicated circuitry. Published by Elsevier B.V. 1. Introduction The process of generating new frequencies from an original oscillatory signal, either up-converting or down-converting the incoming signal, has been traditionally of interest in physics and engineering with applications that include: sensitive optical detection, music synthesis, acoustic and optical resonators, amplitude modulation, satellite communications, image extraction, and phase-noise measurements [1 7]. In recent works, we developed the theoretical foundations, and the first experimental demonstration, of an innovative technique for frequency up-conversion [8], while later, we extended the theory to include frequency downconversion [9,1]. The underlying principle of this new technique is symmetry [11]. For instance, we have shown that symmetrybreaking Hopf bifurcations in a network of two arrays, each with Corresponding author. Tel.: ; fax: address: visarath@sciences.sdsu.edu (V. In). N oscillators, possessing Z N -symmetry (cyclic group of permutations of N objects) can lead to one array oscillating in synchrony but at N times the frequency of the other array. We emphasize that the multifrequency effect is significantly different from that of subharmonic and ultraharmonic motion generated as is described by Hale and Gambill [12] and later by Tiwari and Subramanian [13]. In our case, the multifrequency behavior arises from the mutual interaction of the arrays via Hopf bifurcations. A distinctive feature of this approach is the model independent nature of its foundations, so it can be readily applicable to a wide range of dynamical classes and systems. Also important is that the approach can lead to various frequency up/down-conversion ratios in a single-step process using the dynamical behaviors as opposed to the multistep process typical of heterodyning/superheterodyning methods which involve complicated circuitries and high precision, stable oscillators. In this work, we show that a frequency down-conversion effect can be realized in physical systems comprising electronic circuits in full agreement with previous theoretical work [1]. The frequency down-conversion effect is achieved through a cascade of /$ see front matter. Published by Elsevier B.V. doi:1.116/j.physd

3 72 V. In et al. / Physica D 24 (211) systems are distinct in nature, which shows the generality and applicability of the frequency down-conversion method. We have limited the experimental work in this paper to two cascading arrays in order to keep the complexity of the circuitry to a minimum while allowing us to demonstrate the phenomenology of frequency down-conversion. Analyses of the effects of noise on the signal and the system are important issues and we defer them to future work. 2. Experimental systems and results Fig. 1. Generalized network configuration of M arrays with N oscillators per array. Each array is coupled with a preferred direction but that direction alternates from one array to the next. The λ i s are coupling strengths within array i while C ij is the strength of coupling from array i to array j. Table 1 Down-conversion ratios between the frequencies of the X 1 -array, ω X1, and X 2 -array, ω X2, for a network of two coupled arrays interconnected as is shown in Fig. 1. k is a positive integer. Number of cells ω X1 /ω X k k k k N N 1 2N 1 Nk 1 M arrays of oscillators, with N oscillators per array, as is illustrated in Fig. 1. The efficiency in lowering the frequency from one array to the next can be used in many applications which require shifting the frequency down from high to low for ease of digitizing the signal using an available analog-to-digital converter suitable for the task, as many communication systems and others routinely do. In the absence of an external signal, and for the particular case of M = 2 symmetrical arrays with three elements N = 3 per array, the down-conversion ratio of the collective frequency of the first array can be 1/2, 1/5, or 1/11. Table 1 shows a generalization of the down-conversion ratios for M = 2 arrays, including commensurate ratios, for various values of N. A more general result for M coupled arrays with N elements in each array obtained through various coupling topologies can be found in [1]. Therefore, the focus of this paper is on the experimental implementations to confirm the existence of those patterns for M = 2. The experiments implemented here are based on two distinct systems. The first one consists of coupled overdamped Duffing oscillators, so the individual dynamics of each element in the array is governed by dx = ax bx 3, where x(t) is the state dt variable at time t, and a and b are coefficients of the linear and nonlinear parts, respectively. This system is constructed using discrete