Research Article
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Year 2021, Volume: 3 Issue: 2, 59 - 66, 30.11.2021
https://doi.org/10.51537/chaos.971441

Abstract

References

  • 1999 Chapter 7 - numerical evaluation of fractional derivatives. In Fractional Differential Equations, edited by I. Podlubny, volume 198 of Mathematics in Science and Engineering, pp. 199–221, Elsevier.
  • Ahmad,W., R. El-Khazali, and A. Elwakil, 2001 Fractional-order wien-bridge oscillator. Electronics Letters 37: 1110–1112. Ahmad, W. M. and J. C. Sprott, 2003 Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals 16: 339–351.
  • Arena, P., 2000 Nonlinear noninteger order circuits and systems: an introduction, volume 38. World Scientific.
  • Atangana, A. and B. S. T. Alkahtani, 2015 Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel. Advances in Mechanical Engineering 7: 1687814015591937.
  • Azar, A. T., A. G. Radwan, and S. Vaidyanathan, 2018 Mathematical Techniques of Fractional Order Systems. Elsevier.
  • Caponetto, R., 2010 Fractional order systems: modeling and control applications, volume 72.World Scientific.
  • Carlson, G. and C. Halijak, 1964 Approximation of fractional capacitors (1/s)ˆ(1/n) by a regular newton process. IEEE Transactions on Circuit Theory 11: 210–213.
  • Charef, A., 2006 Analogue realisation of fractional-order integrator, differentiator and fractional piλdμ controller. IEE Proceedings- Control Theory and Applications 153: 714–720.
  • Chen, G. and T. Ueta, 2002 Chaos in circuits and systems˜ world scientific.
  • Chen, L., W. Pan, R. Wu, K. Wang, and Y. He, 2016 Generation and circuit implementation of fractional-order multi-scroll attractors. Chaos, Solitons & Fractals 85: 22–31.
  • Chen, Y. Q. and K. L. Moore, 2002 Discretization schemes for fractional-order differentiators and integrators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49: 363–367.
  • Deniz, F. N., A. Yüce, and N. Tan, 2019 Tuning of pi-pd controller based on standard forms for fractional order systems. Journal of Applied Nonlinear Dynamics 8: 5–21.
  • Elwakil, A. S., 2010 Fractional-order circuits and systems: An emerging interdisciplinary research area. IEEE Circuits and Systems Magazine 10: 40–50.
  • Gómez, F., J. Rosales, and M. Guía, 2013 Rlc electrical circuit of non-integer order. Open Physics 11: 1361–1365.
  • Gómez-Aguilar, J. F., A. Atangana, and V. F. Morales-Delgado, 2017 Electrical circuits rc, lc, and rl described by atangana–baleanu fractional derivatives. International Journal of Circuit Theory and Applications 45: 1514–1533.
  • Gottlieb, H., 1996 Question# 38. what is the simplest jerk function that gives chaos? American Journal of Physics 64: 525–525.
  • Khovanskii, A. N., 1963 The application of continued fractions and their generalizations to problems in approximation theory. Noordhoff Groningen.
  • Kiliç, R., 2010 A practical guide for studying Chua’s circuits, volume 71. World Scientific.
  • Krishna, B. and K. Reddy, 2008 Active and passive realization of fractance device of order 1/2. Active and passive electronic components 2008.
  • Krishna, B. T., 2011 Studies on fractional order differentiators and integrators: A survey. Signal processing 91: 386–426.
  • Matsuda, K. and H. Fujii, 1993 H (infinity) optimized waveabsorbing control-analytical and experimental results. Journal of Guidance, Control, and Dynamics 16: 1146–1153.
  • Maundy, B., A. Elwakil, and S. Gift, 2012 On the realization of multiphase oscillators using fractional-order allpass filters. Circuits, Systems, and Signal Processing 31: 3–17.
  • Oldham, K. and J. Spanier, 1974 The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
  • Oustaloup, A., F. Levron, B. Mathieu, and F. M. Nanot, 2000 Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47: 25–39.
  • Öztürk, I. and R. Kılıç, 2019 Higher dimensional baker map and its digital implementation with lsb-extension method. IEEE Transactions on Circuits and Systems I: Regular Papers 66: 4780–4792.
  • Petráš, I., 2011 Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media.
  • Radwan, A. G., A. S. Elwakil, and A. M. Soliman, 2008 Fractionalorder sinusoidal oscillators: design procedure and practical examples. IEEE Transactions on Circuits and Systems I: Regular Papers 55: 2051–2063.
  • Radwan, A. G., A. S. Elwakil, and A. M. Soliman, 2009 On the generalization of second-order filters to the fractional-order domain. Journal of Circuits, Systems, and Computers 18: 361–386.
  • Radwan, A. G. and K. N. Salama, 2012 Fractional-order rc and rl circuits. Circuits, Systems, and Signal Processing 31: 1901–1915.
  • Sacu, I. E. and M. Alci, 2019 A current mode design of fractional order universal filter. Advances in Electrical and Computer Engineering 19: 71–78.
  • Singh, N., U. Mehta, K. Kothari, and M. Cirrincione, 2020 Optimized fractional low and highpass filters of (1+ α) order on fpaa.
  • Bulletin of the Polish Academy of Sciences. Technical Sciences 68.
  • Sprott, J., 1997 Some simple chaotic jerk functions. American Journal of Physics 65: 537–543.
  • Sprott, J. C., 1994 Some simple chaotic flows. Physical review E 50: R647.
  • Sprott, J. C., 2000a A new class of chaotic circuit. Physics Letters A 266: 19–23.
  • Sprott, J. C., 2000b Simple chaotic systems and circuits. American Journal of Physics 68: 758–763.
  • Stöckmann, H.-J., 1999 Quantum Chaos: An Introduction. Cambridge University Press.
  • Tavazoei, M. S. and M. Haeri, 2007 A necessary condition for double scroll attractor existence in fractional-order systems. Physics Letters A 367: 102–113.
  • Tavazoei, M. S. and M. Haeri, 2008 Chaotic attractors in incommensurate fractional order systems. Physica D: Nonlinear Phenomena 237: 2628–2637.
  • Tlelo-Cuautle, E., A. D. Pano-Azucena, O. Guillén-Fernández, and A. Silva-Juárez, 2020 Analog/digital implementation of fractional order chaotic circuits and applications. Springer.
  • Valsa, J., P. Dvorak, and M. Friedl, 2011 Network model of the cpe. Radioengineering 20: 619–626.

