American Institute of Mathematical Sciences

Advanced
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  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.
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 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. 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.

[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. 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.

[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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[2]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[3]

Brigita Ferčec, Valery G. Romanovski, Yilei Tang, Ling Zhang. Integrability and bifurcation of a three-dimensional circuit differential system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4573-4588. doi: 10.3934/dcdsb.2021243
[4]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
[5]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[6]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[7]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[8]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[9]

Aleksander Ćwiszewski, Wojciech Kryszewski. On a generalized Poincaré-Hopf formula in infinite dimensions. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 953-978. doi: 10.3934/dcds.2011.29.953
[10]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[11]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183
[12]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129
[13]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[14]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013
[15]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[16]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[17]

B. Harbourne, P. Pokora, H. Tutaj-Gasińska. On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces. Electronic Research Announcements, 2015, 22: 103-108. doi: 10.3934/era.2015.22.103
[18]

Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669
[19]

Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571
[20]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (330)
  • HTML views (278)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy