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Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems
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 |
$ \dot x = yz,\quad \dot y = x-y,\quad \dot z = 1-x(\alpha y+\beta x), $ |
$ (x,y,z) \in \mathbb R^3 $ |
$ \alpha, \beta \in[0,1] $ |
$ \alpha = 0 $ |
$ \mathbb R^3 $ |
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. |



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