Abstract
This paper presents a digital hardware implementation of a frequency adaptive Hopf oscillator along with investigation on systematic behavior when they are coupled in a population. The mathematical models of the oscillator are introduced and compared in sense of dynamical behavior by using system-level simulations based on which a piecewise-linear model is developed. It is shown that the model is capable to be implemented digitally with high efficiency. Behavior of the oscillators in different network structures to be used for dynamic Fourier analysis is studied and a structure with more precise operation which is also more efficient for FPGA-based implementation is implemented. Conceptual block-diagram and a high level representation for this network structure are shown where design process and synthesis are explained based on which physical implementation is demonstrated and tested.
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A. Ahmadi, E. Mangieri, K. Maharatna, S. Dasmahapatra, M. Zwolinski, On the VLSI implementation of adaptive-frequency hopf oscillator. IEEE Trans. Circuits Syst. I 58(5), 1076–1088 (2011)
J. Buchli, F. Iida, A. J. Ijspeert, Finding resonance: adaptive frequency oscillators for dynamic legged locomotion. in 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, (2006) pp. 3903–3909
J. Buchli, A.J. Ijspeert, A simple, adaptive locomotion toy-system. From animals to animats 8. in Proceedings of the Eighth International Conference on the Simulation of Adaptive Behavior (SAB’04), (MIT Press, 2004), p. 153–162
J. Buchli, L. Righetti, A.J. Ijspeert, Frequency analysis with coupled nonlinear oscillators. Physica D 237(13), 1705–1718 (2008)
K. Chen, D. Wang, A dynamically coupled neural oscillator network for image segmentation. Neural Netw. 15(3), 423–439 (2002)
M.D. Ercegovac, T. Lang, J.-M. Muller, Reciprocation, square root, inverse square root, and some elementary functions using small multipliers. IEEE Trans. Comput. 49(7), 628–637 (2000)
FPGA and HDL Software Package. (Xilinx Inc., San Jose), http://www.xilinx.com/
T.J. Hamilton, C. Jin, J. Tapson, A. van Schaik, A 2-D cochlea with hopf oscillators. in Biomedical Circuits and Systems Conference, (Nov. 2007), pp. 91–94
D.-G. Han, D. Choi, H. Kim, Improved computation of square roots in specific finite fields. IEEE Trans. Comput. 58(2), 188–196 (2009)
C.F. Hoppensteadt, E.M. Izhikevich, Pattern recognition via synchronization in phase-locked loop neural networks. IEEE Trans. Neural Netw. 11(3), 734–738 (2000)
E. Izhikevich, Y. Kuramoto, Encyclopedia of Mathematical Physics. Ch. Weakly Coupled Oscillators (Academic Press, Salt Lake, 2006), pp. 5–448
H.K. Khalil, Nonlinear Systems (Prentice-Hall, Upper Saddle River, 1996)
L.B. Kier, P.G. Seybold, C.-K. Cheng, Modeling Chemical Systems using Cellular Automata (Springer, Dordrecht, 2005)
N. Kopell, L.N. Howard, Pattern formation in the Belousov reaction. Lectures on Mathematics in the life Series. Am. Math. Soc. 7, 201–216 (1974)
T.-J. Kwon, J. Draper, Floating-point division and square root using a Taylor-series expansion algorithm. Microelectron. J. 40, 1601–1605 (2009)
N. MacDonald, Bifurcation theory applied to a simple model of a biochemical oscillator. J. Theoret. Biol. 65(4), 727–734 (1977)
J. Marsden, M. McCracken, The Hopf Bifurcation and its Applications. Applied Mathematical Sciences, vol. 19 (Springer, New York, 1976)
A.I. Mees, P.E. Rapp, Periodic metabolic systems. J. Math. Biol. 5, 99–114 (1978)
A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Upper Saddle River, 1999)
A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge University Press, Cambridge, 2001)
J.-A. Pineiro, J.D. Bruguera, High-speed double-precision computation of reciprocal, division, square root, and inverse square root. IEEE Trans. Comput. 51(12), 1377–1388 (2002)
C.M. Pinto, M. Golubitsky, Central pattern generators for bipedal locomotion. J. Math. Biol. 53(3), 474–489 (2006)
L. Righetti, J. Buchli, A.J. Ijspeert, Dynamic Hebbian learning in adaptive frequency oscillators. Physica D 216(2), 269–281 (2006)
D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)
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Maleki, M.A., Ahmadi, A., Makki, S.V.AD. et al. Networked Adaptive Non-linear Oscillators: A Digital Synthesis and Application. Circuits Syst Signal Process 34, 483–512 (2015). https://doi.org/10.1007/s00034-014-9863-9
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DOI: https://doi.org/10.1007/s00034-014-9863-9