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A design template for globally stable nonlinear oscillators with multiple dynamics

Year 2024, Volume: 13 Issue: 4, 1192 - 1200, 15.10.2024
https://doi.org/10.28948/ngumuh.1458253

Abstract

In this study, a template is proposed for designing ordinary differential equation-based nonlinear oscillators. The template is a two-dimensional system with two control parameters and an energy function. By choosing the appropriate energy function, it is possible to obtain globally stable systems. These systems can be a gradient system, a Hamiltonian system, or a system with a stable limit cycle depending on the choice of control parameters. Hamiltonian and limit cycle cases can be used as a nonlinear oscillator for various applications. An example system is demonstrated by choosing a simple energy function and the obtained system is simulated to verify its dynamics. Hardware verification of the simulated system is performed with a field programmable gate array (FPGA) implementation.

References

  • P. Kakou, S. K. Gupta, and O. Barry, A nonlinear analysis of a Duffing oscillator with a nonlinear electromagnetic vibration absorber–inerter for concurrent vibration mitigation and energy harvesting. Nonlinear Dynamics, 112(8), 5847–5862, 2024. https://doi.org/10.1007/s11071-023-09163-6.
  • W. Tian, T. Zhao, and Z. Yang, Supersonic meta-plate with tunable-stiffness nonlinear oscillators for nonlinear flutter suppression. International Journal of Mechanical Sciences, 229107533, 2022. https://doi.org/10.1016/j.ijmecsci.2022.107533.
  • Y. Zhu, Y. Wu, Q. Liu, T. Guo, R. Qin, and J. Hui, A backward control based on σ-Hopf oscillator with decoupled parameters for smooth locomotion of bio-inspired legged robot. Robotics and Autonomous Systems, 106165–178, 2018. https://doi.org/10.1016/j.robot.2018.05.009.
  • A. J. Adéchinan, Y. J. F. Kpomahou, L. A. Hinvi, and C. H. Miwadinou, Chaos, coexisting attractors and chaos control in a nonlinear dissipative chemical oscillator. Chinese Journal of Physics, 772684–2697, 2022. https://doi.org/10.1016/j.cjph.2022.03.052.
  • F. Morán and A. Goldbeter, Onset of birhythmicity in a regulated biochemical system. Biophysical Chemistry, 20(1–2), 149–156, 1984. https://doi.org/10.1016/0301-4622(84)80014-9.
  • A. Goldbeter, Computational approaches to cellular rhythms. Nature, 420(6912), 238–245, 2002. https://doi.org/10.1038/nature01259.
  • H. G. Mayr and K. H. Schatten, Nonlinear oscillators in space physics. Journal of Atmospheric and Solar-Terrestrial Physics, 7444–50, 2012. https://doi.org/10.1016/j.jastp.2011.09.008.
  • X. Huang and Q. Cao, A novel nonlinear oscillator consisting torsional springs and rigid rods. International Journal of Non-Linear Mechanics, 161104684, 2024. https://doi.org/10.1016/j.ijnonlinmec.2024.104684.
  • N. Inaba, H. Okazaki, and H. Ito, Nested mixed-mode oscillations in the forced van der Pol oscillator. Communications in Nonlinear Science and Numerical Simulation, 133107932, 2024. https://doi.org/10.1016/j.cnsns.2024.107932.
  • L. Zhu, Dynamics of switching van der Pol circuits. Nonlinear Dynamics, 87(2), 1217–1234, 2017. https://doi.org/10.1007/s11071-016-3111-8.
  • A. Jenkins, Self-oscillation. Physics Reports, 525(2), 167–222, 2013. https://doi.org/10.1016/j.physrep.2012.10.007.
  • E. M. Izhikevich, Simple model of spiking neurons. IEEE Transactions on Neural Networks, 14(6), 1569–1572, 2003. https://doi.org/10.1109/TNN.2003.820440.
  • N. Korkmaz, İ. Öztürk, and R. Kılıç, Modeling, simulation, and implementation issues of CPGs for neuromorphic engineering applications. Computer Applications in Engineering Education, 26(4), 782–803, 2018. https://doi.org/10.1002/cae.21972.
  • N. Dahasert, İ. Öztürk, and R. Kılıç, Experimental realizations of the HR neuron model with programmable hardware and synchronization applications. Nonlinear Dynamics, 70(4), 2343–2358, 2012. https://doi.org/10.1007/s11071-012-0618-5.
  • N. Korkmaz, İ. Öztürk, and R. Kılıç, Multiple perspectives on the hardware implementations of biological neuron models and programmable design aspects. Turkish Journal of Electrical Engineering and Computer Sciences, 24(3), 1729–1746, 2016. https://doi.org/10.3906/elk-1309-5.
  • Yu. A. Tsybina, S. Yu. Gordleeva, A. I. Zharinov, I. A. Kastalskiy, A. V. Ermolaeva, A. E. Hramov, and V. B. Kazantsev, Toward biomorphic robotics: A review on swimming central pattern generators. Chaos, Solitons & Fractals, 165112864, 2022. https://doi.org/10.1016/j.chaos.2022.112864.
  • N. K. Kasabov, Time-Space, Spiking Neural Networks and Brain-Inspired Artificial Intelligence, Springer Series on Bio and Neurosystems, Springer, Berlin, Heidelberg, 2019.
  • E. E. Sel’kov, Self-oscillations in glycolysis 1. A simple kinetic model. European Journal of Biochemistry, 4(1), 79–86, 1968. https://doi.org/10.1111/j.1432-1033.1968.tb00175.x.
  • P. Belardinelli and S. Lenci, A first parallel programming approach in basins of attraction computation. International Journal of Non-Linear Mechanics, 8076–81, 2016. https://doi.org/10.1016/j.ijnonlinmec.2015.10.016.
  • J. K. Moser, Lectures on Hamiltonian systems. In Hamiltonian Dynamical Systems, CRC Press, 1987.
  • S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, 2nd Edn., CRC Press, 2019.
  • J. Cortés, A. J. van der Schaft, and P. E. Crouch, Gradient realization of nonlinear control systems. IFAC Proceedings Volumes, 36(2), 63–68, 2003. https://doi.org/10.1016/S1474-6670(17)38868-7.
  • İ. Öztürk and R. Kılıç, A novel method for producing pseudo random numbers from differential equation-based chaotic systems. Nonlinear Dynamics, 80(3), 1147–1157, 2015. https://doi.org/10.1007/s11071-015-1932-5.
  • P. J. Channell and C. Scovel, Symplectic integration of Hamiltonian systems. Nonlinearity, 3(2), 231, 1990. https://doi.org/10.1088/0951-7715/3/2/001.
  • J. Sundnes, Solving ordinary differential equations in python, Springer Nature, 2024.