An Efficient Design Procedure to Implement the Fractional-Order Chaotic Jerk Systems with the Programmable Analog Platform

Year 2021, Volume: 3 Issue: 2, 59 - 66, 30.11.2021
https://doi.org/10.51537/chaos.971441

Abstract

An effective design procedure has been introduced for implementing the fractional order integrator structures with a modified low pass filters (LPFs) and its functionality is verified by realizing a fractional-order chaotic system. In these applications, the state variables of the fractional-order Sprott’s Jerk system are emulated by these first order LPFs. Since the discrete device based designs have the hard adjustment features and the circuit complexities; the realizations of these LPFs are carried out with the Field Programmable Analog Arrays (FPAAs), sensitively. Hence, the introduced LPF based method has been applied to the fractional order Sprott’s Jerk systems and these fractional-order systems, which are built by the several nonlinear functions, have been implemented with a programmable analog device. In this context, the minimum fractional-orders of the Sprott’s Jerk systems are calculated by considering the stability of the fractional-order nonlinear systems. After that, these systems are simulated by employing the Grünwald-Letnikov (G-L) fractional derivative method by using a common fractional-order. Thus, the stability analyses of the fractional-order Sprott’s Jerk system are supported by the numerical simulation results. After the numerical simulation stage, the design procedures of the FPAA based implementations of the Sprott’s Jerk systems have been dealt with in detail. Finally, thanks to the introduced first-order LPF method, the hardware realizations of the Sprott’s Jerk systems have been achieved successfully with a single FPAA device.

