# American Institute of Mathematical Sciences

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

## Degenerate with respect to parameters fold-Hopf bifurcations

 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.

Citation: Gheorghe Tigan. Degenerate with respect to parameters fold-Hopf bifurcations. Discrete and 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. [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.

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##### 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.
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 $s=1,$ $\theta _{0}<0,$ $\beta_{2}>0$
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) $s=-1,\theta _0<0$
Generic phase portraits of the $2D$ system (38)
Generic phase portraits of the $2D$ system (38)
Bifurcation diagrams for: a) $\theta _{0}>0$ and $s=+1,$ (left); b) $\theta _{0}<0$ and $s=-1$ (right)
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 $\alpha _{1}>0$
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"
The point $A_3$ in 2D and its corresponding limit cycle $C$ in 3D
The circle in 2D and its corresponding torus in 3D
The heteroclinic orbit in 2D and its corresponding sphere in 3D
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" $\beta _{1}=-0.01,\beta _{2}=-0.1,$ $s=1,$ $\theta _{0}=-1$
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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