Research Article
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Year 2022, Volume 4, Issue 4, 234 - 240, 31.12.2022
https://doi.org/10.51537/chaos.1196860

Abstract

References

  • Aguirre-Hernández, B., E. Campos-Cantón, J. A. López-Rentería, and E. C. Díaz-González, 2015 A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems. Chaos, Solitons and Fractals 71: 100–106.
  • Anzo-Hernández, A., H. E. Gilardi-Velázquez, and E. Campos- Cantón, 2018 On multistability behavior of unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 28.
  • Arecchi, F. T., R. Badii, and A. Politi, 1985 Generalized multistability and noise-induced jumps in a nonlinear dynamical system. Physical Review A 32.
  • Attneave, F., 1971 Multistability in perception. Scientific American 225: 63–71.
  • Campos-Cantón, E., G. Barajas-Ramírez, J. G.and Solís-Perales, and R. Femat, 2010 Multiscroll attractors by switching systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 20: 013116.
  • Campos-Cantón, E., R. Femat, and G. Chen, 2012 Attractors generated from switching unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 22.
  • Díaz-González, E. C., B. Aguirre-Herández, J. A. López-Rentería, E. Campos-Cantón, and C. A. Loredo-Villalobos, 2017 Stability and multiscroll attractors of control systems via the abscissa. Mathematical Problems in Engineering 2017.
  • Echenausía-Monroy, J., H. Gilardi-Velázquez, N.Wang, R. Jaimes- Reátegui, J. García-López, et al., 2022 Multistability route in a pwl multi-scroll system through fractional-order derivatives. Chaos, Solitons and Fractals 161: 112355.
  • Feudel, U., 2008 Complex dynamics in multistable systems. International Journal of Bifurcation and Chaos 18: 1607–1626.
  • Geist, K., U. Parlitz, and Lauterborn, 1990 Comparison of different methods for computing lyapunov exponents. Progress of Theoretical Physics 83: 875–893.
  • Gilardi-Velázquez, H. E., J. L. Echenausia-Monroy, R. J. Escalante- González, B. B. Cassal-Quiroga, and G. Huerta-Cuellar, 2022 On the relationship between integer and fractional pwl systems with multistable behavior. In Complex Systems and Their Applications, edited by G. Huerta Cuéllar, E. Campos Cantón, and E. Tlelo- Cuautle, pp. 113–129, Cham, Springer International Publishing.
  • Gilardi-Velázquez, H. E., L. J. Ontañón García, D. G. Hurtado- Rodriguez, and E. Campos-Cantón, 2017 Multistability in piecewise linear systems versus eigenspectra variation and round function. International Journal of Bifurcation and Chaos 27: 1730031.
  • Hartman, P., 1964 Ordinary Differential Equations. Wiley, New York. Lorenz, E. N., 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20: 130–141.
  • Lynch, S., 2004 Dynamical systems with applications using MATLAB. Birkhäuser, Boston.
  • Madan, R. N., 1993 Chua’s Circuit: A paradigm for Chaos. World Scientific, Singapore.
  • Pisarchik, A. N. and U. Feudel, 2014 Control of multistability. Physics Reports 540: 167–218.
  • Sparrow, C., 1982 The Lorenz Equation: Bifurcations, Chaos and Strange Attractors. Springer-Verlag, New York.
  • Uspensky, J. V., 1987 Teoría de ecuaciones. Limusa, México.

Generation of Multistability through Unstable Systems

Year 2022, Volume 4, Issue 4, 234 - 240, 31.12.2022
https://doi.org/10.51537/chaos.1196860

Abstract

In this work, we propose an approach to generate multistability based on a class of unstable systems that have all their roots in the right complex half-plane. Multistability is the coexistence of multiple stable states for a set of system parameters. The approach is realized by using linear third order differential equations that consists of two parameters. The first bifurcation parameter transforms the unstable system with all its roots in the right complex half-plane into an unstable system with one root in the left complex half-plane and two roots remaining in the right complex half-plane. With this first transformation, the system is capable of generating attractors by means of a piecewise linear function and the system presents monostability. We then use the another bifurcation parameter to switch from a monostable multiscroll attractor to several multistable states showing a single-scroll attractor.

