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Networked Adaptive Non-linear Oscillators: A Digital Synthesis and Application

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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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References

  1. 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)

    Article  MathSciNet  Google Scholar 

  2. 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

  3. 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

  4. J. Buchli, L. Righetti, A.J. Ijspeert, Frequency analysis with coupled nonlinear oscillators. Physica D 237(13), 1705–1718 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Chen, D. Wang, A dynamically coupled neural oscillator network for image segmentation. Neural Netw. 15(3), 423–439 (2002)

    Article  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. FPGA and HDL Software Package. (Xilinx Inc., San Jose), http://www.xilinx.com/

  8. 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

  9. D.-G. Han, D. Choi, H. Kim, Improved computation of square roots in specific finite fields. IEEE Trans. Comput. 58(2), 188–196 (2009)

    Article  MathSciNet  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. E. Izhikevich, Y. Kuramoto, Encyclopedia of Mathematical Physics. Ch. Weakly Coupled Oscillators (Academic Press, Salt Lake, 2006), pp. 5–448

    Google Scholar 

  12. H.K. Khalil, Nonlinear Systems (Prentice-Hall, Upper Saddle River, 1996)

    Google Scholar 

  13. L.B. Kier, P.G. Seybold, C.-K. Cheng, Modeling Chemical Systems using Cellular Automata (Springer, Dordrecht, 2005)

    Google Scholar 

  14. 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)

    MathSciNet  Google Scholar 

  15. T.-J. Kwon, J. Draper, Floating-point division and square root using a Taylor-series expansion algorithm. Microelectron. J. 40, 1601–1605 (2009)

    Article  Google Scholar 

  16. N. MacDonald, Bifurcation theory applied to a simple model of a biochemical oscillator. J. Theoret. Biol. 65(4), 727–734 (1977)

    Article  Google Scholar 

  17. J. Marsden, M. McCracken, The Hopf Bifurcation and its Applications. Applied Mathematical Sciences, vol. 19 (Springer, New York, 1976)

    Book  Google Scholar 

  18. A.I. Mees, P.E. Rapp, Periodic metabolic systems. J. Math. Biol. 5, 99–114 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Upper Saddle River, 1999)

    Google Scholar 

  20. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge University Press, Cambridge, 2001)

    Book  Google Scholar 

  21. 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)

    Article  MathSciNet  Google Scholar 

  22. C.M. Pinto, M. Golubitsky, Central pattern generators for bipedal locomotion. J. Math. Biol. 53(3), 474–489 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Righetti, J. Buchli, A.J. Ijspeert, Dynamic Hebbian learning in adaptive frequency oscillators. Physica D 216(2), 269–281 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)

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Correspondence to Arash Ahmadi.

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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

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