ارائه یک سیستم فوق‌آشوب جدید و تحقق فیزیکی آن از طریق طراحی و ساخت یک مدار الکترونیکی آنالوگ

نوع مقاله : مقاله پژوهشی فارسی

نویسندگان

1 بخش کنترل و الکترونیک، دانشکده مهندسی برق، دانشگاه یزد، یزد، ایران.

2 بخش مهندسی قدرت و کنترل، دانشکده مهندسی برق و کامپیوتر، دانشگاه شیراز، شیراز، ایران

چکیده

در این مقاله، سیستم فوق‌آشوبی جدیدی تنها با یک نقطه تعادل مبدا ارائه می‌گردد که بر پایه‌ی سیستم‌ آشوبی معمولی لیو ساخته شده است. به‌منظور نشان دادن وجود پدیده فوق‌آشوب در این سیستم‌، تعدادی از معیارهای محاسباتی و ترسیمی مورد بررسی و تحلیل قرار می‌گیرند. از مهمترین این معیارها و شاخص‌ها می‌توان به بررسی اتلافی بودن، اثبات ناپایداری نقطه تعادل، ترسیم صفحه‌های فاز جاذب عجیب سیستم، بررسی پاسخ‌های زمانی‌متغیرهای حالت، محاسبه نماهای لیاپانوف، محاسبه بعد کسری سیستم و آنالیز حساسیت شدید پاسخ‌های متغیرهای حالت به تغییرات اندک در شرایط اولیه اشاره کرد. در ادامه نشان داده می‌شود که سیستم فوق‌آشوب پیشنهادی دارای دو نمای لیاپانوف مثبت بسیار بزرگ در مقایسه با سیستم‌های فوق‌آشوبی دیگر است. بررسی‌ها بر روی سیستم معرفی‌شده، این نتیجه را نمایان می‌کند که تغییر هر کدام از پارامترهای سیستم، باعث ایجاد رفتارهای گوناگون دینامیکی از جمله آشوب معمولی (آشوب بعد پایین)، سیکل‌حدی، شبه‌پریودیک و فوق‌آشوبی می‌شود. برای تحقق فیزیکی سیستم فوق‌آشوبی، مدار الکترونیکی آنالوگی طراحی می‌شود که از عناصر ساده‌ای همچون مقاومت‌های خطی، خازن‌های خطی، تقویت‌کننده‌های عملیاتی و ضرب‌کننده‌های آنالوگ تشکیل شده است. در ادامه، ابتدا با استفاده از نرم‌افزار ORCAD16.6 مدار را شبیه‌سازی کرده و در مرحله بعد مدار به صورت عملی در آزمایشگاه ساخته شده و مورد تست واقعی قرار می‌گیرد. نتایج حاصل از شبیه‌سازی با نرم‌افزار ORCAD16.6 و داده‌های آزمایشگاهی نشان می‌دهند که پدیده‌ی غیرخطی فوق‌آشوب در مدار آنالوگ رخ می‌دهد.

