doi: 10.3934/dcdsb.2021217
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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

* Corresponding author: Andrus Giraldo (a.giraldo@auckland.ac.nz)

Received  May 2021 Revised  July 2021 Early access September 2021

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.

Citation: Andrus Giraldo, Neil G. R. Broderick, Bernd Krauskopf. Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021217
References:
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show all references

References:
[1]

A. AbadR. BarrioF. 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.  Google Scholar

[2]

M. AlbiezR. GatiJ. FöllingS. HunsmannM. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[5]

R. BarrioM. CarvalhoL. 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.  Google Scholar

[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.  Google Scholar

[7]

R. BarrioA. 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.  Google Scholar

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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.  Google Scholar

[9]

M. Brunstein, Nonlinear Dynamics in III-V Semiconductor Photonic Crystal Nano-Cavities, PhD thesis, Université Paris Sud - Paris XI, 2011. Google Scholar

[10]

R. C. CallejaE. J. DoedelA. R. HumphriesA. 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.  Google Scholar

[11]

B. CaoK. W. Mahmud and M. Hafezi, Two coupled nonlinear cavities in a driven-dissipative environment, Phys. Rev. A, 94 (2016), 063805.   Google Scholar

[12]

H. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs, Springer, Berlin, 1993. Google Scholar

[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.  Google Scholar

[14]

W. CasteelsF. StormeA. Le Boité and C. Ciuti, Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators, Phys. Rev. A, 93 (2016), 033824.   Google Scholar

[15]

A. R. ChampneysV. KirkE. KnoblochB. 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.  Google Scholar

[16]

A. R. ChampneysY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. Coullet and N. Vandenberghe, Chaotic self-trapping of a weakly irreversible double Bose condensate, Phys. Rev. E, 64 (2001), 025202.   Google Scholar

[20]

M. Dellnitz and C. Heinrich, Admissible symmetry increasing bifurcations, Nonlinearity, 8 (1995), 1039-1066.   Google Scholar

[21]

E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.   Google Scholar

[22]

E. J. DoedelB. 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.  Google Scholar

[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/. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26] H. M. Gibbs, Optical Bistability: Controlling Light with Light, Academic Press, 1985.   Google Scholar
[27]

R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, Wiley-Interscience, 2002.  Google Scholar

[28]

A. GiraldoB. KrauskopfN. G. R. BroderickA. 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.  Google Scholar

[29]

A. GiraldoB. 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.  Google Scholar

[30]

A. GiraldoB. 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.  Google Scholar

[31]

P. Glendinning, Bifurcations near homoclinic orbits with symmetry, Phys. Lett. A, 103 (1984), 163-166.  doi: 10.1016/0375-9601(84)90242-1.  Google Scholar

[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.  Google Scholar

[33]

C. GrebogiE. 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.  Google Scholar

[34]

S. HaddadiP. HamelG. BeaudoinI. SagnesC. SauvanP. LalanneJ. A. Levenson and A. M. Yacomotti, Photonic molecules: Tailoring the coupling strength and sign, Optics Express, 22 (2014), 12359.   Google Scholar

[35]

P. HamelS. HaddadiF. RaineriP. MonnierG. BeaudoinI. SagnesA. Levenson and A. M. Yacomotti, Spontaneous mirror–symmetry breaking in coupled photonic–crystal nanolasers, Nature Photonics, 9 (2015), 311-315.   Google Scholar

[36]

C. Heinrich, Symmetry increasing bifurcations via collisions of attractors, Rocky Mountain J. Math., 29 (2008), 559-608.  doi: 10.1216/rmjm/1181071652.  Google Scholar

[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. Google Scholar

[38] J. D. JoannopoulosS. G. JohnsonJ. N. Winn and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2$^{nd}$ edition, Princeton University Press, 2008.   Google Scholar
[39]

G. P. King and S. T. Gaito, Bistable chaos. II. Bifurcation analysis M., Phys. Rev. A, 46 (1992), 3100-3110.   Google Scholar

[40]

J. KnoblochJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

A. Lohse and A. Rodrigues, Boundary crisis for degenerate singular cycles, Nonlinearity, 30 (2017), 2211-2245.  doi: 10.1088/1361-6544/aa675f.  Google Scholar

