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Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $
1. | Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin 9054, New Zealand, Department of Mathematics, University of Auckland, Auckland 1010, New Zealand |
2. | Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin 9054, New Zealand, Department of Physics, University of Auckland, Auckland 1010, New Zealand |
The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power $ f $ and the detuning $ \delta $ of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the $ (f, \delta) $-plane is organized by two points of codimension-two bifurcations——a $ \mathbb{Z}_2 $-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the $ (f, \delta) $-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.
References:
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M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani and M. K. Oberthaler,
Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction, Phys. Rev. Lett., 95 (2005), 010402.
doi: 10.1103/PhysRevLett.95.010402. |
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V. Arnold, S. Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps: Volume I: The Classification of Critical Points Caustics and Wave Fronts, Monographs in Mathematics, Birkhäuser Boston, 1985.
doi: 10.1007/978-1-4612-5154-5. |
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P. Ashwin,
Symmetric chaos in systems of three and four forced oscillators, Nonlinearity, 3 (1990), 603-617.
doi: 10.1088/0951-7715/3/3/004. |
[5] |
R. Barrio, M. Carvalho, L. Castro and A. A. P. Rodrigues,
Experimentally accessible orbits near a Bykov cycle, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2030030.
doi: 10.1142/S021812742030030X. |
[6] |
R. Barrio and A. Shilnikov, Parameter–sweeping techniques for temporal dynamics of neuronal systems: Case study of Hindmarsh–Rose model, J. Math. Neurosci., 1 (2011), 20pp.
doi: 10.1186/2190-8567-1-6. |
[7] |
R. Barrio, A. Shilnikov and L. Shilnikov,
Kneadings, symbolic dynamics and painting Lorenz chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1230016.
doi: 10.1142/S0218127412300169. |
[8] |
A. Ben-Tal,
Symmetry restoration in a class of forced oscillators, Phys. D, 171 (2002), 236-248.
doi: 10.1016/S0167-2789(02)00623-1. |
[9] |
M. Brunstein, Nonlinear Dynamics in III-V Semiconductor Photonic Crystal Nano-Cavities, PhD thesis, Université Paris Sud - Paris XI, 2011. |
[10] |
R. C. Calleja, E. J. Doedel, A. R. Humphries, A. Lemus-Rodríguez and E. B. Oldeman,
Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem, Celest. Mech. Dyn. Astron., 114 (2012), 77-106.
doi: 10.1007/s10569-012-9434-y. |
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B. Cao, K. W. Mahmud and M. Hafezi,
Two coupled nonlinear cavities in a driven-dissipative environment, Phys. Rev. A, 94 (2016), 063805.
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H. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs, Springer, Berlin, 1993. |
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W. Casteels and C. Ciuti,
Quantum entanglement in the spatial–symmetry-breaking phase transition of a driven-dissipative Bose–Hubbard dimer, Phys. Rev. A, 95 (2017), 013812.
doi: 10.1103/physreva.95.013812. |
[14] |
W. Casteels, F. Storme, A. Le Boité and C. Ciuti,
Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators, Phys. Rev. A, 93 (2016), 033824.
|
[15] |
A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. D. M. Rademacher,
Unfolding a tangent equilibrium–to–periodic heteroclinic cycle, SIAM J. Appl. Dyn. Syst., 8 (2009), 1261-1304.
doi: 10.1137/080734923. |
[16] |
A. R. Champneys, Y. Kuznetsov and B. Sandstede,
A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.
doi: 10.1142/S0218127496000485. |
[17] |
P. Chossat and M. Golubitsky,
Symmetry–increasing bifurcation of chaotic attractors, Physica D, 32 (1988), 423-436.
doi: 10.1016/0167-2789(88)90066-8. |
[18] |
F. Christiansen and H. H. Rugh,
Computing Lyapunov spectra with continuous Gram–Schmidt orthonormalization, Nonlinearity, 10 (1997), 1063-1072.