component electronics capable of oscillating up to the kilohertz range. The second experiments consists of coupled bistable systems constructed from integrated circuits where the individual dynamics of each unit cell is described by τ dx = gx + dt I s tanh(c s x), where τ is the time constant of the entire dynamics, g is a linear coefficient, and I s and c s are coefficients controlling the (nonlinearity) bistability of the dynamics. These two experimental. A circuit version of Fig. 1, using a two-array cascade, was constructed using coupled overdamped Duffing oscillators. To simplify the use of the index notation for identifying the unit cells of each array, we will label the first array as the X-array and the second array as the Y -array from here on, and a single index i will suffice. Each array consists of subunits x 1, x 2, and x 3 for the X-array and y 1, y 2, and y 3 for the Y -array. The dynamics of the coupled arrays system are then described by ẋ i = ax i bx i 3 + λ xi (x i+1 ) ẏ i = ay i by i 3 + λ yi (y i+2 ) + C xy x i, where i = 1, 2, 3 mod 3, and x i and y i are the state variables of the ith subunit in the X-array and Y -array, respectively. λ xi is the coupling coefficient of the ith subunit within the X-array, λ yi is the coupling coefficient of the ith subunit within the Y -array, and C xy is the cross-coupling from the X-array to the Y -array. The circuit for each unit consists mainly of operational amplifiers, which act as summing inverting amplifiers, and integrators, and produce the linear and nonlinear terms, which represent the dynamics of a subunit; see Fig. 2. Additional operational amplifiers are used for the unidirectional function of coupling between the oscillators within an array and the terms for the cross-coupling between the arrays, which connect the X-array to the Y -array as shown in the diagram of Fig. 1. The complete diagram of the electronic network of the two arrays is given in Fig. 3. Careful attention is given to the selection of components in order to match the subunits in each array. Even after this is done, we still notice that they differ considerably. This is due to the availability of parts and to the difficulty in producing the nonlinear term, which in the electronic circuit is implemented as a piecewise linear function. All values of the components used in the circuit are given in Table 2. They were selected through hardware simulations, via the SPICE program (an analog electronic circuit simulator), of a model of the circuit. Such hardware simulations are critical for minimizing the trial and error process of testing various resistor and capacitor values that set the coefficients a, b, λ xi, λ yi, and C xy to the appropriate operational regime before fabricating the actual experimental system. When the circuit is first powered up with a ±5. V power supply, λ yi is set to a value slightly below the critical value, λ c, for a sustained oscillation. This critical coupling value was determined in [14] to be λ c = a when the arrays are decoupled from 2 each other. From the circuit diagram, the linear coefficient a is determined to be R 1 which sets R 3 λ c = 1. Setting the coupling 12 value is done by choosing the particular value in the feedback resistors, R cy, where λ cy = R 5. As a result, all the unit cells R cy in the array remain in stationary states. The X-array is set, with λ xi = R 5, to slightly greater than the critical value to initiate the R cx oscillation in an out-of-phase pattern. It may take several trials to get to the correct pattern because the behavior is dependent on the initial condition of the coupled systems. In one instance a different pattern may show up where the oscillations of the unit cells are (1)

4 V. In et al. / Physica D 24 (211) a b Fig. 2. Circuit diagram of the unit cells used in the X-array and the Y -array. (a) shows the detailed circuit diagram of the unit cell (x i ) in the X-array and the other unit cells in this array are constructed similarly. (b) shows the detailed circuit diagram of the unit cell (y i ) in the Y -array and the other unit cells in the array are similar to this. The coupling connections within the arrays and across the array conform to what is illustrated in Fig. 1. Table 2 Component values used in the experimental network circuit of Fig. 3. Component Value Units R 1 1 K R 2 1 K R 3 6 K R 4 1 K R 5 1 K R R 7 1 K R 8 1 K R 9 3 K R cx 5 K R cy 15 K R g 8 K C 1 1 nf in phase, because they share the basin of attraction and the slight variation of the startup conditions. To get to the correct behavior, the initial conditions or states of the array need to be changed. This is done by briefly pinning the voltage at x 1, x 2, or x 3 with a separate power supply; see Fig. 3. In the experiment, x 