[46] A. Matsko, Practical Applications of Microresonators in Optics and Photonics, 1$^{nd}$ edition, CRC Press, Inc., USA, 2009.   Google Scholar
[47]

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Figure 1.  One-parameter bifurcation diagrams in $ f $ of systemk (1) for $ \kappa = 2 $ at different values of $ \delta $ as stated, where solutions are represented by the intensities $ |A|^2 $ and $ |B|^2 $. Shown are stable equilibria (blue), saddle equilibrium with a single unstable eigenvalue (cyan) and with two unstable eigenvalues (orange), stable periodic solutions (dark green), and saddle periodic solution with one unstable Floquet multiplier (light green). Regions in colored frames are enlarged in the corresponding subpanels
Figure 2.  Pairs of connecting orbits of systemk (1) for $ \kappa = 2 $ and $ \delta = -5 $, shown in the $ (|A|^2, |B|^2) $-plane and as time series of $ |A|^2 $ and $ |B|^2 $, respectively. Panelsk (a) show the Shilnikov homoclinic orbits of the symmetric saddle equilibrium $ p $ at $ f \approx 2.6227 $, labeled $ \bf{HOM} $ in Fig.k 1(d3); and panelsk (b) show the nearby heteroclinic EtoP connections from $ p $ to two asymmetric saddle periodic orbits at $ f \approx 2.5900 $
Figure 3.  Chaotic attractors (blue curves) of system (1) for $ \kappa = 2 $ and $ \delta = -5 $ at $ f = 3.06 $ (a) and at $ f = 3.3 $ (b), shown in the $ (|A|^2, |B|^2) $-plane together with the branch $ W_+^u(p) $ (red curve) and as time series of $ |A|^2 $ and $ |B|^2 $, respectively; the insets show the respective power spectrum on a logarithmic scale
Fig. 1. The inset (a2) shows an enlargement of the area indicated by the black frame in panel (a1), after the linear transformation $ \tilde{\delta} = \delta-(-2.6047(f-3.087)-8.632) $, to showcase the organization of the bifurcation curves near the points $ \bf{T} $ and $ \bf{Fl} $">Figure 4.  Bifurcation diagram in the $ (f, \delta) $-plane of system (1) for $ \kappa = 2 $. Shown are curves of saddle-node bifurcations of symmetry equilibria $ \bf{S} $ (brown) and asymmetric equilibria $ \bf{S^*} $ (orange), pitchfork $ \bf{P} $ (purple) and Andronov--Hopf $ \bf{H} $ (green) bifurcations, saddle-node bifurcations of periodic solutions $ \bf{SNP} $ (dark-green), period-doubling bifurcations $ \bf{PD} $ (red), Shilnikov bifurcations $ \bf{Hom} $ to symmetric focus equilibria (black), homoclinic bifurcations to real symmetric equilibria $ \bf{HOM^1_q} $ and $ \bf{HOM^2_q} $ (lilac and dark-lilac), and EtoP connections (cyan); also shown are codimension-two points of cusp $ \bf{CP} $, saddle-node-pitchfork $ \bf{SP} $, generalised Andronov--Hopf $ \bf{GH} $, Bykov T-point $ \bf{T} $ and homoclinic flip $ \bf{Fl} $ bifurcations; the horizontal dashed lines correspond to the panels of Fig. 1. The inset (a2) shows an enlargement of the area indicated by the black frame in panel (a1), after the linear transformation $ \tilde{\delta} = \delta-(-2.6047(f-3.087)-8.632) $, to showcase the organization of the bifurcation curves near the points $ \bf{T} $ and $ \bf{Fl} $