doi: 10.1088/0951-7715/10/5/004. |
[19] |
P. Coullet and N. Vandenberghe,
Chaotic self-trapping of a weakly irreversible double Bose condensate, Phys. Rev. E, 64 (2001), 025202.
|
[20] |
M. Dellnitz and C. Heinrich,
Admissible symmetry increasing bifurcations, Nonlinearity, 8 (1995), 1039-1066.
|
[21] |
E. J. Doedel,
AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.
|
[22] |
E. J. Doedel, B. Krauskopf and H. M. Osinga,
Global invariant manifolds in the transition to preturbulence in the lorenz system, Indag. Math., 22 (2011), 222-240.
doi: 10.1016/j.indag.2011.10.007. |
[23] |
E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, Department of Computer Science, Concordia University, Montreal, Canada, 2010, available at http://www.cmvl.cs.concordia.ca/. |
[24] |
P. D. Drummond and D. F. Walls,
Quantum theory of optical bistability. I. Nonlinear polarisability model, J. Phys. A: Math. Gen, 13 (1980), 725-741.
|
[25] |
C. Emary and T. Brandes,
Chaos and the quantum phase transition in the Dicke model, Phys. Rev. E, 67 (2003), 066203.
doi: 10.1103/PhysRevE.67.066203. |
[26] |
H. M. Gibbs, Optical Bistability: Controlling Light with Light, Academic Press, 1985.
![]() |
[27] |
R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, Wiley-Interscience, 2002. |
[28] |
A. Giraldo, B. Krauskopf, N. G. R. Broderick, A. M. Yacomotti and J. A. Levenson,
The driven–dissipative Bose–Hubbard dimer: Phase diagram and chaos, New J. Phys., 22 (2020), 043009.
doi: 10.1088/1367-2630/ab7539. |
[29] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Syst., 16 (2017), 640-686.
doi: 10.1137/16M1097419. |
[30] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Cascades of global bifurcations and chaos near a homoclinic flip bifurcation: A case study, SIAM J. Appl. Dyn. Syst., 17 (2018), 2784-2829.
doi: 10.1137/17M1149675. |
[31] |
P. Glendinning,
Bifurcations near homoclinic orbits with symmetry, Phys. Lett. A, 103 (1984), 163-166.
doi: 10.1016/0375-9601(84)90242-1. |
[32] |
A. Golmakani and A. J. Homburg,
Lorenz attractors in unfoldings of homoclinic–flip bifurcations, Dyn. Syst., 26 (2011), 61-76.
doi: 10.1080/14689367.2010.503186. |
[33] |
C. Grebogi, E. Ott and J. A. Yorke,
Crises, sudden changes in chaotic attractors, and transient chaos, Phys. D, 7 (1983), 181-200.
doi: 10.1016/0167-2789(83)90126-4. |
[34] |
S. Haddadi, P. Hamel, G. Beaudoin, I. Sagnes, C. Sauvan, P. Lalanne, J. A. Levenson and A. M. Yacomotti,
Photonic molecules: Tailoring the coupling strength and sign, Optics Express, 22 (2014), 12359.
|
[35] |
P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson and A. M. Yacomotti,
Spontaneous mirror–symmetry breaking in coupled photonic–crystal nanolasers, Nature Photonics, 9 (2015), 311-315.
|
[36] |
C. Heinrich,
Symmetry increasing bifurcations via collisions of attractors, Rocky Mountain J. Math., 29 (2008), 559-608.
doi: 10.1216/rmjm/1181071652. |
[37] |
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, In Handbook of Dynamical Systems, (eds. H. W. Broer, B. Hasselblatt and F. Takens), Elsevier, New York, 3 (2010), 381–509. |
[38] |
J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2$^{nd}$ edition, Princeton University Press, 2008.
