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April  2017, 37(4): 2115-2140. doi: 10.3934/dcds.2017091

Degenerate with respect to parameters fold-Hopf bifurcations

Gheorghe Tigan

Department of Mathematics, Politehnica University of Timisoara, Pta Victoriei, No. 2,300006, Timisoara, Timis, Romania

Received  May 2016 Revised  November 2016 Published  December 2016

Fund Project: The author is supported by grant FP7-PEOPLE-2012-IRSES-316338.

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.

Keywords: Dynamical systems, normal forms, fold-Hopf bifurcations, degenerate bifurcations, bifurcation diagrams.
Mathematics Subject Classification: Primary:37G05, 37G10;Secondary:37G15.
Citation: Gheorghe Tigan. Degenerate with respect to parameters fold-Hopf bifurcations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2115-2140. doi: 10.3934/dcds.2017091
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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J. Lu and G. Chen, A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12 (2012), 659-661.  doi: 10.1142/S0218127402004620.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. C. Sprott, Some simple chaotic flows, Physical Review E, 50 (1994), 647-650.  doi: 10.1103/PhysRevE.50.R647.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. C. Sprott, Some simple chaotic flows, Physical Review E, 50 (1994), 647-650.  doi: 10.1103/PhysRevE.50.R647.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

$s=-1,$ $\theta _{0}<0,$ $ \beta _{2}<0, $ "d2" to $s=-1,$ $\theta _{0}<0,$ $\beta _{2}>0,$ "d3" to $s=-1,$ $\theta _{0}>0,$ $\beta _{2}<0,$ "d4" to $s=-1,$ $\theta _{0}>0,$ $\beta _{2}>0,$ "d5" to $ s=1,$ $\theta _{0}>0,$ $\beta _{2}>0,$ "d6" to $s=1,$ $ \theta _{0}>0,$ $\beta _{2}<0,$ "d7" to $s=1,$ $ \theta _{0}<0,$ $\beta _{2}<0$ and "d8" to $s=1,$ $\theta _{0}<0,$ $\beta_{2}>0$">Figure 2.  Generic phase portraits of the 2D system (38) on the bifurcation curve $\delta :\beta _{1}+\xi _{3}^{2}=0; $ "d1" corresponds to $s=-1,$ $\theta _{0}<0,$ $ \beta _{2}<0, $ "d2" to $s=-1,$ $\theta _{0}<0,$ $\beta _{2}>0,$ "d3" to $s=-1,$ $\theta _{0}>0,$ $\beta _{2}<0,$ "d4" to $s=-1,$ $\theta _{0}>0,$ $\beta _{2}>0,$ "d5" to $ s=1,$ $\theta _{0}>0,$ $\beta _{2}>0,$ "d6" to $s=1,$ $ \theta _{0}>0,$ $\beta _{2}<0,$ "d7" to $s=1,$ $ \theta _{0}<0,$ $\beta _{2}<0$ and "d8" to $s=1,$ $\theta _{0}<0,$ $\beta_{2}>0$
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Figure 1.  Generic phase portraits of the 2D system (38) at $ \alpha_1=\alpha_2=0$ and (a) $s=+1,\theta_0>0,$ (b) $s=+1, \theta_0<0,$ (c) $s=-1,\theta_0>0,$ (d) $s=-1,\theta _0<0$
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Figure 3.  Generic phase portraits of the $2D$ system (38)
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Figure 4.  Generic phase portraits of the $2D$ system (38)
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Figure 5.  Bifurcation diagrams for: a) $\theta _{0}>0$ and $s=+1,$ (left); b) $\theta _{0}<0$ and $s=-1$ (right)