Fig. 2. Panelsk (a) show the homoclinic orbits to the real symmetric saddle $ q $ on $ \bf{HOM^2_q} $ at $ \delta = -9.5 $ and $ f \approx 3.4139 $; and panels (b) show the heteroclinic cycle at the Bykov T-point $ \bf{T} $ at $ \delta \approx -8.5888 $ and $ f \approx 3.0705 $, consisting of a pair of codimension-two connections from $ p $ to $ q $ (see (b2) and (b3) for the time series) and a single structurally stable connection in $ \text{Fix}(\eta) $ from $ q $ to $ p $ (see (b4) and (b5) for the time series)">Figure 5.  Pairs of connecting orbits of system (1) for $ \kappa = 2 $, represented as in Fig. 2. Panelsk (a) show the homoclinic orbits to the real symmetric saddle $ q $ on $ \bf{HOM^2_q} $ at $ \delta = -9.5 $ and $ f \approx 3.4139 $; and panels (b) show the heteroclinic cycle at the Bykov T-point $ \bf{T} $ at $ \delta \approx -8.5888 $ and $ f \approx 3.0705 $, consisting of a pair of codimension-two connections from $ p $ to $ q $ (see (b2) and (b3) for the time series) and a single structurally stable connection in $ \text{Fix}(\eta) $ from $ q $ to $ p $ (see (b4) and (b5) for the time series)
Figure 6.  The branch $ W^u_+(p) $ (red curve) of the symmetric equilibrium $ p $ of system (1) for $ \kappa = 2 $ and $ \delta = -7 $ as it spirals to one of a pair of stable asymmetric equilibria for $ f = 3.5 $ (a) and for $ f = 4.0 $ (b), shown in the $ (|A|^2, |B|^2) $-plane (top row) as temporal trace of $ |B|^2-|A|^2 $ (bottom row). The maxima (light green rhombi) and minima (dark green squares) of this time series define the respective shown kneading sequence $ S $
Fig. 4 (not labelled) and additionally the Shilnikov bifurcation curve $ \bf{HOM_1} $ (black curve). The inset (b) shows $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{HOM_1} $ for $ (f, \delta) \approx (3.1000, -7.7740) $, as indicated by the white asterisk in panel (a)">Figure 7.  Coloring by finite kneading sequences $ S^2 $ and associated Shilnikov bifurcation near the Bykov T-point $ \bf{T} $. Panel (a) shows the $ (f, \delta) $-plane of system (1) with the curves of local bifurcation of equilibria and periodic orbits from Fig. 4 (not labelled) and additionally the Shilnikov bifurcation curve $ \bf{HOM_1} $ (black curve). The inset (b) shows $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{HOM_1} $ for $ (f, \delta) \approx (3.1000, -7.7740) $, as indicated by the white asterisk in panel (a)
Fig. 7 and additionally the Shilnikov bifurcation curves $ \bf{HOM^1_2} $ and $ \bf{HOM^2_2} $ (grey curves). The inset (a2) is an enlargement of the region inside the white frame in panel (a1). Panelsk (b1) and (b2) show $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{HOM^1_2} $ for $ (f, \delta) \approx (3.1000, -6.2153) $ and at $ \bf{HOM^2_2} $ for $ (f, \delta) \approx (3.10, -7.9449) $, respectively, as indicated by the white asterisks in panel (a1)">Figure 8.  Coloring by finite kneading sequences $ S^3 $ and associated Shilnikov bifurcations near $ \bf{T} $. Panel (a1) shows the $ (f, \delta) $-plane of system (1) with the bifurcation curves from Fig. 7 and additionally the Shilnikov bifurcation curves $ \bf{HOM^1_2} $ and $ \bf{HOM^2_2} $ (grey curves). The inset (a2) is an enlargement of the region inside the white frame in panel (a1). Panelsk (b1) and (b2) show $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{HOM^1_2} $ for $ (f, \delta) \approx (3.1000, -6.2153) $ and at $ \bf{HOM^2_2} $ for $ (f, \delta) \approx (3.10, -7.9449) $, respectively, as indicated by the white asterisks in panel (a1)