Çoklu dinamiklere sahip global kararlı lineer olmayan osilatörler için bir tasarım şablonu

Year 2024, Volume: 13 Issue: 4, 1192 - 1200, 15.10.2024
https://doi.org/10.28948/ngumuh.1458253

Abstract

Bu çalışmada adi diferansiyel denklem tabanlı lineer olmayan osilatörlerin tasarımı için bir şablon önerilmektedir. Önerilen şablon iki adet kontrol parametresine ve bir adet enerji fonksiyonuna sahip iki boyutlu bir sistemdir. Uygun enerji fonksiyonu seçimi ile global olarak kararlı sistemler elde etmek mümkün olmaktadır. Bu sistemler farklı kontrol parametreleri için gradyan sistem, Hamilton sistemi veya kararlı bir limit çevrime sahip bir sistem olabilmektedir. Hamilton sistemi ve limit çevrim durumlarında sistem farklı uygulamalar için lineer olmayan osilatör olarak kullanılabilir. Basit bir enerji fonksiyonu seçimiyle örnek bir sistem elde edilip sistemin dinamikleri simülasyonlarla doğrulanmıştır. Simülasyonu yapılan sistemin donanımsal doğrulaması ise alanda programlanabilir kapı dizisi (FPGA) kullanılarak gerçekleştirilmiştir.

