Symbolic Methods for Analysing Bifurcations and Chaos of Two Five-Parameter Families of Planar Quadratic Maps
DOI:
https://doi.org/10.25271/sjuoz.2020.8.2.723Keywords:
Bifurcations, Chaos, Planar quadratic maps, Symbolic computation, Snapback repellerAbstract
In this work, we analyze the dynamical behaviors of two five-parameter families of planar quadratic maps by utilizing strategies of symbolic computation. We are going to use computer algebra methods to clarify how to detect the stability of equilibrium points to analyze chaos and also the bifurcation of planar maps. Based on strategies for solving the systems in types of semi-algebraic and by utilizing an algorithmic approach, we obtain respectively for the two maps, sufficient conditions on the parameters to have a prescribed number of (stable) equilibrium points; necessary conditions on the parameters to undergo a certain type of bifurcation or to have chaotic behavior induced by snapback repeller.
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Lazard, D., & Rouillier, F. (2007). Solving parametric polynomial systems. Journal of Symbolic Computation, 42(6), 636-667.
Li, C., & Chen, G. (2003). An improved version of the Marotto theorem. Chaos, Solitons & Fractals, 18(1), 69-77.
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Marotto, F. R., & FR, M. (1978). Snap-Back Repellers Imply Chaos In RN.
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Sang, B., & Huang, B. (2017). Bautin bifurcations of a
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Wang, D. (2001). Elimination methods. Springer Science & Business Media.
Wen, G. (2005). Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Physical Review E, 72(2), 026201.
Wu, W. T. (2000). Mathematics mechanization: mechanical geometry theorem-proving, mechanical geometry problem-solving, and polynomial equations-solving. Beijing: Science Press.
Yang, L., & Xia, B. (2005). Real Solution Classification for Parametric Semi-Algebraic Systems. In Algorithmic Algebra and Logic (pp. 281-289).
Alexandra, B., Jean-claude, C., Laura, G., & Christian, M. (1996). Chaotic dynamics in two-dimensional noninvertible maps (Vol. 20). World Scientific.
Bistritz, Y. U. V. A. L. (1984). Zero location with respect to the unit circle of discrete-time linear system polynomials. Proceedings of the IEEE, 72(9), 1131-1142.
Buchberger, B. (1985). Gröbner bases: An algorithmic method in polynomial ideal theory. Multidimensional systems theory.
Collins, G. E., & Hong, H. (1991). Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3), 299-328.
Davidchack, R. L., Lai, Y. C., Klebanoff, A., & Bollt, E. M. (2001). Towards complete detection of unstable periodic orbits in chaotic systems. Physics Letters A, 287(1-2), 99-104.
Din, Q. (2017). Complexity and chaos control in a discrete-time prey-predator model. Communications in Nonlinear Science and Numerical Simulation, 49, 113-134.
Faugère, J. C. (2002, July). A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5). In Proceedings of the 2002 international symposium on Symbolic and algebraic computation (pp. 75-83).
Galor, O. (2007). Discrete dynamical systems. Springer Science & Business Media.
He, Z., & Lai, X. (2011). Bifurcation and chaotic behaviour of a discrete-time predator–prey system. Nonlinear Analysis: Real World Applications, 12(1), 403-417.
Hénon, M. (1976). A two-dimensional mapping with a strange attractor. In The Theory of Chaotic Attractors (pp. 94-102). Springer, New York, NY.
Hong, H., Liska, R., & Steinberg, S. (1997). Testing stability by quantifier elimination. Journal of Symbolic Computation, 24(2), 161-187.
Hong, H., Tang, X., & Xia, B. (2015). Special algorithm for stability analysis of multistable biological regulatory systems. Journal of Symbolic Computation, 70, 112-135.
Huang, B., & Niu, W. (2019a). Analysis of snapback repellers using methods of symbolic computation. International Journal of Bifurcation and Chaos, 29(04), 1950054.
Huang, B., & Niu, W. (2019b). Algebraic approach to chaos induced by snapback repeller. ACM Communications in Computer Algebra, 53(3), 122-125.
Huang, B., & Niu, W. (2020c). Algebraic Analysis of Bifurcations and Chaos for Discrete Dynamical Systems. In International Conference on Mathematical Aspects of Computer and Information Sciences. MACIS 2019. Lecture Notes in Computer Science, pp. 169-184. Springer Nature.
Kaslik, E., & Balint, S. (2009). Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture. Neural Networks, 22(10), 1411-1418.
Lazard, D., & Rouillier, F. (2007). Solving parametric polynomial systems. Journal of Symbolic Computation, 42(6), 636-667.
Li, C., & Chen, G. (2003). An improved version of the Marotto theorem. Chaos, Solitons & Fractals, 18(1), 69-77.
Li, X., Mou, C., Niu, W., & Wang, D. (2011). Stability analysis for discrete biological models using algebraic methods. Mathematics in Computer Science, 5(3), 247-262.
Marotto, F. R., & FR, M. (1978). Snap-Back Repellers Imply Chaos In RN.
Marotto, F. R. (2005). On redefining a snap-back repeller. Chaos, Solitons & Fractals, 25(1), 25-28.
Niu, W., Shi, J., & Mou, C. (2016). Analysis of codimension 2 bifurcations for high-dimensional discrete systems using symbolic computation methods. Applied Mathematics and Computation, 273, 934-947.
Sang, B., & Huang, B. (2017). Bautin bifurcations of a
financial system. Electronic Journal of Qualitative
Theory of Differential Equations, 2017(95), 1-22.
Wang, D. (2001). Elimination methods. Springer Science & Business Media.
Wen, G. (2005). Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Physical Review E, 72(2), 026201.
Wu, W. T. (2000). Mathematics mechanization: mechanical geometry theorem-proving, mechanical geometry problem-solving, and polynomial equations-solving. Beijing: Science Press.
Yang, L., & Xia, B. (2005). Real Solution Classification for Parametric Semi-Algebraic Systems. In Algorithmic Algebra and Logic (pp. 281-289).
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Published
2020-06-30
How to Cite
Mikaeel, S. H., & Othman, B. H. (2020). Symbolic Methods for Analysing Bifurcations and Chaos of Two Five-Parameter Families of Planar Quadratic Maps. Science Journal of University of Zakho, 8(2), 72–79. https://doi.org/10.25271/sjuoz.2020.8.2.723
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Science Journal of University of Zakho
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