Fig. 8 and additionally the Shilnikov bifurcation curves $ \bf{HOM^1_3} $ to $ \bf{HOM^4_3} $ (grey curves). Panelsk (b1)--(b4) show $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{HOM^1_3} $ to $ \bf{HOM^4_3} $ for $ (f, \delta) \approx (3.1000, -5.1716) $; $ (3.1000, -6.8865) $; $ (3.7000, -7.7238) $; $ (3.7000, -8.2547) $ as indicated by the white asterisks in panel (a)">Figure 9.  Coloring by finite kneading sequences $ S^4 $ and associated Shilnikov bifurcations near $ \bf{T} $. Panel (a) shows the $ (f, \delta) $-plane of system (1) with the bifurcation curves from Fig. 8 and additionally the Shilnikov bifurcation curves $ \bf{HOM^1_3} $ to $ \bf{HOM^4_3} $ (grey curves). Panelsk (b1)--(b4) show $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{HOM^1_3} $ to $ \bf{HOM^4_3} $ for $ (f, \delta) \approx (3.1000, -5.1716) $; $ (3.1000, -6.8865) $; $ (3.7000, -7.7238) $; $ (3.7000, -8.2547) $ as indicated by the white asterisks in panel (a)
Fig. 7(a) to Fig. 9(a). Panel (b1) shows additionally curves $ \bf{Hep} $, $ \bf{Hep_1^-} $ and $ \bf{Hep^{+/-}_i} $, $ \bf{i} = \bf{2} , .., \bf{4} $ of codimension-one heteroclinic EtoP connections between $ p $ and the orientable saddle periodic orbits $ \Gamma_o $ and $ \Gamma_o^* $. In panel (b1) parameter values are indicated for which the curves $ W^u_+(p) $, $ \Gamma_o $ and $ \Gamma_o^* $ are shown in separate figures, namely: for the white asterisk along the curve $ \bf{Hep} $ see Fig. 2(b); and for the yellow asterisks along the curves $ \bf{Hep_3^+} $, $ \bf{Hep_3^-} $ and $ \bf{Hep_4^+} $ and the grey squares inside the respective bounded regions with constants kneading sequence see Fig. 11. The inset (b2) shows $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{Hep^-_1} $ at the parameter values indicated by the black asterisk in panel (b1)">Figure 10.  Coloring by finite kneading sequences $ S^n $ and representative curves of Shilnikov bifurcations (grey and unlabelled) in the $ (f, \delta) $-plane near $ \bf{T} $ for $ n = 7 $ (a) and for $ n = 12 $ (b1); compare with Fig. 7(a) to Fig. 9(a). Panel (b1) shows additionally curves $ \bf{Hep} $, $ \bf{Hep_1^-} $ and $ \bf{Hep^{+/-}_i} $, $ \bf{i} = \bf{2} , .., \bf{4} $ of codimension-one heteroclinic EtoP connections between $ p $ and the orientable saddle periodic orbits $ \Gamma_o $ and $ \Gamma_o^* $. In panel (b1) parameter values are indicated for which the curves $ W^u_+(p) $, $ \Gamma_o $ and $ \Gamma_o^* $ are shown in separate figures, namely: for the white asterisk along the curve $ \bf{Hep} $ see Fig. 2(b); and for the yellow asterisks along the curves $ \bf{Hep_3^+} $, $ \bf{Hep_3^-} $ and $ \bf{Hep_4^+} $ and the grey squares inside the respective bounded regions with constants kneading sequence see Fig. 11. The inset (b2) shows $ W^u_+(p) $ in the $ (|A|^2, |B|^2) $-plane at $ \bf{Hep^-_1} $ at the parameter values indicated by the black asterisk in panel (b1)
Figure 11.  Heteroclinic orbits along the curves $ \bf{Hep_3^+} $, $ \bf{Hep_3^-} $ and $ \bf{Hep_4^+} $ (left column) and the behavior of $ W^u_+(p) $ inside the respective enclosed regions of constant kneading sequence (right column); shown in the $ (|A|^2, |B|^2) $-plane are $ W^u_+(p) $ (red curve), $ \Gamma_o $ and $ \Gamma_o^* $ (cyan curves), and the different equilibria. Panels (a1), (b1) and (c1) are for $ (f, \delta) \approx (3.3000, -6.1106) $ on $ \bf{Hep_3^+} $, $ (f, \delta) \approx (3.0000, -5.9816) $ on $ \bf{Hep_3^-} $, and $ (f, \delta) \approx (2.9200, -5.7506) $ on $ \bf{Hep_4^+} $, respectively. Panels (a2)--(c2) are for $ (f, \delta) = (3.2, -6.25) $; $ (3.05, -6.1) $; $ (2.93, -5.9) $
