Dynamics of a Switched by Hysteresis Linear System in R2

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Jhon Ceron http://orcid.org/0000-0002-6385-5060 Jorge Amador http://orcid.org/0000-0002-4296-6271 Gerard Olivar http://orcid.org/0000-0003-1862-4842


This paper examines the local and global behavior of a two-variable piece-wise smooth dynamic system by studying the flow of two nonhomogeneous linear dynamic systems that switch on the boundaries of a hysteresis band. Both equilibrium solutions and parametric domains that guarantee the existence of periodic orbits within the band, were determined analytically. While, phase portraits and basins of attraction of the system for different values of the parameters were performed through numerical simulations. Results show the coexistence of multiple stationary states of different types when different parameter values are used.


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CERON, Jhon; AMADOR, Jorge; OLIVAR, Gerard. Dynamics of a Switched by Hysteresis Linear System in R2. Ingeniería y Ciencia | ing.cienc., [S.l.], v. 13, n. 25, p. 9-28, apr. 2017. ISSN 2256-4314. Available at: <http://publicaciones.eafit.edu.co/index.php/ingciencia/article/view/4014>. Date accessed: 20 sep. 2017. doi: https://doi.org/10.17230/ingciencia.13.25.1.
Hysteresis; periodic orbit; piece-wise dynamic systems; basins of attraction.


I. Tsypkin, Relay Control Systems. Cambridge University Press, 1984.

G. Bertotti, I. D. Mayergoyz, and I. Mayergoyz, The Science
of Hysteresis. Elsevier, Mathematical modeling and applications.,
2006. [Online]. Available: http://www.sciencedirect.com/science/article/pii/B9780124808744500002

K. J. Å ström, G. C. Goodwin, and P. R. Kumar, Adaptive Control, Filtering, and Signal Processing, ser. The IMA Volumes in Mathematics and its Applications. New York, NY: Springer New York, The IMA Volumes in Mathematics and Its Applications Series., 1995, vol. 74. [Online]. Available: http://link.springer.com/10.1007/978-1-4419-8568-2

J. F. Revelo, “Diseño de una técnica de control con banda de histéresis adaptativa para corrección del factor de potencia,” Master’s thesis, Universidad Nacional de Colombia - Sede Manizales, 2009.

M. Dawande, V. Kanetkar, and G. Dubey, “Three-phase switch mode rectifier with hysteresis current control,” IEEE Transactions on Power Electronics, vol. 11, no. 3, pp. 466–471, May 1996. [Online]. Available: http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=491640

H. Lee and V. I. Utkin, “Chattering suppression methods in sliding mode control systems,” Annual Reviews in Control, vol. 31, no. 2, pp. 179–188, jan 2007. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1367578807000363

B. K. Bose, “An adaptive hysteresis-band current control technique of a voltage-fed pwm inverter for machine drive system,” Industrial Electronics, IEEE Transactions on, vol. 37, no. 5, pp. 402–408, 1990. [Online]. Available: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=103436

J. N. Hincapié, A. Trejos, M. E. Moncada, and A. Escobar, “Electrónica de potencia para el calentamiento por inducción doméstico: revisión del estado del arte,” Ingeniería y ciencia, vol. 9, no. 18, pp. 237–262, 2013.

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, ser. Applied Mathematical Sciences. New York, NY: Springer New York, 1996, vol. 121. [Online]. Available: http://link.springer.com/10.1007/978-1-4612-4048-8

L. Ontañón-García, E. Jiménez-López, E. Campos-Cantón, and M. Basin, “A family of hyperchaotic multi-scroll attractors in Rn,” Applied Mathematics and Computation, vol. 233, pp. 522 – 533, 2014. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0096300314001799

F. Han, X. Yu, Y. Feng, and J. Hu, “On multiscroll chaotic attractors in hysteresis-based piecewise-linear systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 11, pp. 1004–1008, Nov 2007.

M. D. Bernardo, K. H. Johansson, and F. Vasca, “Self-Oscillations and Sliding in Relay Feedback Systems: Symetry and Bifurcations,” International Journal of Bifurcation and Chaos, vol. 11, no. 04, pp. 1121–1140, Apr. 2001. [Online]. Available: http://www.worldscientific.com/doi/abs/10.1142/ S0218127401002584

Y. A. Kuznetsov, S. Rinaldi, and A. Gragnani, “One-Parameter Bifurcations in Planar Filippov Systems,” International Journal of Bifurcation and Chaos, vol. 13, no. 08, pp. 2157–2188, Aug. 2003. [Online]. Available: http://www.worldscientific.com/doi/abs/10.1142/S0218127403007874

L. Perko, Differential Equations and Dynamical Systems, ser. Texts in Applied Mathematics. New York, NY: Springer-Verlag New York, 2001, vol. 7.

H. Minkowski, Geometrie der zahlen. Leipzig-Berlin: R. G. Teubner, 1910. [Online]. Available: https://archive.org/details/geometriederzahl00minkrich

U. Moreno, P. Peres, and I. Bonatti, “Analysis of piecewise-linear oscillators with hysteresis,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 50, no. 8, pp. 1120–1124, Aug. 2003. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.

Z. Zhusubaliyev and E. Soukhoterin, “Oscillations in a relay control system with hysteresis and time dead zone,” Mathematics and Computers in Simulation, vol. 58, no. 4-6, pp. 329–350, mar 2002. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0378475401003767

M. Spivak, Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus. Benjamin-Cummings, 1965. [Online]. Available: http://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)

Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, ser. Texts in Applied Mathematics. New York: Springer-Verlag, Texts in Applied Mathematics Volume 2., 2003, vol. 2. [Online]. Available: http://link.springer.com/10.1007/b97481

C. S. Hsu, Cell-to-Cell Mapping, A Method of Global Analysis for Nonlinear Systems, ser. Applied Mathematical Sciences. New York, NY: Springer New York, Applied Mathematical Sciences., 1987, vol. 64. [Online]. Available: http://www.springerlink.com/index/10.1007/978-1-4757-3892-6