References

  • P. Kakou, S. K. Gupta, and O. Barry, A nonlinear analysis of a Duffing oscillator with a nonlinear electromagnetic vibration absorber–inerter for concurrent vibration mitigation and energy harvesting. Nonlinear Dynamics, 112(8), 5847–5862, 2024. https://doi.org/10.1007/s11071-023-09163-6.
  • W. Tian, T. Zhao, and Z. Yang, Supersonic meta-plate with tunable-stiffness nonlinear oscillators for nonlinear flutter suppression. International Journal of Mechanical Sciences, 229107533, 2022. https://doi.org/10.1016/j.ijmecsci.2022.107533.
  • Y. Zhu, Y. Wu, Q. Liu, T. Guo, R. Qin, and J. Hui, A backward control based on σ-Hopf oscillator with decoupled parameters for smooth locomotion of bio-inspired legged robot. Robotics and Autonomous Systems, 106165–178, 2018. https://doi.org/10.1016/j.robot.2018.05.009.
  • A. J. Adéchinan, Y. J. F. Kpomahou, L. A. Hinvi, and C. H. Miwadinou, Chaos, coexisting attractors and chaos control in a nonlinear dissipative chemical oscillator. Chinese Journal of Physics, 772684–2697, 2022. https://doi.org/10.1016/j.cjph.2022.03.052.
  • F. Morán and A. Goldbeter, Onset of birhythmicity in a regulated biochemical system. Biophysical Chemistry, 20(1–2), 149–156, 1984. https://doi.org/10.1016/0301-4622(84)80014-9.
  • A. Goldbeter, Computational approaches to cellular rhythms. Nature, 420(6912), 238–245, 2002. https://doi.org/10.1038/nature01259.
  • H. G. Mayr and K. H. Schatten, Nonlinear oscillators in space physics. Journal of Atmospheric and Solar-Terrestrial Physics, 7444–50, 2012. https://doi.org/10.1016/j.jastp.2011.09.008.
  • X. Huang and Q. Cao, A novel nonlinear oscillator consisting torsional springs and rigid rods. International Journal of Non-Linear Mechanics, 161104684, 2024. https://doi.org/10.1016/j.ijnonlinmec.2024.104684.
  • N. Inaba, H. Okazaki, and H. Ito, Nested mixed-mode oscillations in the forced van der Pol oscillator. Communications in Nonlinear Science and Numerical Simulation, 133107932, 2024. https://doi.org/10.1016/j.cnsns.2024.107932.
  • L. Zhu, Dynamics of switching van der Pol circuits. Nonlinear Dynamics, 87(2), 1217–1234, 2017. https://doi.org/10.1007/s11071-016-3111-8.
  • A. Jenkins, Self-oscillation. Physics Reports, 525(2), 167–222, 2013. https://doi.org/10.1016/j.physrep.2012.10.007.
  • E. M. Izhikevich, Simple model of spiking neurons. IEEE Transactions on Neural Networks, 14(6), 1569–1572, 2003. https://doi.org/10.1109/TNN.2003.820440.
  • N. Korkmaz, İ. Öztürk, and R. Kılıç, Modeling, simulation, and implementation issues of CPGs for neuromorphic engineering applications. Computer Applications in Engineering Education, 26(4), 782–803, 2018. https://doi.org/10.1002/cae.21972.
  • N. Dahasert, İ. Öztürk, and R. Kılıç, Experimental realizations of the HR neuron model with programmable hardware and synchronization applications. Nonlinear Dynamics, 70(4), 2343–2358, 2012. https://doi.org/10.1007/s11071-012-0618-5.
  • N. Korkmaz, İ. Öztürk, and R. Kılıç, Multiple perspectives on the hardware implementations of biological neuron models and programmable design aspects. Turkish Journal of Electrical Engineering and Computer Sciences, 24(3), 1729–1746, 2016. https://doi.org/10.3906/elk-1309-5.
  • Yu. A. Tsybina, S. Yu. Gordleeva, A. I. Zharinov, I. A. Kastalskiy, A. V. Ermolaeva, A. E. Hramov, and V. B. Kazantsev, Toward biomorphic robotics: A review on swimming central pattern generators. Chaos, Solitons & Fractals, 165112864, 2022. https://doi.org/10.1016/j.chaos.2022.112864.
  • N. K. Kasabov, Time-Space, Spiking Neural Networks and Brain-Inspired Artificial Intelligence, Springer Series on Bio and Neurosystems, Springer, Berlin, Heidelberg, 2019.
  • E. E. Sel’kov, Self-oscillations in glycolysis 1. A simple kinetic model. European Journal of Biochemistry, 4(1), 79–86, 1968. https://doi.org/10.1111/j.1432-1033.1968.tb00175.x.
  • P. Belardinelli and S. Lenci, A first parallel programming approach in basins of attraction computation. International Journal of Non-Linear Mechanics, 8076–81, 2016. https://doi.org/10.1016/j.ijnonlinmec.2015.10.016.
  • J. K. Moser, Lectures on Hamiltonian systems. In Hamiltonian Dynamical Systems, CRC Press, 1987.
  • S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, 2nd Edn., CRC Press, 2019.
  • J. Cortés, A. J. van der Schaft, and P. E. Crouch, Gradient realization of nonlinear control systems. IFAC Proceedings Volumes, 36(2), 63–68, 2003. https://doi.org/10.1016/S1474-6670(17)38868-7.
  • İ. Öztürk and R. Kılıç, A novel method for producing pseudo random numbers from differential equation-based chaotic systems. Nonlinear Dynamics, 80(3), 1147–1157, 2015. https://doi.org/10.1007/s11071-015-1932-5.
  • P. J. Channell and C. Scovel, Symplectic integration of Hamiltonian systems. Nonlinearity, 3(2), 231, 1990. https://doi.org/10.1088/0951-7715/3/2/001.
  • J. Sundnes, Solving ordinary differential equations in python, Springer Nature, 2024.
There are 25 citations in total.