کلیدواژه‌ها


 [1] B. Samardzic, and B. M. Zlatkovic, “Analysis of spatial chaos appearance in cascade connected nonlinear electrical circuits,” Chaos, Solitons & Fractals, vol. 95, no. 1, pp. 14-20, 2017.
[2] C. Volos, J. O. Maaita, S. Vaidyanathan, V. T. Pham, I. Stouboulos, and I. Kyprianidis, “A novel four-dimensional hyperchaotic four-wing system with a saddle–focus equilibrium,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 64, no. 3, pp. 339-343, 2017.
[3] I. Bodale, and V. A. Oancea, “Chaos control for Willamowski–Rössler model of chemical reactions,” Chaos, Solitons & Fractals, vol. 78, no. 1, pp. 1-9, 2015.
[4] A. Buscarino, C. Corradino, L. Fortuna, M. Frasca, and J. C. Sprott, “Nonideal behavior of analog multipliers for chaos generation,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 63, no. 4, pp. 396-400, 2016.
[5] N. Zamani, M. Ataei, and M. Niroomand, “Analysis and control of chaotic behavior in boost converter by ramp compensation based on Lyapunov exponents assignment: theoretical and experimental investigation,” Chaos, Solitons & Fractals, vol. 81, no. 1, pp. 20-29, 2015.
[6] X. Wu, D. Wang, J. Kurths, and H. Kan, “A novel lossless color image encryption scheme using 2D DWT and 6D hyperchaotic system,” Information Sciences, vol. 349-350, no. 1, pp. 137-153, 2016.
[7] C. Xue, N. Jiang, Y. Lv, and K. Qiu, “Secure key distribution based on dynamic chaos synchronization of cascaded semiconductor laser systems,” IEEE Transactions on Communications, vol. 65, no. 1, pp. 312-319, 2017.
[8] Y. Scharf, “A chaotic outlook on biological systems,” Chaos, Solitons & Fractals, vol. 95, no. 1, pp. 42-47, 2017.
[9] H. Yu, G. Cai, and Y. Li, “Dynamic analysis and control of a new hyperchaotic finance system,” Nonlinear Dynamics, vol. 67, no. 3, pp. 2171-2182, 2012.
[10] G.A. Leonov, N.V. Kuznetsov, N.A. Korzhemanova, and D.V. Kusakin, “Lyapunov dimension formula for the global attractor of the Lorenz system,” Communications in Nonlinear Science and Numerical Simulation, vol. 41, no. 1, pp. 84-103, 2016.
[11] O. E. Rössler, “An equation for continuous chaos,” Physics Letters A, vol. 57, no. 1, pp. 397-398, 1976.
[12] Z. Chen, Y. Yuang, G. Qi, and Z. Yuan, “A novel hyperchaos system only with one equilibrium,” Physics Letters A, vol. 36, no. 6, pp. 696-701, 2007.
[13] D. Cafagna, and G. Grassi, “New 3D scroll attractors in hyperchaotic chua’s circuit forming a ring,” International Journal of Bifurcation and Chaos, vol. 13, no. 10, pp. 2889-2903, 2003.
[14] Y. Li, W. K. S. Tang, and G. Chen, “Hyperchaos evolved from the generalized Lorenz equation,” International Journal of Circuit Theory and Applications, vol. 33, no. 4, pp. 235-251, 2005.
[15] C. Liu, T. Liu, L. Liu, and K. Liu, “A novel chaotic attractor,” Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1031-1038, 2004.
[16] A. Abooee, and M. R. Jahed-Motlagh, “Analysis and circuitry realization of a novel three-dimensional chaotic system,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 5, pp. 1235-1245, 2013.
[17] Z. Shen, and J. Li, “Chaos control for a unified chaotic system using output feedback controllers,” Mathematics and Computers in Simulation, vol. 132, no. 1, pp. 208-219, 2017.
[18] R. Barrio, M. A. Martínez, S. Serrano, and D. Wilczak, “When chaos meets hyperchaos: 4D Rössler model,” Physics Letters A, vol. 379, no. 38, pp. 2300-2305, 2015.
[19] W. Wu, Z. Chen and Z. Yuan, “The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos, and hyperchaos,” Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2340-2356, 2009.
[21] A. Abooee, and M. R. Jahed-Motlagh, “A new hyperchaotic secure communication scheme and its circuitry realization,” In Proceeding of 22th Iranian Conference on Electrical Engineering (ICEE2014), 20-22 May 2014, Shahid Beheshti University, Tehran, Iran, pp. 1295-1300, 2014.
[22] Y. Li, G. Chen, and W. K. S. Tang, “Controlling a unified chaotic system to hyperchaotic,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 52, no. 4, pp. 204-207, 2005.
[23] C. Huang, and J. Cao, “Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system,” Physica A: Statistical Mechanics and its Applications, vol. 473, no. 1, pp. 262-275, 2017.
[24] L. X. Yang, and J. Jiang, “Complex dynamical behavior and modified projective synchronization in fractional-order hyper-chaotic complex Lü system,” Chaos, Solitons & Fractals, vol. 78, no. 1, pp. 267-276, 2015.
[25] Y. Liu, “Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system,” Nonlinear Dynamics, vol. 67, no. 1, pp. 89-96, 2012.