1 is momentarily set to 1.7 V. After a few tries the array can be induced into its outof-phase state, after which the state is stable and self-sustaining. Soon after this is done, the cross-couplings between the arrays are connected via R g, where C xy = R 3 R g, and the Y -array begins to display out-of-phase oscillation at a frequency 1/5 the value of the X-array frequency. In the experiment the nominal frequency of the X-array is set to run at 189. Hz. When the cross-coupling is connected via R g at the values stated in Table 2, the Y -array is induced to display out-of-phase oscillation amongst its unit cells where the individual frequency is at 37.8 Hz, which is 1/5 of the X- array as shown in Fig. 4. In this figure, the left column shows the traveling wave dynamics of oscillators x 1, x 2, and x 3 in the X- array. The right column shows the dynamics of the corresponding oscillators in the Y -array. The two graphs at the bottom depict the power spectral density plots of the corresponding time series represented in each column. The bottom right graph clearly illustrates that the Y -array is oscillating, within the experimental error, at a fifth of the oscillation of the X-array. As is the case in the numerical simulations shown in [1], the frequency downconversion behavior can only occur within a certain window in parameter space. Other behaviors of the cascade arrays were also investigated by varying λ xi, λ yi, and C xy through their corresponding feedback resistors R cx, R cy and R g, respectively. We observed, besides the 1/5 down-conversion ratio, also 1/2 down-conversion and additional frequency conversion ratios that appear as the parameter R cy changes, as is shown in Fig. 5. In one particular instance, when R cy < 2KΩ, the Y -array oscillates faster than the X-array. These other ratios are expected on the basis of the simulation results obtained in previous work [1], which show good agreement between theory and experiments. The previous experimental system explores the emergence of oscillations and the frequency down-conversion resulting from the inter-array coupling of two arrays, in the absence of an external signal. In a second set of experiments we address the effects of an external signal in a system that was designed and constructed in microelectronics where the oscillation frequency is in the range of 24 MHz to 1.3 GHz, in the range of some RF applications. A diagram of the full experimental system is shown in Fig. 6 where it consists of two cascade arrays, an X-array and a Y -array coupled in a similar fashion to the first system, but with an input signal applied directly to the X-array. Due to the high frequency oscillations, the entire experimental system has to be designed on an integrated circuit board where impedance matching is necessary to apply the external signal to the bistable circuit of the X-array without affecting the Y -array. Fig. 7 shows the experimental board. At the center of the board is the custom-made microcircuit (chip) fabricated using a standard.35 µm feature size chip fabrication process. The rationale for this experiment is to explore further the downconversion effect on the applied signal, in particular, effects of resonance between the natural frequency of oscillation of the individual arrays and that of the external signal. Each subunit in each array is described by an overdamped bistable system containing the hyperbolic tangent function as the nonlinear term given as τ ẋ = gx+i s tanh(c s x). The behavior of the entire system is suitably described by τ ẋ i = gx i + I s tanh(c s x i ) I c tanh(c c x i+1 ) + I g tanh(c g ϵ(t)) τ ẏ i = gy 1 + I s tanh(c s y 1 ) I c tanh(c c y i+2 ) + I g tanh(c g x i ), where the entire dynamics is derived from Kirchhoff s junction law of circuit design and the I c and c c control the coupling strength inside the arrays, the I g and c g control the gains of the input signal in the X-array and the inter-array coupling in the Y -array, τ is proportional to the system time constant which is controlled by the total node parasitic capacitance, and ϵ(t) is the input signal, mainly a sinusoidal or FM modulated signal for the experiment. The parameters c s, c c, c g are the process dependent parameters and they cannot be changed once the microcircuit is designed and (2)

5 74 V. In et al. / Physica D 24 (211) Fig. 3. Circuit implementation of the complete network configuration shown in Fig. 1. The detailed circuit of each unit cell in the X-array and Y -array is given in Fig. 2. Fig. 4. Time series recordings of voltage outputs in the network circuit of Fig. 3. Voltage outputs oscillate in traveling wave patterns in both arrays but the frequencies of the oscillations in the X-array are five times those of the Y -array.