Fig. 13; and for the grey and red squares see Fig. 14">Figure 12.  Bifurcations associated with S-invariant periodic orbits. Panel (a) shows the coloring of the $ (f, \delta) $-plane by finite kneading sequences $ S^{12} $ with associated curves of saddle node of periodic orbit bifurcation $ \bf{SNP^s} $ (light-green) that creates the orientable S-invariant periodic orbits $ \Gamma^s_a $ and $ \Gamma^s_o $, of symmetry breaking bifurcation $ \bf{SB^s} $ (yellow) of $ \Gamma^s_a $, of successive period-doubling bifurcations $ \bf{PD_i^s} $ (brown), of symmetry increasing bifurcation $ \bf{SI^s} $, and of EtoP connections $ \bf{Hep^{s}_i} $ with $ \bf{i} = \bf{3} , \ldots, \bf{6} $ (dark cyan) from $ p $ to $ \Gamma^s_o $. The periodic orbits $ \Gamma^s_a $ (green) and $ \Gamma^s_o $ (cyan) at the right-most triangle with $ (f, \delta) = (3.3, -5.4) $ are shown in the $ (|A|^2, |B|^2) $-plane in panel (b1) and as time series of $ |A|^2 $ and $ |B|^2 $ in panels (b2) and (b3), respectively. In panel (a) parameter values are indicated for which the curves $ W^u_+(p) $, $ \Gamma^s_a $ and $ \Gamma^s_o $ are shown in separate figures, namely: for the grey triangles and the white asterisks along the curves $ \bf{Hep_3^s} $, $ \bf{Hep_4^s} $ and $ \bf{Hep_5^s} $ see Fig. 13; and for the grey and red squares see Fig. 14
Fig. 12(a) inside regions of constant kneading sequence (left column), and the associated heteroclinic orbits at the white asterisk in Fig. 12(a) along the bounding curves $ \bf{Hep_3^s} $ to $ \bf{Hep_5^s} $ (right column). Shown in the $ (|A|^2, |B|^2) $-plane are $ W^u_+(p) $ (red curve), the periodic orbits $ \Gamma^s_a $ (green curve) and $ \Gamma^s_o $ (cyan curve), and the different equilibria. Panels (a1)--(c1) are for $ (f, \delta) = (3.30, -5.4) $; $ (2.85, -5.2) $; $ (2.73, -5.2) $. Panels (a2), (b2) and (c2) are for $ (f, \delta) \approx (3.2000, -5.2902) $ on $ \bf{Hep_3^s} $, $ (f, \delta) \approx (2.8500, -5.0840) $ on $ \bf{Hep_4^s} $, and $ (f, \delta) \approx (2.7200, -5.0877) $ on $ \bf{Hep_5^s} $, respectively">Figure 13.  The behavior of $ W^u_+(p) $ at the grey triangles in Fig. 12(a) inside regions of constant kneading sequence (left column), and the associated heteroclinic orbits at the white asterisk in Fig. 12(a) along the bounding curves $ \bf{Hep_3^s} $ to $ \bf{Hep_5^s} $ (right column). Shown in the $ (|A|^2, |B|^2) $-plane are $ W^u_+(p) $ (red curve), the periodic orbits $ \Gamma^s_a $ (green curve) and $ \Gamma^s_o $ (cyan curve), and the different equilibria. Panels (a1)--(c1) are for $ (f, \delta) = (3.30, -5.4) $; $ (2.85, -5.2) $; $ (2.73, -5.2) $. Panels (a2), (b2) and (c2) are for $ (f, \delta) \approx (3.2000, -5.2902) $ on $ \bf{Hep_3^s} $, $ (f, \delta) \approx (2.8500, -5.0840) $ on $ \bf{Hep_4^s} $, and $ (f, \delta) \approx (2.7200, -5.0877) $ on $ \bf{Hep_5^s} $, respectively