[26] A. P. Misra, D. Ghosh, and A. R. Chowdhury, “A novel hyperchaos in the quantum Zakharov system for plasmas,” Physics Letters A, vol. 372, no. 9, pp. 1469-1476, 2008.
[27] M. Sun, L. Tian, and C. Zeng, “The energy resources system with parametric perturbations and its hyperchaos control,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2620-2626, 2009.
[29] X. Huang, Z. Zhao, Z. Wang, and Y. Li, “Chaos and hyperchaos in fractional-order cellular neural networks,” Neurocomputing, vol. 91, no. 1, pp. 13-21, 2012.
[30] A. S. Elwakil, and M. P. Kennedy, “Inductorless hyperchaos generator,” Microelectronics Journal, vol. 30, no. 8, pp. 739-743, 1999.
[31] G. Ibrahim, and S. S. E. H. Elnashaie, “Hyperchaos in acetylcholinesterase enzyme systems,” Chaos, Solitons & Fractals, vol. 8, no. 12, pp. 1977-2007, 1997.
[32] C. Di, X. Yang, and D. Huang, “A new water resources supply-demand system and its hyperchaos control,” Procedia Engineering, vol. 15, no. 1, pp. 734-738, 2011.
[33] A. M. A. El-Sayed, H. M. Nour, A. Elsaid, A. E. Matouk, and A. Elsonbaty, “Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system,” Applied Mathematical Modelling, vol. 40, no. 5-6, pp. 3516-3534, 2016.
[34] O. S. Ojoniyi, and A. N. Njah, “A 5D hyperchaotic Sprott B system with coexisting hidden attractors,” Chaos, Solitons & Fractals, vol. 87, no. 2, pp. 172-181, 2016.
[35] T. Gao, Z. Chen, Q. Gu, and Z. Yuan, “A new hyper-chaos generated from generalized Lorenz system via nonlinear feedback,” Chaos, Solitons & Fractals, vol. 35, no. 2, pp. 390-397, 2008.
[36] F. Wang, and C. Liu, “A new criterion for chaos and hyperchaos synchronization using linear feedback control,” Physics Letters A, vol. 360, no. 2, pp. 274-278, 2006.
[37] T. Gao, G. Chen, Z. Chen, and S. Chen, “The generation and circuit implementation of a new hyperchaos upon Lorenz system,” Physics Letters A, vol. 361, no. 1-2, pp. 78-86, 2007.
[38] Y. L. Wu, C. H. Yang, and C. H. Wu, “Chip implementation of a new hyperchaotic oscillator,” Electronics Letters, vol. 53, no. 4, pp. 226-228, 2017.
[39] A. M. A. El-Sayed, H. M. Nour, A. Elsaid, A .E. Matouk, and A. Elsonbaty, “Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system,” Applied Mathematics and Computation, vol. 239, no. 1, pp. 333-342, 2014.
[40] C. Shen, S. Yu, J. Lü, and G. Chen, “A systematic methodology for constructing hyperchaotic systems with multiple positive Lyapunov exponents and circuit implementation,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 61, no. 3, pp. 854-864, 2014.
[41] J. P. Singh, and B. K. Roy, “A novel hyperchaotic system with stable and unstable line of equilibria and sigma shaped poincare map,” IFAC-PapersOnLine, vol. 49, no. 1, pp. 526-531, 2016.
[42] Q. Yang, K. Zhang, and G. Chen, “Hyperchaotic attractors from a linearly controlled Lorenz system,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1601-1617, 2009.
[43] J. Ma, A. B. Li, Z. S. Pu, and L. J. Yang, “A time-varying hyperchaotic system and its realization in circuit,” Nonlinear Dynamics, vol. 62, no. 3, pp. 535-541, 2010.
[44] L. M. Tam, J. H. Chen, H. K. Chen, and W. M. S. Tou, “Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 826-839, 2008.
[45] N. Yujun, W. Xingyuan, W. Mingjun, and Z. Huaguang, “A new hyperchaotic system and its circuit implementation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3518-3524, 2010.
[46] Y. Chen, and Q.Yang, “A new Lorenz-type hyperchaotic system with a curve of equilibria,” Mathematics and Computers in Simulation, vol. 112, no. 1, pp. 40-55, 2015.
[47] S. Chen, H. R. Momeni, G. Qi, and Z. L. Wang, “Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form,” Nonlinear Dynamics, vol. 67, no. 2, pp. 1161-1173, 2012.
[48] Z. Wang, J. Ma, S. Cang, Z. Wang, and Z. Chen, “Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractors,” Optik-International Journal for Light and Electron Optics, vol. 127, no. 5, pp. 2424-2431, 2016.
[49] C. Shen, S. Yu, J. Lü, and G. Chen, “Designing hyperchaotic systems with any desired number of positive Lyapunov exponents via a simple model,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 61, no. 8, pp. 2380-2389, 2014.
[50] Z. Wei, R. Wang, and A. Liu, “A new finding of the existence of hidden hyperchaotic attractors with no equilibria,” Mathematics and Computers in Simulation, vol. 100, no. 1, pp. 13-23, 2014.
[51] V. T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, and S. T. Kingni, “A no-equilibrium hyperchaotic system with a cubic nonlinear term,” Optik-International Journal for Light and Electron Optics, vol. 127, no. 6, pp. 3259-3265, 2016.