References

  • Aguirre-Hernández, B., E. Campos-Cantón, J. A. López-Rentería, and E. C. Díaz-González, 2015 A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems. Chaos, Solitons and Fractals 71: 100–106.
  • Anzo-Hernández, A., H. E. Gilardi-Velázquez, and E. Campos- Cantón, 2018 On multistability behavior of unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 28.
  • Arecchi, F. T., R. Badii, and A. Politi, 1985 Generalized multistability and noise-induced jumps in a nonlinear dynamical system. Physical Review A 32.
  • Attneave, F., 1971 Multistability in perception. Scientific American 225: 63–71.
  • Campos-Cantón, E., G. Barajas-Ramírez, J. G.and Solís-Perales, and R. Femat, 2010 Multiscroll attractors by switching systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 20: 013116.
  • Campos-Cantón, E., R. Femat, and G. Chen, 2012 Attractors generated from switching unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 22.
  • Díaz-González, E. C., B. Aguirre-Herández, J. A. López-Rentería, E. Campos-Cantón, and C. A. Loredo-Villalobos, 2017 Stability and multiscroll attractors of control systems via the abscissa. Mathematical Problems in Engineering 2017.
  • Echenausía-Monroy, J., H. Gilardi-Velázquez, N.Wang, R. Jaimes- Reátegui, J. García-López, et al., 2022 Multistability route in a pwl multi-scroll system through fractional-order derivatives. Chaos, Solitons and Fractals 161: 112355.
  • Feudel, U., 2008 Complex dynamics in multistable systems. International Journal of Bifurcation and Chaos 18: 1607–1626.
  • Geist, K., U. Parlitz, and Lauterborn, 1990 Comparison of different methods for computing lyapunov exponents. Progress of Theoretical Physics 83: 875–893.
  • Gilardi-Velázquez, H. E., J. L. Echenausia-Monroy, R. J. Escalante- González, B. B. Cassal-Quiroga, and G. Huerta-Cuellar, 2022 On the relationship between integer and fractional pwl systems with multistable behavior. In Complex Systems and Their Applications, edited by G. Huerta Cuéllar, E. Campos Cantón, and E. Tlelo- Cuautle, pp. 113–129, Cham, Springer International Publishing.
  • Gilardi-Velázquez, H. E., L. J. Ontañón García, D. G. Hurtado- Rodriguez, and E. Campos-Cantón, 2017 Multistability in piecewise linear systems versus eigenspectra variation and round function. International Journal of Bifurcation and Chaos 27: 1730031.
  • Hartman, P., 1964 Ordinary Differential Equations. Wiley, New York. Lorenz, E. N., 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20: 130–141.
  • Lynch, S., 2004 Dynamical systems with applications using MATLAB. Birkhäuser, Boston.
  • Madan, R. N., 1993 Chua’s Circuit: A paradigm for Chaos. World Scientific, Singapore.
  • Pisarchik, A. N. and U. Feudel, 2014 Control of multistability. Physics Reports 540: 167–218.
  • Sparrow, C., 1982 The Lorenz Equation: Bifurcations, Chaos and Strange Attractors. Springer-Verlag, New York.
  • Uspensky, J. V., 1987 Teoría de ecuaciones. Limusa, México.

Details

Primary Language English
Subjects Physics, Mathematical
Journal Section Research Articles
Authors

Edgar DİAZ-GONZALEZ>
Instituto Potosino de Investigacion Cientifica y Tecnologica
0000-0001-8387-3437
Mexico


Arturo GUERRA-LÓPEZ>
Instituto Potosino de Investigación Científica y Tecnologica
0000-0003-4954-3789
Mexico


Baltazar Aguirre HERNANDEZ>
Universidad Autonoma Metropolitana
0000-0002-6227-5232
Mexico


Eric CAMPOS> (Primary Author)
Instituto Potosino de Investigación Científica y Tecnológica
0000-0002-1098-1610
Mexico

Supporting Institution CONACYT
Project Number A1-S-30433.
Thanks Díaz-González Edgar Cristian also wants to thank the support of CONACYT trough the postdoctoral fellowship and the IPICYT by the support in the realization of this paper. A. Guerra-López wants to thank to CONACYT for its Master’s degree scholarship. E. Campos-Cantón acknowledges CONACYT for the financial support through Project no. A1-S-30433.
Publication Date December 31, 2022
Published in Issue Year 2022, Volume 4, Issue 4

Cite

APA Diaz-gonzalez, E. , Guerra-lópez, A. , Hernandez, B. A. & Campos, E. (2022). Generation of Multistability through Unstable Systems . Chaos Theory and Applications , Dissemination and Research in the Study of Complex Systems and Their Applications (EDIESCA 2022) , 234-240 . DOI: 10.51537/chaos.1196860

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