6 V. In et al. / Physica D 24 (211) Fig. 5. Different frequency down-conversion ratios detected in the experimental system as the coupling coefficient λ yi is swept via changing the resistor value R cy. Similar ratios are also observed in numerical simulations of the corresponding model equations. Input Input SigIn SigIn SigIn X1 X2 X2 SigIn SigIn SigIn Y1 Y2 Y3 Fig. 6. Diagram of the microcircuit design for the network configuration of Fig. 1. Power Supply Signal Generator Oscilloscope Fig. 7. Experimental board of a microcircuit implementation of the network configuration of Fig. 1. The board has the ability to apply an external signal to the X-array. Outputs Fig. 8. Two-parameter bifurcation diagram obtained numerically from Eq. (2). It shows different regions of multifrequency patterns in the parameter space (I g2, I c2 ). Regions labeled 1/2, 1/3, and 1/5 correspond to the actual frequency downconversion ratios between the voltage oscillations of two interconnected arrays of three oscillators per array. Region 2IP represents a pattern of oscillation where two units of each array are phase-locked with the same amplitude, with the third unit being out of phase by π. Region 3IP represents full entrainment, frequency and phase locking of voltage oscillations between corresponding oscillatory units of each array. The fixed parameters are: τ =.285 pf, g =.2 Siemens, c s = c c = c g = 7 1, I V c = 195 µa and I s = 384 µa for both X- and Y -arrays, with I g = 95 µa for the X-array, and I g2 and I c2 are the sweep parameters for I g and I c, respectively, in the Y -array. fabricated. In our circuit these parameters are determined to be the same (c s = c c = c g ), with a determined value of approximately 7 1 V. Fig. 8 shows a two-parameter bifurcation diagram obtained, numerically, through Eq. (2). The diagram reveals various regions of different frequency down-conversion ratios that match previous theoretical work [1], in particular, 1/2, 1/3, and 1/5 ratios, and a couple of more subtle patterns identified as 2IP and 3IP. In the 2IP pattern, two oscillatory units (of each array) share the same phase and same amplitude, but the third one is out of phase by π. In the 3IP pattern, there is a complete entrainment between voltage oscillations in the two arrays, i.e., frequency and phase locking of each individual y i element to its corresponding x i element. Fig. 9 illustrates in more detail the layout of a subunit within the X-array with various parts labeled according to their interconnections. A similar design is employed for the subunit in the Y -array with a slight change where the label external signal is replaced with an input from the X-array. So element y 1 receives an input from x 1, y 2 receives input from x 2, and so on. The experimental system is set so that the X-array (receiving array) oscillates at about 365 MHz, and the Y -array is in a region of locking onto 1/5th of that frequency with the proper setting of the system parameters τ, g, I s, I c, and I g. There are a great variety of down-conversion patterns, as mentioned earlier in this paper and in previous theoretical work [1], including 1/2 and 1/5 down-conversion ratios. To aid in setting up the correct parameters that lead to the X-array oscillating out of phase and the Y -array oscillating at 1/5th of that frequency, a model of the experiment was simulated in a SPICE program. Once those parameters are determined and set, the behavior is readily established when powering the circuit. The down-converter board was powered with a 3.3 V power supply. I c, I s, and I g were set by adjusting potentiometers and were measured directly with a multimeter. I c and I s were set to 195 µa

7 76 V. In et al. / Physica D 24 (211) Differential Input Nodes (from adjacent element) External signal Non-linear OTA Block a Non-linear OTA Block b Total node parasitic capacitance Differential Outputt Nodes (to adjacent element) Fig. 9. Illustration of a unit cell in the microcircuit of Fig. 6. Part (a) shows the block diagram of a unit cell in the X-array where the input signal V sig is the applied external signal. Similar construction is used for building the unit cells in the Y -array, but the V sig input point is used for the cross-coupling between an element in the X-array and an element in the Y -array. Part (b) shows the nonlinear OTA block which is constructed from three distinct OTAs where each OTA is represented by the hyperbolic tangent function described in Eq. (2). and 384 µa, respectively, on both the X- and Y -arrays. On the X- array, I g was set to 96 µa, and on the Y -array, it was set to 6.2 