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On the local C1, α solution of ideal magneto-hydrodynamical equations
Degenerate with respect to parameters fold-Hopf bifurcations
Department of Mathematics, Politehnica University of Timisoara, Pta Victoriei, No. 2,300006, Timisoara, Timis, Romania |
In this work we study degenerate with respect to parameters fold-Hopfbifurcations in three-dimensional differential systems. Such degeneraciesarise when the transformations between parameters leading to a normal formare not regular at some points in the parametric space. We obtain newgeneric results for the dynamics of the systems in such a degenerateframework. The bifurcation diagrams we obtained show that in a degeneratecontext the dynamics may be completely different than in a non-degenerateframework.
References:
[1] |
G. Chen and T. Ueta,
Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9 (1999), 1465-1466.
doi: 10.1142/S0218127499001024. |
[2] |
J.D. Crawford and E. Knobloch,
Classification and unfolding of degenerate Hopf bifurcations with O(2) symmetry: No distinguished parameter, Physica D: Nonlinear Phenomena, 31 (1988), 1-48.
doi: 10.1016/0167-2789(88)90011-5. |
[3] |
F. Dumortier, S. Ibanez, H. Kokubu and C. Simo,
About the unfolding of a Hopf-zero singularity, Discrete and Continuous Dynamical Systems -Series A, 33 (2013), 4435-4471.
doi: 10.3934/dcds.2013.33.4435. |
[4] |
I. Garcia and C. Valls,
The three-dimensional center problem for the zero-Hopf singularity, Discrete and Continuous Dynamical Systems -Series A, 36 (2016), 2027-2046.
doi: 10.3934/dcds.2016.36.2027. |
[5] |
X. He, C. Li and Y. Shu,
Triple-zero bifurcation in van der Pol's oscillator with delayed feedback, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 5229-5239.
doi: 10.1016/j.cnsns.2012.05.001. |
[6] |
J. Huang and Y. Zhao,
Bifurcation of isolated closed orbits from degenerated singularity, Discrete and Continuous Dynamical Systems -Series A, 33 (2013), 2861-2883.
doi: 10.3934/dcds.2013.33.2861. |
[7] |
W. Jiang and B. Niu,
On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 464-477.
doi: 10.1016/j.cnsns.2012.08.004. |
[8] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory Springer-Verlag, 1995.
doi: 10.1007/978-1-4757-2421-9. |
[9] |
J. S. W. Lamb, M.-A. Teixeira and K. N. Webster,
Heteroclinic bifurcations near Hopf-zero
bifurcation in reversible vector fields in $\mathbb{R}^{3}$, Journal of Differential Equations, 219 (2005), 78-115.
doi: 10.1016/j.jde.2005.02.019. |
[10] |
C. Lazureanu and T. Binzar,
On the symmetries of a Rikitake type system, C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 529-533.
doi: 10.1016/j.crma.2012.04.016. |
[11] |
C. Lazureanu and T. Binzar,
On a new chaotic system, Mathematical Methods in the Applied Sciences, 38 (2015), 1631-1641.
doi: 10.1002/mma.3174. |
[12] |
J. Lu and G. Chen,
A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12 (2012), 659-661.
doi: 10.1142/S0218127402004620. |
[13] |
M. Perez-Molina and M. F. Perez-Polo,
Fold-Hopf bifurcation, steady state, self-oscillating and chaotic behavior in an electromechanical transducer with nonlinear control, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 5172-5188.
doi: 10.1016/j.cnsns.2012.06.004. |
[14] |
E. Ponce, J. Ros and E. Vela,
Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry, Physica D: Nonlinear Phenomena, 250 (2013), 34-46.
doi: 10.1016/j.physd.2013.01.010. |
[15] |
R. Qesmi, M. Ait Babram and M. L. Hbid,
Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity, Applied Mathematics and Computation, 181 (2006), 220-246.
doi: 10.1016/j.amc.2006.01.030. |
[16] |
J. C. Sprott,
Some simple chaotic flows, Physical Review E, 50 (1994), 647-650.
doi: 10.1103/PhysRevE.50.R647. |
[17] |
G. Tigan,
Analysis of a dynamical system derived from the Lorenz system, Scientific Bulletin of the Politehnica University of Timisoara, 50 (2005), 61-72.
|
[18] |
G. Tigan and D. Opris,
Analysis of a 3D dynamical system, Chaos, Solitons and Fractals, 36 (2008), 1315-1319.