Figure 14.  Attracting symmetry-broken periodic orbits $ \widehat{\Gamma}_a $ (green curve) and chaotic attractors (blue curves) of system (1) for $ \kappa = 2 $, $ f = 3.3 $, and $ \delta = -5.52 $ in row (a), $ \delta = -5.61 $ in row (b) and $ \delta = -5.63 $ in row (c); shown in the $ (|A|^2, |B|^2) $-plane with the branch $ W_+^u(p) $ (red curve) and as time series of $ |A|^2 $ and $ |B|^2 $, respectively; the insets show the power spectra of the chaotic attractors
Fig. 16">Figure 15.  Bifurcations associated with the symmetric attractors arising from $ \Gamma_a $ and $ \Gamma^*_a $. Shown is the coloring of the $ (f, \delta) $-plane by finite kneading sequences $ S^{12} $ with curves of period-doubling bifurcations $ \bf{PD_i} $ (brown), of Andronov–Hopf bifurcation $ \bf{H} $ where the saddles $ \Gamma_o $ and $ \Gamma_o^* $ bifurcate with the two stable equilibria to create the pair of saddle equilibria $ q $ and $ q^* $, and of folds $ \bf{F} $ (blue) and $ \bf{F^*} $ where $ W^s(p) $ is tangent to $ W^u(\Gamma_o) $ and $ W^u(\Gamma_o^*) $, and to $ W^u(q) $ and $ W^u(q^*) $, respectively. Also labelled are the codimension-two points $ \bf{CF} $ where $ \bf{F} $ intersects the curve $ \bf{Hep} $, and $ \bf{HBF} $ where the tangency of $ W^s(p) $ coincides with the Andronov–Hopf bifurcation. The yellow squares and white asterisks indicate the parameter points chosen for the panels in Fig. 16
Fig. 15, and panels (a2) and (b2) show the respective tangent heteroclinic orbits at the asterisks on the curves $ \bf{F} $ and $ \bf{F^*} $; here $ (f, \delta) = (2.80, -5.0) $ in (a1), $ (f, \delta) \approx (2.800, -4.8752) $ in (a2), $ (f, \delta) = (3.25, -5.10) $ in (b1), and $ (f, \delta) \approx (3.2500, -5.0192) $ in (b2)">Figure 16.  Heteroclinic orbits in the $ (|A|^2, |B|^2) $-plane between the symmetric saddle equilibrium $ p $ and the periodic orbit $ \Gamma^s_o $ (a) and the asymmetric saddle equilibrium $ q_o $ (b). Panelsk (a1) and (b1) show pairs of structurally stable heteroclinic orbits at the squares in Fig. 15, and panels (a2) and (b2) show the respective tangent heteroclinic orbits at the asterisks on the curves $ \bf{F} $ and $ \bf{F^*} $; here $ (f, \delta) = (2.80, -5.0) $ in (a1), $ (f, \delta) \approx (2.800, -4.8752) $ in (a2), $ (f, \delta) = (3.25, -5.10) $ in (b1), and $ (f, \delta) \approx (3.2500, -5.0192) $ in (b2)
Figure 17.  Symmetry increasing bifurcation of chaotic attractors at the transition of system (1) through the heteroclinic fold curve $ \bf{F} $, before $ \bf{F} $ at $ (f, \delta) = (2.8, -4.8) $ in panel (a), at $ \bf{F} $ at $ (f, \delta)\approx (2.8000, -4.8752) $ in panel (b), and after $ \bf{F} $ at $ (f, \delta) = (2.8, -5.0) $ in panel (c). Shown in $ (|A|^2, |B|^2, |A|\, |B|\sin(\phi_A-\phi_B)) $-space are the saddle equilibrium $ p $, the asymmetric attracting equilibria (blue dots), the saddle periodic orbits $ \Gamma_o $ and $ \Gamma^*_o $ (cyan curves) and the parts of their unstable manifolds $ W^{u}(\Gamma_o) $ (gold surface) and $ W^{u}(\Gamma^*_o) $ (red surface) that accumulate on the respective chaotic attractors; panel (b) also shows the tangent heteroclinic orbits in $ W^s(p) \cap W^{u}(\Gamma_o) $ (brown curves) and $ W^s(p) \cap W^{u}(\Gamma_o^*) $ (red curves), and panel (c) also shows the pair of transversal heteroclinic orbits in $ W^s(p) \cap W^{u}(\Gamma_o) $ (pink and maroon curves)