![]() |
[39] |
G. P. King and S. T. Gaito,
Bistable chaos. II. Bifurcation analysis M., Phys. Rev. A, 46 (1992), 3100-3110.
|
[40] |
J. Knobloch, J. S. Lamb and K. N. Webster,
Using Lin's method to solve Bykov's problems, J. Differential Equations, 257 (2014), 2984-3047.
doi: 10.1016/j.jde.2014.06.006. |
[41] |
B. Krauskopf and B. E. Oldeman,
A planar model system for the saddle-node Hopf bifurcation with global reinjection, Nonlinearity, 17 (2004), 1119-1151.
doi: 10.1088/0951-7715/17/4/001. |
[42] |
B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, In Numerical Continuation Methods for Dynamical Systems, (2007), 117–154.
doi: 10.1007/978-1-4020-6356-5_4. |
[43] |
B. Krauskopf and T. Rieß,
A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690.
doi: 10.1088/0951-7715/21/8/001. |
[44] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{nd}$ edition, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[45] |
A. Lohse and A. Rodrigues,
Boundary crisis for degenerate singular cycles, Nonlinearity, 30 (2017), 2211-2245.
doi: 10.1088/1361-6544/aa675f. |
[46] |
A. Matsko, Practical Applications of Microresonators in Optics and Photonics, 1$^{nd}$ edition, CRC Press, Inc., USA, 2009.
![]() |
[47] |
T. Matsumoto, L. O. Chua and M. Komuro,
Birth and death of the double scroll, Physica D, 24 (1987), 97-124.
doi: 10.1016/0167-2789(87)90069-8. |
[48] |
I. Melbourne, M. Dellnitz and M. Golubitsky,
The structure of symmetric attractors, Arch. Ration. Mech. Anal., 123 (1993), 75-98.
doi: 10.1007/BF00386369. |
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J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York-Berlin, 1982. |
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J. D. M. Rademacher,
Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443.
doi: 10.1016/j.jde.2005.03.016. |
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J. D. M. Rademacher,
Lyapunov–Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Differential Equations, 249 (2010), 305-348.
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show all references
References:
[1] |
A. Abad, R. Barrio, F. Blesa and M. Rodríguez,
Algorithm 924: Tides, a Taylor series Integrator for Differential Equations, ACM Trans. Math. Software, 39 (2012), 1-28.
doi: 10.1145/2382585.2382590. |
[2] |
M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani and M. K. Oberthaler,
Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction, Phys. Rev. Lett., 95 (2005), 010402.
doi: 10.1103/PhysRevLett.95.010402. |
[3] |
V. Arnold, S. Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps: Volume I: The Classification of Critical Points Caustics and Wave Fronts, Monographs in Mathematics, Birkhäuser Boston, 1985.
doi: 10.1007/978-1-4612-5154-5. |
[4] |
P. Ashwin,
Symmetric chaos in systems of three and four forced oscillators, Nonlinearity, 3 (1990), 603-617.
doi: 10.1088/0951-7715/3/3/004. |
[5] |
R. Barrio, M. Carvalho, L. Castro and A. A. P. Rodrigues,
Experimentally accessible orbits near a Bykov cycle, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2030030.
doi: 10.1142/S021812742030030X. |
[6] |
R. Barrio and A. Shilnikov, Parameter–sweeping techniques for temporal dynamics of neuronal systems: Case study of Hindmarsh–Rose model, J. Math. Neurosci., 1 (2011), 20pp.
doi: 10.1186/2190-8567-1-6. |
[7] |
R. Barrio, A. Shilnikov and L. Shilnikov,
Kneadings, symbolic dynamics and painting Lorenz chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1230016.
doi: 10.1142/S0218127412300169. |
[8] |
A. Ben-Tal,
Symmetry restoration in a class of forced oscillators, Phys. D, 171 (2002), 236-248.
