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March  2021, 26(3): 1653-1673. doi: 10.3934/dcdsb.2020177

Dynamic aspects of Sprott BC chaotic system

Marcos C. Mota and  Regilene D. S. Oliveira ,

Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São–carlense, 400, Centro, 13.566-590, São Carlos, SP, Brazil

* Corresponding author: regilene@icmc.usp.br

Communicated by Dongmei Xiao

Received  April 2019 Published  June 2020

Fund Project: This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Níıvel Superior - Brasil (CAPES) - Finance Code 001 and FAPESP grant number 2017/20854-5

In this paper we study global dynamic aspects of the quadratic system
$ \dot x = yz,\quad \dot y = x-y,\quad \dot z = 1-x(\alpha y+\beta x), $
where
$ (x,y,z) \in \mathbb R^3 $
 and
$ \alpha, \beta \in[0,1] $
 are two parameters. It contains the Sprott B and the Sprott C systems at the two extremes of its parameter spectrum and we call it Sprott BC system. Here we present the complete description of its singularities and we show that this system passes through a Hopf bifurcation at
$ \alpha = 0 $
. Using the Poincaré compactification of a polynomial vector field in
$ \mathbb R^3 $
 we give a complete description of its dynamic on the Poincaré sphere at infinity. We also show that such a system does not admit a polynomial first integral, nor algebraic invariant surfaces, neither Darboux first integral.
Keywords: Sprott BC chaotic system, Hopf bifurcation, Poincaré compactification, invariant algebraic surfaces, Darboux integrability.
Mathematics Subject Classification: Primary: 34C05, 34C45; Secondary: 37D10, 37D45.
Citation: Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177
References:
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Z. Elhadj and C. J. Sprott, The unified chaotic system describing the Lorenz and Chua systems, Facta Univ., Electron. Energ., 3 (2010), 345-355.  doi: 10.2298/fuee1003345e.  Google Scholar

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J. Llibre, A. Mahdi and C. Valls, Darboux integrability of the Lü system, J. Geom. Phys., 63 (2013), 118-128.  doi: 10.1016/j.geomphys.2012.10.003.  Google Scholar

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J. Llibre and C. Valls, Analytic integrability of a Chua system, J. Math. Phys., 49 (2008), 102701. doi: 10.1063/1.2992481.  Google Scholar

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J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in $\mathbb{R}^n$ taking into account the multiplicity at infinity, Bull. Sci. Math., 133 (2009), 765-778.  doi: 10.1016/j.bulsci.2009.06.002.  Google Scholar

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J. Llibre and X. Zhang, Darboux theory of integrability in $\mathbb{C}^n$ taking into account the multiplicity, J. Diff. Eqs., 246 (2009), 541-551.  doi: 10.1016/j.jde.2008.07.020.  Google Scholar

[12]

J. Lü and G. Chen, A new chaotic attractor coined, Int. J. Bifurcat. Chaos., 3 (2002), 659-661.  doi: 10.1142/s0218127402004620.  Google Scholar

[13]

J. Lü et al., Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcat. Chaos., 12 (2002), 2917-2926.  doi: 10.1142/s021812740200631x.  Google Scholar

[14]

A. Mahdi and C. Valls, Integrability of the Nosé–Hoover equation, J. Geom. Phys., 61 (2011), 1348-1352.  doi: 10.1016/j.geomphys.2011.02.018.  Google Scholar

[15]

R. Oliveira and C. Valls, Chaotic behavior of a generalized Sprott E differential system, Int. J. Bifurcat. Chaos., 5 (2016), 1650083. doi: 10.1142/s0218127416500838.  Google Scholar

[16]

J. C. Sprott, Some simple chaotic flows, Phys. Rev. E., 50 (1994), R647–R650. doi: 10.1103/physreve.50.r647.  Google Scholar

[17]

Z. Wei and Q. Yang, Dynamical analysis of the generalized Sprott C system with only two stable equilibria, Nonlinear Dyn., 4 (2012), 543-554.  doi: 10.1007/s11071-011-0235-8.  Google Scholar

show all references

References:
[1] D. Bleecker and G. Csordas, Basic Partial Differential Equations, International Press, Cambridge, MA, 1996.  doi: 10.1201/9781351070089.  Google Scholar
[2]