Figure 18.  The singular EtoP cycle between $ p $ and $ \Gamma_o $ of system (1) at the codimension-two bifurcation point $ \bf{CF} $ with $ (f, \delta) \approx (2.577, -4.952) $ where the curves $ \bf{F} $ and $ \bf{Hep} $ intersect, shown in $ (|A|^2, |B|^2, |A|\, |B|\sin(\phi_A-\phi_B)) $-space in panel (a) and as a compact time connection (CTC) plot in panel (b). It consists of the codimension-one EtoP connection formed by $ W_u^+(p) $ (red curve) and the tangent return connection (orange curve) that converges forward in time to $ p $ and backward in time to $ \Gamma_o $
Fig. 15 and additionally the curves $ \bf{SB^s} $ (yellow) of symmetry breaking bifurcation, $ \bf{PD_i^s} $ (brown) of successive period-doubling bifurcations, $ \bf{SI^s} $ (light blue) of symmetry increasing bifurcation, $ \bf{F^s} $ (blue) of fold bifurcation where $ W^s(p) $ and $ W^u(\Gamma^s) $ are tangent, and $ \bf{HT^s_o} $ (magenta) of homoclinic tangency of $ \Gamma^s_o $; see also the inset (a2) and compare with Fig. 12. At the red squares in panels (a) one finds the chaotic attractors from Fig. 14(b) and (c), and at the blue square at $ (f, \delta) = (3.3, -5.75) $ there is the chaotic attractor shown in panels (b); see also Fig. 20">Figure 19.  Bifurcations associated with the symmetric attractors arising from $ \widehat{\Gamma}_a $ and $ \widehat{\Gamma}^*_a $. Panel (a1) shows the coloring of the $ (f, \delta) $-plane by finite kneading sequences $ S^{12} $ with the curves from Fig. 15 and additionally the curves $ \bf{SB^s} $ (yellow) of symmetry breaking bifurcation, $ \bf{PD_i^s} $ (brown) of successive period-doubling bifurcations, $ \bf{SI^s} $ (light blue) of symmetry increasing bifurcation, $ \bf{F^s} $ (blue) of fold bifurcation where $ W^s(p) $ and $ W^u(\Gamma^s) $ are tangent, and $ \bf{HT^s_o} $ (magenta) of homoclinic tangency of $ \Gamma^s_o $; see also the inset (a2) and compare with Fig. 12. At the red squares in panels (a) one finds the chaotic attractors from Fig. 14(b) and (c), and at the blue square at $ (f, \delta) = (3.3, -5.75) $ there is the chaotic attractor shown in panels (b); see also Fig. 20
Fig. 19. Shown in $ (|A|^2, |B|^2, |A|\, |B|\sin(\phi_A-\phi_B)) $-space are the saddle equilibrium $ p $, the asymmetric attracting equilibria (blue dots), the S-invariant saddle periodic orbit $ \Gamma^s $ (cyan curves) and the two sides of its unstable manifolds $ W^{u}(\Gamma^s) $ (gold and red surfaces)">Figure 20.  Symmetry increasing and interior crisis bifurcations of chaotic attractors of system (1) due to crossing the curves $ \bf{SI^s} $ and $ \bf{F^s} $, shown before the curve $ \bf{SI^s} $ at $ (f, \delta) = (3.3, -5.61) $ in panel (a), after $ \bf{SI^s} $ but before $ \bf{F^s} $ at $ (f, \delta) = (3.3, -5.63) $ in panel (b), and after $ \bf{F^s} $ at $ (f, \delta) = (3.3, -5.75) $ in panel (c); see the stars in Fig. 19. Shown in $ (|A|^2, |B|^2, |A|\, |B|\sin(\phi_A-\phi_B)) $-space are the saddle equilibrium $ p $, the asymmetric attracting equilibria (blue dots), the S-invariant saddle periodic orbit $ \Gamma^s $ (cyan curves) and the two sides of its unstable manifolds $ W^{u}(\Gamma^s) $ (gold and red surfaces)
Figure 21.  The singular heteroclinic cycle between $ p $, $ \Gamma_o $ and $ \Gamma^s $ at the codimension-two bifurcation point $ \bf{CF_1^s} $ with $ (f, \delta) \approx (2.6932, -5.347) $ where the curves $ \bf{F^s} $ and $ \bf{Hep} $ intersect, shown in $ (|A|^2, |B|^2, |A|\, |B|\sin(\phi_A-\phi_B)) $-space in panel (a) and as a CTC plot in panel (b). It consists of the codimension-one EtoP connection from $ p $ to $ \Gamma_o $ (red curve), a structurally stable connection from $ \Gamma_o $ to $ \Gamma^s $ (orange curve), and a tangent connection from $ \Gamma^s $ back to $ p $ (blue curve)