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$ \alpha _{1}<0,$ respectively, "snun" for $\alpha _{1}>0$">Figure 6.  Bifurcation diagrams for: a) $0<\theta _{0}\leq \frac{1}{2}$ and $s=-1,$ (left); b) $\theta _{0}>\frac{1}{2}$ and $s=-1$ (right). The generic portraits on the lines $k^{1},k^{2}$ are the same as in their neighborhood corresponding to $A_{3}$ a node, namely "nssn" for $ \alpha _{1}<0,$ respectively, "snun" for $\alpha _{1}>0$
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Figure 7.  Bifurcation diagram for $\theta _{0}<0$ and $s=+1.$ The generic portraits on the lines $k^{1},k^{2}$ are the same as in their neighborhood corresponding to $A_{3}$ a node, namely "ssun" or "sssn"
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Figure 8.  The point $A_3$ in 2D and its corresponding limit cycle $C$ in 3D
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Figure 9.  The circle in 2D and its corresponding torus in 3D
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Figure 10.  The heteroclinic orbit in 2D and its corresponding sphere in 3D
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$\beta _{1}=-0.5,\beta _{2}=-0.001,$ $s=-1,$ $ \theta _{0}=1;$ "sn" $\beta _{1}=-0.01,\beta _{2}=-0.2,$ $s=1,$ $\theta _{0}=1;$ "ss" $\beta _{1}=-0.12, \beta _{2}=-0.14,$ $s=-1,$ $\theta _{0}=-1;$ "sns" $ \beta _{1}=-0.01,\beta _{2}=-0.24,$ $s=-1,$ $\theta _{0}=-1,$ "snuf" $\beta _{1}=-0.01,\beta _{2}=-0.24,$ $s=-1,$ $ \theta _{0}=1;$ "sssn" $\beta _{1}=-0.01,\beta _{2}=-0.1,$ $s=1,$ $\theta _{0}=-1$">Figure 11.  Generic phase portraits of the 3D system. In all cases $ \omega_1=0.1$ and $\omega_2=-0.1$ The other numeric values are as follows: "nn" $\beta _{1}=-0.5,\beta _{2}=-0.001,$ $s=-1,$ $ \theta _{0}=1;$ "sn" $\beta _{1}=-0.01,\beta _{2}=-0.2,$ $s=1,$ $\theta _{0}=1;$ "ss" $\beta _{1}=-0.12, \beta _{2}=-0.14,$ $s=-1,$ $\theta _{0}=-1;$ "sns" $ \beta _{1}=-0.01,\beta _{2}=-0.24,$ $s=-1,$ $\theta _{0}=-1,$ "snuf" $\beta _{1}=-0.01,\beta _{2}=-0.24,$ $s=-1,$ $ \theta _{0}=1;$ "sssn" $\beta _{1}=-0.01,\beta _{2}=-0.1,$ $s=1,$ $\theta _{0}=-1$
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$\beta _{1}=-0.01,\beta _{2}=0.02,$ $ s=1,$ $\theta _{0}=-1;$ "ssc" $\beta _{1}=-0.01, \beta _{2}=-0.002, $ $s=1,$ $\theta _{0}=-1;$ "ssh" $\beta _{1}=-0.01,\beta _{2}=-0.0028,$ $s=1,$ $\theta _{0}=-1$ and "nns" $\beta _{1}=-0.01,\beta _{2}=-0.011,$ $s=1,$ $ \theta _{0}=1$">Figure 12.  Generic phase portraits of the 3D system. In all cases $ \omega _{1}=0.1$ and $\omega _{2}=-0.1$ The other numeric values are as follows: "ssuf" $\beta _{1}=-0.01,\beta _{2}=0.02,$ $ s=1,$ $\theta _{0}=-1;$ "ssc" $\beta _{1}=-0.01, \beta _{2}=-0.002, $ $s=1,$ $\theta _{0}=-1;$ "ssh" $\beta _{1}=-0.01,\beta _{2}=-0.0028,$ $s=1,$ $\theta _{0}=-1$ and "nns" $\beta _{1}=-0.01,\beta _{2}=-0.011,$ $s=1,$ $ \theta _{0}=1$
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