References

  • 1999 Chapter 7 - numerical evaluation of fractional derivatives. In Fractional Differential Equations, edited by I. Podlubny, volume 198 of Mathematics in Science and Engineering, pp. 199–221, Elsevier.
  • Ahmad,W., R. El-Khazali, and A. Elwakil, 2001 Fractional-order wien-bridge oscillator. Electronics Letters 37: 1110–1112. Ahmad, W. M. and J. C. Sprott, 2003 Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals 16: 339–351.
  • Arena, P., 2000 Nonlinear noninteger order circuits and systems: an introduction, volume 38. World Scientific.
  • Atangana, A. and B. S. T. Alkahtani, 2015 Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel. Advances in Mechanical Engineering 7: 1687814015591937.
  • Azar, A. T., A. G. Radwan, and S. Vaidyanathan, 2018 Mathematical Techniques of Fractional Order Systems. Elsevier.
  • Caponetto, R., 2010 Fractional order systems: modeling and control applications, volume 72.World Scientific.
  • Carlson, G. and C. Halijak, 1964 Approximation of fractional capacitors (1/s)ˆ(1/n) by a regular newton process. IEEE Transactions on Circuit Theory 11: 210–213.
  • Charef, A., 2006 Analogue realisation of fractional-order integrator, differentiator and fractional piλdμ controller. IEE Proceedings- Control Theory and Applications 153: 714–720.
  • Chen, G. and T. Ueta, 2002 Chaos in circuits and systems˜ world scientific.
  • Chen, L., W. Pan, R. Wu, K. Wang, and Y. He, 2016 Generation and circuit implementation of fractional-order multi-scroll attractors. Chaos, Solitons & Fractals 85: 22–31.
  • Chen, Y. Q. and K. L. Moore, 2002 Discretization schemes for fractional-order differentiators and integrators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49: 363–367.
  • Deniz, F. N., A. Yüce, and N. Tan, 2019 Tuning of pi-pd controller based on standard forms for fractional order systems. Journal of Applied Nonlinear Dynamics 8: 5–21.
  • Elwakil, A. S., 2010 Fractional-order circuits and systems: An emerging interdisciplinary research area. IEEE Circuits and Systems Magazine 10: 40–50.
  • Gómez, F., J. Rosales, and M. Guía, 2013 Rlc electrical circuit of non-integer order. Open Physics 11: 1361–1365.
  • Gómez-Aguilar, J. F., A. Atangana, and V. F. Morales-Delgado, 2017 Electrical circuits rc, lc, and rl described by atangana–baleanu fractional derivatives. International Journal of Circuit Theory and Applications 45: 1514–1533.
  • Gottlieb, H., 1996 Question# 38. what is the simplest jerk function that gives chaos? American Journal of Physics 64: 525–525.
  • Khovanskii, A. N., 1963 The application of continued fractions and their generalizations to problems in approximation theory. Noordhoff Groningen.
  • Kiliç, R., 2010 A practical guide for studying Chua’s circuits, volume 71. World Scientific.
  • Krishna, B. and K. Reddy, 2008 Active and passive realization of fractance device of order 1/2. Active and passive electronic components 2008.
  • Krishna, B. T., 2011 Studies on fractional order differentiators and integrators: A survey. Signal processing 91: 386–426.
  • Matsuda, K. and H. Fujii, 1993 H (infinity) optimized waveabsorbing control-analytical and experimental results. Journal of Guidance, Control, and Dynamics 16: 1146–1153.
  • Maundy, B., A. Elwakil, and S. Gift, 2012 On the realization of multiphase oscillators using fractional-order allpass filters. Circuits, Systems, and Signal Processing 31: 3–17.
  • Oldham, K. and J. Spanier, 1974 The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
  • Oustaloup, A., F. Levron, B. Mathieu, and F. M. Nanot, 2000 Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47: 25–39.
  • Öztürk, I. and R. Kılıç, 2019 Higher dimensional baker map and its digital implementation with lsb-extension method. IEEE Transactions on Circuits and Systems I: Regular Papers 66: 4780–4792.
  • Petráš, I., 2011 Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media.
  • Radwan, A. G., A. S. Elwakil, and A. M. Soliman, 2008 Fractionalorder sinusoidal oscillators: design procedure and practical examples. IEEE Transactions on Circuits and Systems I: Regular Papers 55: 2051–2063.
  • Radwan, A. G., A. S. Elwakil, and A. M. Soliman, 2009 On the generalization of second-order filters to the fractional-order domain. Journal of Circuits, Systems, and Computers 18: 361–386.
  • Radwan, A. G. and K. N. Salama, 2012 Fractional-order rc and rl circuits. Circuits, Systems, and Signal Processing 31: 1901–1915.
  • Sacu, I. E. and M. Alci, 2019 A current mode design of fractional order universal filter. Advances in Electrical and Computer Engineering 19: 71–78.
  • Singh, N., U. Mehta, K. Kothari, and M. Cirrincione, 2020 Optimized fractional low and highpass filters of (1+ α) order on fpaa.
  • Bulletin of the Polish Academy of Sciences. Technical Sciences 68.
  • Sprott, J., 1997 Some simple chaotic jerk functions. American Journal of Physics 65: 537–543.
  • Sprott, J. C., 1994 Some simple chaotic flows. Physical review E 50: R647.
  • Sprott, J. C., 2000a A new class of chaotic circuit. Physics Letters A 266: 19–23.
  • Sprott, J. C., 2000b Simple chaotic systems and circuits. American Journal of Physics 68: 758–763.
  • Stöckmann, H.-J., 1999 Quantum Chaos: An Introduction. Cambridge University Press.
  • Tavazoei, M. S. and M. Haeri, 2007 A necessary condition for double scroll attractor existence in fractional-order systems. Physics Letters A 367: 102–113.
  • Tavazoei, M. S. and M. Haeri, 2008 Chaotic attractors in incommensurate fractional order systems. Physica D: Nonlinear Phenomena 237: 2628–2637.
  • Tlelo-Cuautle, E., A. D. Pano-Azucena, O. Guillén-Fernández, and A. Silva-Juárez, 2020 Analog/digital implementation of fractional order chaotic circuits and applications. Springer.
  • Valsa, J., P. Dvorak, and M. Friedl, 2011 Network model of the cpe. Radioengineering 20: 619–626.
There are 41 citations in total.

Details

Primary Language English
Subjects Electrical Engineering
Journal Section Research Articles
Authors

Nimet Korkmaz 0000-0002-7419-1538

İbrahim Ethem Saçu 0000-0002-8627-8278

Publication Date November 30, 2021
Published in Issue Year 2021 Volume: 3 Issue: 2

Cite

APA Korkmaz, N., & Saçu, İ. E. (2021). An Efficient Design Procedure to Implement the Fractional-Order Chaotic Jerk Systems with the Programmable Analog Platform. Chaos Theory and Applications, 3(2), 59-66. https://doi.org/10.51537/chaos.971441

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830 28903   

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