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

Dynamic aspects of Sprott BC chaotic system

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  March 2021 Early access  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.
Citation: Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177
References:
[1] D. Bleecker and G. Csordas, Basic Partial Differential Equations, International Press, Cambridge, MA, 1996.  doi: 10.1201/9781351070089.
[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

J. LlibreA. 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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

show all references

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

[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.

[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.

[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.

[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.

[7]

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

[8]

J. LlibreA. 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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

Figure 1.  Local behavior of orbits around the finite singularities of Sprott B (in 1(A)) and Sprott C (in 1(B)) systems
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
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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