Details

Primary Language English
Subjects Circuits and Systems, Numerical Design
Journal Section Research Articles
Authors

İsmail Öztürk 0000-0001-9561-4651

Early Pub Date September 4, 2024
Publication Date October 15, 2024
Submission Date March 25, 2024
Acceptance Date July 25, 2024
Published in Issue Year 2024 Volume: 13 Issue: 4

Cite

APA Öztürk, İ. (2024). A design template for globally stable nonlinear oscillators with multiple dynamics. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, 13(4), 1192-1200. https://doi.org/10.28948/ngumuh.1458253
AMA Öztürk İ. A design template for globally stable nonlinear oscillators with multiple dynamics. NOHU J. Eng. Sci. October 2024;13(4):1192-1200. doi:10.28948/ngumuh.1458253
Chicago Öztürk, İsmail. “A Design Template for Globally Stable Nonlinear Oscillators With Multiple Dynamics”. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi 13, no. 4 (October 2024): 1192-1200. https://doi.org/10.28948/ngumuh.1458253.
EndNote Öztürk İ (October 1, 2024) A design template for globally stable nonlinear oscillators with multiple dynamics. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi 13 4 1192–1200.
IEEE İ. Öztürk, “A design template for globally stable nonlinear oscillators with multiple dynamics”, NOHU J. Eng. Sci., vol. 13, no. 4, pp. 1192–1200, 2024, doi: 10.28948/ngumuh.1458253.
ISNAD Öztürk, İsmail. “A Design Template for Globally Stable Nonlinear Oscillators With Multiple Dynamics”. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi 13/4 (October 2024), 1192-1200. https://doi.org/10.28948/ngumuh.1458253.
JAMA Öztürk İ. A design template for globally stable nonlinear oscillators with multiple dynamics. NOHU J. Eng. Sci. 2024;13:1192–1200.
MLA Öztürk, İsmail. “A Design Template for Globally Stable Nonlinear Oscillators With Multiple Dynamics”. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, vol. 13, no. 4, 2024, pp. 1192-00, doi:10.28948/ngumuh.1458253.
Vancouver Öztürk İ. A design template for globally stable nonlinear oscillators with multiple dynamics. NOHU J. Eng. Sci. 2024;13(4):1192-200.

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