doi: 10.1016/S0167-2789(02)00623-1. |
[9] |
M. Brunstein, Nonlinear Dynamics in III-V Semiconductor Photonic Crystal Nano-Cavities, PhD thesis, Université Paris Sud - Paris XI, 2011. |
[10] |
R. C. Calleja, E. J. Doedel, A. R. Humphries, A. Lemus-Rodríguez and E. B. Oldeman,
Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem, Celest. Mech. Dyn. Astron., 114 (2012), 77-106.
doi: 10.1007/s10569-012-9434-y. |
[11] |
B. Cao, K. W. Mahmud and M. Hafezi,
Two coupled nonlinear cavities in a driven-dissipative environment, Phys. Rev. A, 94 (2016), 063805.
|
[12] |
H. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs, Springer, Berlin, 1993. |
[13] |
W. Casteels and C. Ciuti,
Quantum entanglement in the spatial–symmetry-breaking phase transition of a driven-dissipative Bose–Hubbard dimer, Phys. Rev. A, 95 (2017), 013812.
doi: 10.1103/physreva.95.013812. |
[14] |
W. Casteels, F. Storme, A. Le Boité and C. Ciuti,
Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators, Phys. Rev. A, 93 (2016), 033824.
|
[15] |
A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. D. M. Rademacher,
Unfolding a tangent equilibrium–to–periodic heteroclinic cycle, SIAM J. Appl. Dyn. Syst., 8 (2009), 1261-1304.
doi: 10.1137/080734923. |
[16] |
A. R. Champneys, Y. Kuznetsov and B. Sandstede,
A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.
doi: 10.1142/S0218127496000485. |
[17] |
P. Chossat and M. Golubitsky,
Symmetry–increasing bifurcation of chaotic attractors, Physica D, 32 (1988), 423-436.
doi: 10.1016/0167-2789(88)90066-8. |
[18] |
F. Christiansen and H. H. Rugh,
Computing Lyapunov spectra with continuous Gram–Schmidt orthonormalization, Nonlinearity, 10 (1997), 1063-1072.
doi: 10.1088/0951-7715/10/5/004. |
[19] |
P. Coullet and N. Vandenberghe,
Chaotic self-trapping of a weakly irreversible double Bose condensate, Phys. Rev. E, 64 (2001), 025202.
|
[20] |
M. Dellnitz and C. Heinrich,
Admissible symmetry increasing bifurcations, Nonlinearity, 8 (1995), 1039-1066.
|
[21] |
E. J. Doedel,
AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.
|
[22] |
E. J. Doedel, B. Krauskopf and H. M. Osinga,
Global invariant manifolds in the transition to preturbulence in the lorenz system, Indag. Math., 22 (2011), 222-240.
doi: 10.1016/j.indag.2011.10.007. |
[23] |
E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, Department of Computer Science, Concordia University, Montreal, Canada, 2010, available at http://www.cmvl.cs.concordia.ca/. |
[24] |
P. D. Drummond and D. F. Walls,
Quantum theory of optical bistability. I. Nonlinear polarisability model, J. Phys. A: Math. Gen, 13 (1980), 725-741.
|
[25] |
C. Emary and T. Brandes,
Chaos and the quantum phase transition in the Dicke model, Phys. Rev. E, 67 (2003), 066203.
doi: 10.1103/PhysRevE.67.066203. |
[26] |
H. M. Gibbs, Optical Bistability: Controlling Light with Light, Academic Press, 1985.
![]() |
[27] |
R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, Wiley-Interscience, 2002. |
[28] |
A. Giraldo, B. Krauskopf, N. G. R. Broderick, A. M. Yacomotti and J. A. Levenson,
The driven–dissipative Bose–Hubbard dimer: Phase diagram and chaos, New J. Phys., 22 (2020), 043009.
doi: 10.1088/1367-2630/ab7539. |
[29] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Syst., 16 (2017), 640-686.