doi: 10.1016/j.chaos.2006.07.052. |
[19] |
G. Tigan, Analysis of degenerate fold-Hopf bifurcation in a three-dimensional differential system, Qualitative Theory of Dynamical Systems, to appear. Google Scholar |
[20] |
P. D. Woods and A. R. Champneys,
Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation, Physica D: Nonlinear Phenomena, 129 (1999), 147-170.
doi: 10.1016/S0167-2789(98)00309-1. |
show all references
References:
[1] |
G. Chen and T. Ueta,
Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9 (1999), 1465-1466.
doi: 10.1142/S0218127499001024. |
[2] |
J.D. Crawford and E. Knobloch,
Classification and unfolding of degenerate Hopf bifurcations with O(2) symmetry: No distinguished parameter, Physica D: Nonlinear Phenomena, 31 (1988), 1-48.
doi: 10.1016/0167-2789(88)90011-5. |
[3] |
F. Dumortier, S. Ibanez, H. Kokubu and C. Simo,
About the unfolding of a Hopf-zero singularity, Discrete and Continuous Dynamical Systems -Series A, 33 (2013), 4435-4471.
doi: 10.3934/dcds.2013.33.4435. |
[4] |
I. Garcia and C. Valls,
The three-dimensional center problem for the zero-Hopf singularity, Discrete and Continuous Dynamical Systems -Series A, 36 (2016), 2027-2046.
doi: 10.3934/dcds.2016.36.2027. |
[5] |
X. He, C. Li and Y. Shu,
Triple-zero bifurcation in van der Pol's oscillator with delayed feedback, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 5229-5239.
doi: 10.1016/j.cnsns.2012.05.001. |
[6] |
J. Huang and Y. Zhao,
Bifurcation of isolated closed orbits from degenerated singularity, Discrete and Continuous Dynamical Systems -Series A, 33 (2013), 2861-2883.
doi: 10.3934/dcds.2013.33.2861. |
[7] |
W. Jiang and B. Niu,
On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 464-477.
doi: 10.1016/j.cnsns.2012.08.004. |
[8] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory Springer-Verlag, 1995.
doi: 10.1007/978-1-4757-2421-9. |
[9] |
J. S. W. Lamb, M.-A. Teixeira and K. N. Webster,
Heteroclinic bifurcations near Hopf-zero
bifurcation in reversible vector fields in $\mathbb{R}^{3}$, Journal of Differential Equations, 219 (2005), 78-115.
doi: 10.1016/j.jde.2005.02.019. |
[10] |
C. Lazureanu and T. Binzar,
On the symmetries of a Rikitake type system, C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 529-533.
doi: 10.1016/j.crma.2012.04.016. |
[11] |
C. Lazureanu and T. Binzar,
On a new chaotic system, Mathematical Methods in the Applied Sciences, 38 (2015), 1631-1641.
doi: 10.1002/mma.3174. |
[12] |
J. Lu and G. Chen,
A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12 (2012), 659-661.
doi: 10.1142/S0218127402004620. |
[13] |
M. Perez-Molina and M. F. Perez-Polo,
Fold-Hopf bifurcation, steady state, self-oscillating and chaotic behavior in an electromechanical transducer with nonlinear control, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 5172-5188.
doi: 10.1016/j.cnsns.2012.06.004. |
[14] |
E. Ponce, J. Ros and E. Vela,
Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry, Physica D: Nonlinear Phenomena, 250 (2013), 34-46.
doi: 10.1016/j.physd.2013.01.010. |
[15] |
R. Qesmi, M. Ait Babram and M. L. Hbid,
Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity, Applied Mathematics and Computation, 181 (2006), 220-246.
doi: 10.1016/j.amc.2006.01.030. |
[16] |
J. C. Sprott,
Some simple chaotic flows, Physical Review E, 50 (1994), 647-650.
doi: 10.1103/PhysRevE.50.R647. |
[17] |
G. Tigan,
Analysis of a dynamical system derived from the Lorenz system, Scientific Bulletin of the Politehnica University of Timisoara, 50 (2005), 61-72.
|
[18] |
G. Tigan and D. Opris,
Analysis of a 3D dynamical system, Chaos, Solitons and Fractals, 36 (2008), 1315-1319.
doi: 10.1016/j.chaos.2006.07.052. |
[19] |
G. Tigan, Analysis of degenerate fold-Hopf bifurcation in a three-dimensional differential system, Qualitative Theory of Dynamical Systems, to appear. Google Scholar |
[20] |
P. D. Woods and A. R. Champneys,
Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation, Physica D: Nonlinear Phenomena, 129 (1999), 147-170.
doi: 10.1016/S0167-2789(98)00309-1. |











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