µa. The value of τ is set by the total node parasitic capacitances, C 1 and C 2, of.1 pf, and g is set to.2 Siemens. Fig. 1 shows the resulting time series of the X-array (left column) and the time series of the Y -array (right column) along with their associated power spectral density plots in the lower figures. Notice that each individual array is oscillating in a traveling wave pattern with a phase lag of 2π/3. The voltage in the X-array oscillates at 365 MHz and, as expected, the voltage in the Y -array oscillates at 1/5th of those of the X-array. As predicted by the theoretical work [1], the electronic microcircuit can also exhibit other frequency down-conversion ratios, in particular 1/2 and 1/3, and a couple of more subtle patterns that we label as 2IP and 3IP. Fig. 11 shows a twoparameter bifurcation diagram which was obtained directly from the electronic microcircuit for the regions of existence of these patterns in parameter space (I g, I c ). In the 2IP pattern, two oscillatory units of each array share the same phase and amplitude, but the third one is out of phase by π. In the 3IP pattern, there is a complete entrainment between voltage oscillations in the two arrays, i.e., frequency and phase locking of each individual y i element to its corresponding x i element. When an input signal is applied to the X-array with the peakto-peak amplitude of.244 V, the array responds by locking onto that signal at 1/3 of the frequency of the incoming signal while the oscillations of the subunits in the X-array are still out of phase with each other by 2π/3 degrees or 2π/N for larger arrays, and the amplitude response of the array is a constant (maximum) voltage swing of the two-state dynamics of the circuit which in this case is.244 V peak to peak; see Fig. 12 in the left column. The input signal can go from mv ( 7 dbm) down to 1.13 mv peak to peak ( 55 dbm) to still have this lock-on characteristic. For N elements in the array, the phase difference between the elements is 2π/N in each case, and the frequency of each element is 1/N of the input signal. This behavior has been observed and reported.2 X Array.2 Y Array.1.1 Voltage. Voltage MHz MHz Frequency(GHz) Frequency(MHz) Fig. 1. Top: voltage measurements of the electronic microcircuit (see Fig. 7) show oscillatory behavior, in the form of a traveling wave pattern with a phase lag of 2π/3, in the dynamics of two interconnected arrays as it appears in Fig. 1. Bottom: the power spectral density plot confirms that the frequency of the voltage oscillations in the Y -array is, approximately, 1/5th of those in the X-array.

8 V. In et al. / Physica D 24 (211) IP 8 2IP /3 1/5 1/2 Ig 1 = 95µA Ic 1 = 225µA Is 1 = 382µA Is 2 = 382µA Fig. 11. Two-parameter bifurcation diagram obtained directly from the electronic microcircuit of Fig. 7 depicting the boundaries of several multifrequency patterns in parameter space (I g, I c ) of the Y -array. Regions labeled 1/2, 1/3, 1/5, 2IP, and 3IP represent actual frequency down-conversion ratios between the voltage oscillations of two interconnected arrays of three oscillators per array. These regions correspond to those of the numerical bifurcation diagram shown in Fig. 8. Voltage(mV) Input Signal x PSD of Input Signal 72Mhz Frequency(GHz) 1. Voltage(V) X array x PSD of X array 24Mhz Frequency(GHz) 1. Voltage(V) Y Array x PSD of Y array 48Mhz Frequency(GHz) 1. Fig. 12. Voltage measurements of the electronic microcircuit of Fig. 7 modeling the network in Fig. 1 subject to an externally applied signal. When the external signal is applied to the X-array, that array responds by locking its voltage oscillations to 1/3 of the frequency of the incoming signal. Since the two-array network is operating in a region with a frequency down-conversion ratio of 1/5, the combined effect on the signal as it passes through the Y -array is a down-conversion ratio, factor 1/15. The power spectral density plot in the right column validates the 1/15 ratio comparing the top and the bottom figures. in our previous work on the coupled sensor systems [15,1]. Since the first array down-converts the input signal s frequency by 1/3 (for N = 3) and the second array down-converts the output of the first array by a factor of 1/5, the combined effect of the signal passing through the two coupled arrays has a down-conversion factor of 1/15. Fig. 12 confirms a 1/15th down-conversion factor in the electronic microcircuit of Fig. 7 for an input signal at 72 MHz as it travels from the X-array into the Y -array. The power spectral density plot of the voltage oscillations in the Y -array verifies this assertion.

