
-
Previous Article
A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems
- DCDS Home
- This Issue
-
Next Article
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. |
[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. |
[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. |











[1] |
Héctor Barge, José M. R. Sanjurjo. Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2585-2601. doi: 10.3934/dcds.2021204 |
[2] |
Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545 |
[3] |
Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 |
[4] |
Yancong Xu, Deming Zhu, Xingbo Liu. Bifurcations of multiple homoclinics in general dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 945-963. doi: 10.3934/dcds.2011.30.945 |
[5] |
Ruyuan Zhang. Hopf bifurcations of ODE systems along the singular direction in the parameter plane. Communications on Pure and Applied Analysis, 2005, 4 (2) : 445-461. doi: 10.3934/cpaa.2005.4.445 |
[6] |
Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure and Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703 |
[7] |
Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287 |
[8] |
Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092 |
[9] |
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 |
[10] |
Niclas Kruff, Sebastian Walcher. Coordinate-independent criteria for Hopf bifurcations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1319-1340. doi: 10.3934/dcdss.2020075 |
[11] |
Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897 |
[12] |
C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289 |
[13] |
Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362 |
[14] |
Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 |
[15] |
Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025 |
[16] |
Cinzia Soresina. Hopf bifurcations in the full SKT model and where to find them. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022120 |
[17] |
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 |
[18] |
Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63 |
[19] |
Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345 |
[20] |
Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]