C. J. Christopher, Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh Sect. A, 6 (1994), 1209-1229.  doi: 10.1017/s0308210500030213.  Google Scholar

[3]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer–Verlag, Berlin, 2006. doi: 10.1007/978-3-540-32902-2.  Google Scholar

[4]

Z. Elhadj and C. J. Sprott, The unified chaotic system describing the Lorenz and Chua systems, Facta Univ., Electron. Energ., 3 (2010), 345-355.  doi: 10.2298/fuee1003345e.  Google Scholar

[5]

Y. Feng and Z. Wei, Delayed feedback control and bifurcation analysis of the generalized Sprott B system with hidden attractors, Eur. Phys. J-Spec. Top., 224 (2015), 1619-1636.  doi: 10.1140/epjst/e2015-02484-9.  Google Scholar

[6]

F. R. Gantmakher, The Theory of Matrices, Vol. 1. Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation. AMS Chelsea Publishing, Providence, RI, 1998. doi: ISBN:0-8218-1376-5.  Google Scholar

[7]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. doi: 10.1007/b98848.  Google Scholar

[8]

J. Llibre, A. Mahdi and C. Valls, Darboux integrability of the Lü system, J. Geom. Phys., 63 (2013), 118-128.  doi: 10.1016/j.geomphys.2012.10.003.  Google Scholar

[9]

J. Llibre and C. Valls, Analytic integrability of a Chua system, J. Math. Phys., 49 (2008), 102701. doi: 10.1063/1.2992481.  Google Scholar

[10]

J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in $\mathbb{R}^n$ taking into account the multiplicity at infinity, Bull. Sci. Math., 133 (2009), 765-778.  doi: 10.1016/j.bulsci.2009.06.002.  Google Scholar

[11]

J. Llibre and X. Zhang, Darboux theory of integrability in $\mathbb{C}^n$ taking into account the multiplicity, J. Diff. Eqs., 246 (2009), 541-551.  doi: 10.1016/j.jde.2008.07.020.  Google Scholar

[12]

J. Lü and G. Chen, A new chaotic attractor coined, Int. J. Bifurcat. Chaos., 3 (2002), 659-661.  doi: 10.1142/s0218127402004620.  Google Scholar

[13]

J. Lü et al., Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcat. Chaos., 12 (2002), 2917-2926.  doi: 10.1142/s021812740200631x.  Google Scholar

[14]

A. Mahdi and C. Valls, Integrability of the Nosé–Hoover equation, J. Geom. Phys., 61 (2011), 1348-1352.  doi: 10.1016/j.geomphys.2011.02.018.  Google Scholar

[15]

R. Oliveira and C. Valls, Chaotic behavior of a generalized Sprott E differential system, Int. J. Bifurcat. Chaos., 5 (2016), 1650083. doi: 10.1142/s0218127416500838.  Google Scholar

[16]

J. C. Sprott, Some simple chaotic flows, Phys. Rev. E., 50 (1994), R647–R650. doi: 10.1103/physreve.50.r647.  Google Scholar

[17]

Z. Wei and Q. Yang, Dynamical analysis of the generalized Sprott C system with only two stable equilibria, Nonlinear Dyn., 4 (2012), 543-554.  doi: 10.1007/s11071-011-0235-8.  Google Scholar

Figure 1.  Local behavior of orbits around the finite singularities of Sprott B (in 1(A)) and Sprott C (in 1(B)) systems
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Figure 2.  Phase portrait of system (3) on the Poincaré sphere. In Figure 2(A) there exist two closed curves filled up with singularities and one pair of distinguished singularities. These distinguished singularities possess two parabolic attractor sectors and two parabolic repelling sectors. In Figure 2(B) there exist one closed curve filled up with singularities and one pair of center type singularities
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Figure 3.  Phase portrait of system (3) on the Poincaré sphere. In Figure 3(A) there exist a pair of cusp type singularities and a pair of node type singularities (being one attractor and other repelling). In Figure 3(B) there exist a pair of saddles, a pair of centers and a pair of nodes (being one attractor and other repelling)
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