9 78 V. In et al. / Physica D 24 (211) Conclusion In previous theoretical work we demonstrated that certain frequency down-conversion patterns can be induced by the topology of connections of a cascade of arrays of oscillators. The fundamental ideas and methods are dictated by symmetry and, in this regard, they are significantly different from those of standard techniques such as subharmonic synchronization and subharmonic excitation. In this work, we demonstrate the experimental realization of the fundamental ideas that lead to the frequency down-conversion effect through two systems of electronic circuits for different frequency ranges (the low frequency range and megahertz ranges). The experimental results show very good agreement with theoretical work reported in the past. In particular, the bifurcation diagrams obtained directly from the electronic components of the circuit show similar regions of existence of robust patterns of oscillation and of frequency downconversion ratios. We wish to emphasize that the high frequency signal down-conversion to a lower frequency is desirable for easier signal digitization via commonly available analog-to-digital converters (ADCs). Experimental results show that the downconversion is very efficient in the sense that with only two arrays of three oscillators per array, the circuit can already achieve a 1/15th down-conversion ratio for an input signal. This effect has a direct application in some high frequency RF systems where sampling a signal at such high frequency is a major technological problem with the current state-of-the-art ADCs without resorting to some type of heterodyning method. An analysis of the effects of noise on the circuits is beyond the scope of the present work; the analysis is deferred to future work. Acknowledgements We wish to acknowledge support from SPAWAR Systems Center s S&T Program (internal funding), US Airforce, and the Office of Naval Research (Code 3). A.P. was supported in part by the National Science Foundation grant CMMI References [1] K. Otsuka, Pattern recognition with a bidirectionally coupled nonlinearoptical-element system, Opt. Lett. 14 (1989) 925. [2] C. Poynton, Digital Video and HDTV: Algorithms and Interfaces, Morgan Kaufmann Publishers, 23. [3] A. Pikovsky, M. Rosenbleum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge, University Press, UK, 21. [4] E. Mosekilde, Y. Maistrenko, D. Postnov, Chaotic Synchronization: Applications to Living Systems, World Scientific, 22. [5] A. Balanov, N. Janson, D. Postnov, O. Sosnovtseva, Synchronization: From Simple to Complex, Springer, 29. [6] A. Ishida, Y. Inuishi, Time and field variations of acoustic frequency spectrum in amplifying CdS revealed by Brillouin scattering measurements, Phys. Lett. A 27 (1968) 442. [7] E.M. Conwell, A.K. Ganguly, Mixing of acoustic waves in piezoelectric semiconductors, Phys. Rev. B 4 (1971) [8] V. In, A. Palacios, A. Bulsara, P. Longhini, A. Kho, J. Neff, S. Baglio, B. Ando, Complex behavior in driven unidirectionally coupled overdamped Duffing elements, Phys. Rev. E 73 (26) 66121; A. Palacios, R. Carretero, P. Longhini, N. Renz, V. In, A. Kho, J. Neff, B. Meadows, A. Bulsara, Multifrequency synthesis using two coupled nonlinear oscillator arrays, Phys. Rev. E 72 (25) [9] V. In, A. Kho, J. Neff, A. Palacios, P. Longhini, B. Meadows, Experimental observation of multifrequency patterns of arrays of coupled nonlinear oscillators, Phys. Rev. Lett. 91 (24) (23) [1] P. Longhini, A. Palacios, V. In, J. Neff, A. Kho, A. Bulsara, Exploiting dynamical symmetry in coupled nonlinear elements for efficient frequency downconversion, Phys. Rev. E 76 (27) [11] M. Golubitsky, I. Stewart, Patterns of oscillations in coupled systems in geometry, dynamics, and mechanics, in: P. Holmes, P. Newton, A. Weinstein (Eds.), 6th Birthday Volume for J.E. Marsden, Springer-Verlag, 22, pp ; M. Golubitsky, M. Nicol, I. Stewart, Some curious phenomena in coupled cell systems, J. Nonlinear Sci. 14 (24) 27; D. Armbruster, P. Chossat, Remarks on multi-frequency oscillations in symmetrically coupled oscillators, Phys. Lett. A 254 (1999) 269; M. Golubitsky, I.N. Stewart, Interior symmetries in coupled cell networks motivated by the leech heart, Preprint. [12] R. Gambill, J. Hale, Subharmonic and ultraharmonic solutions for weakly nonlinear systems, J. Ration. Mech. Anal. 5 (2) (1956) [13] R. Tiwari, R. Subramanian, Subharmonic and superharmonic synchronization in weakly non-linear systems, J. Sound Vib. 47 (1976) [14] V. In, A. Bulsara, A. Palacios, P. Longhini, A. Kho, J. Neff, Coupling-induced oscillations in overdamped bistable systems, Phys. Rev. E 68 (23) 4512(R). [15] V. In, A.R. Bulsara, A. Palacios, P. Longhini, A. Kho, Complex dynamics in unidirectionally coupled overdamped bistable systems subject to a timeperiodic external signal, Phys. Rev. E 72 (25) 4514(R).