doi: 10.1137/16M1097419. |
[30] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Cascades of global bifurcations and chaos near a homoclinic flip bifurcation: A case study, SIAM J. Appl. Dyn. Syst., 17 (2018), 2784-2829.
doi: 10.1137/17M1149675. |
[31] |
P. Glendinning,
Bifurcations near homoclinic orbits with symmetry, Phys. Lett. A, 103 (1984), 163-166.
doi: 10.1016/0375-9601(84)90242-1. |
[32] |
A. Golmakani and A. J. Homburg,
Lorenz attractors in unfoldings of homoclinic–flip bifurcations, Dyn. Syst., 26 (2011), 61-76.
doi: 10.1080/14689367.2010.503186. |
[33] |
C. Grebogi, E. Ott and J. A. Yorke,
Crises, sudden changes in chaotic attractors, and transient chaos, Phys. D, 7 (1983), 181-200.
doi: 10.1016/0167-2789(83)90126-4. |
[34] |
S. Haddadi, P. Hamel, G. Beaudoin, I. Sagnes, C. Sauvan, P. Lalanne, J. A. Levenson and A. M. Yacomotti,
Photonic molecules: Tailoring the coupling strength and sign, Optics Express, 22 (2014), 12359.
|
[35] |
P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson and A. M. Yacomotti,
Spontaneous mirror–symmetry breaking in coupled photonic–crystal nanolasers, Nature Photonics, 9 (2015), 311-315.
|
[36] |
C. Heinrich,
Symmetry increasing bifurcations via collisions of attractors, Rocky Mountain J. Math., 29 (2008), 559-608.
doi: 10.1216/rmjm/1181071652. |
[37] |
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, In Handbook of Dynamical Systems, (eds. H. W. Broer, B. Hasselblatt and F. Takens), Elsevier, New York, 3 (2010), 381–509. |
[38] |
J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2$^{nd}$ edition, Princeton University Press, 2008.
![]() |
[39] |
G. P. King and S. T. Gaito,
Bistable chaos. II. Bifurcation analysis M., Phys. Rev. A, 46 (1992), 3100-3110.
|
[40] |
J. Knobloch, J. S. Lamb and K. N. Webster,
Using Lin's method to solve Bykov's problems, J. Differential Equations, 257 (2014), 2984-3047.
doi: 10.1016/j.jde.2014.06.006. |
[41] |
B. Krauskopf and B. E. Oldeman,
A planar model system for the saddle-node Hopf bifurcation with global reinjection, Nonlinearity, 17 (2004), 1119-1151.
doi: 10.1088/0951-7715/17/4/001. |
[42] |
B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, In Numerical Continuation Methods for Dynamical Systems, (2007), 117–154.
doi: 10.1007/978-1-4020-6356-5_4. |
[43] |
B. Krauskopf and T. Rieß,
A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690.
doi: 10.1088/0951-7715/21/8/001. |
[44] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{nd}$ edition, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[45] |
A. Lohse and A. Rodrigues,
Boundary crisis for degenerate singular cycles, Nonlinearity, 30 (2017), 2211-2245.
doi: 10.1088/1361-6544/aa675f. |
[46] |
A. Matsko, Practical Applications of Microresonators in Optics and Photonics, 1$^{nd}$ edition, CRC Press, Inc., USA, 2009.
![]() |
[47] |
T. Matsumoto, L. O. Chua and M. Komuro,
Birth and death of the double scroll, Physica D, 24 (1987), 97-124.
doi: 10.1016/0167-2789(87)90069-8. |
[48] |
I. Melbourne, M. Dellnitz and M. Golubitsky,
The structure of symmetric attractors, Arch. Ration. Mech. Anal., 123 (1993), 75-98.
doi: 10.1007/BF00386369. |
[49] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York-Berlin, 1982. |
[50] |
J. D. M. Rademacher,
Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443.
doi: 10.1016/j.jde.2005.03.016. |
[51] |
J. D. M. Rademacher,
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