Fig. 19. The intersection points of these fold curves with the curve $ \bf{Hep} $ are denoted $ \bf{DSC_+^s} $, $ \bf{DSC_+^+} $ and $ \bf{DSC_+^-} $, respectively; the curve $ \bf{TC_+^s} $ ends at the codimension-two point $ \bf{HTC_+^s} $ on the Andronov--Hopf bifurcation curve $ \bf{H} $">Figure 22.  Bifurcation curves of first tangencies of periodic orbits in the $ (f, \delta) $-plane, namely the curve $ \bf{TC_+^s} $ of heteroclinic tangency between $ W^u(\Gamma_o) $ and $ W^s\Gamma^s_o) $, the curve $ \bf{TC_+^+} $ of homoclinic tangency between $ W^u(\Gamma_o) $ and $ W^s\Gamma_o) $, and the curve $ \bf{TC_+^-} $ of heteroclinic tangency between $ W^u(\Gamma_o) $ and $ W^s\Gamma_o^*) $; also shown are the fold curves $ \bf{F} $, $ \bf{F^*} $ and $ \bf{F^s} $ from Fig. 19. The intersection points of these fold curves with the curve $ \bf{Hep} $ are denoted $ \bf{DSC_+^s} $, $ \bf{DSC_+^+} $ and $ \bf{DSC_+^-} $, respectively; the curve $ \bf{TC_+^s} $ ends at the codimension-two point $ \bf{HTC_+^s} $ on the Andronov--Hopf bifurcation curve $ \bf{H} $
Fig. 22: in panel (a) between $ p $, $ \Gamma_o $ and $ \Gamma_o^s $ at $ \bf{DSC_+^s} $ with $ (f, \delta) \approx (2.6219, -5.1092) $; in panel (b) between $ p $ and $ \Gamma_o $ with a homoclinic loop of $ \Gamma_o^s $ at $ \bf{DSC_+^+} $ with $ (f, \delta) \approx (2.7822, -5.6551) $; and in panel (c) between $ p $, $ \Gamma_o $ and $ \Gamma_o^* $ at $ \bf{DSC_+^-} $ at $ (f, \delta) \approx (2.7923, -5.6952) $. These cycles consists of the EtoP connection from $ p $ to the first periodic orbit (red curve), a tangent connection between the two respective periodic orbits (orange curve), and a structurally stable connection from the second periodic orbit back to $ p $ (blue curve)">Figure 23.  CTC plots of singular heteroclinic cycles on the curve $ \bf{Hep} $ at the codimension-two points labelled in Fig. 22: in panel (a) between $ p $, $ \Gamma_o $ and $ \Gamma_o^s $ at $ \bf{DSC_+^s} $ with $ (f, \delta) \approx (2.6219, -5.1092) $; in panel (b) between $ p $ and $ \Gamma_o $ with a homoclinic loop of $ \Gamma_o^s $ at $ \bf{DSC_+^+} $ with $ (f, \delta) \approx (2.7822, -5.6551) $; and in panel (c) between $ p $, $ \Gamma_o $ and $ \Gamma_o^* $ at $ \bf{DSC_+^-} $ at $ (f, \delta) \approx (2.7923, -5.6952) $. These cycles consists of the EtoP connection from $ p $ to the first periodic orbit (red curve), a tangent connection between the two respective periodic orbits (orange curve), and a structurally stable connection from the second periodic orbit back to $ p $ (blue curve)
Fig. 3, at red and blue squares see Fig. 20. Panelsk (b) and (c) show $ W^u_+(p) $ at the green triangles in panel (a2) at $ (f, \delta) = (2.6, -4.89) $ and $ (f, \delta) = (2.6, -4.91) $, respectively. Panelsk (d1) and (d2) show the four transversal homoclinic orbits to $ \Gamma_o $, labeled $ \gamma_1 $ to $ \gamma_4 $, that exist simulaneously for $ (f, \delta) = (2.65, -4.985) $, the yellow square in panel (a2)">Figure 24.  Panel (a1) shows the maximum Lyapunov exponent in the $ (f, \delta) $-plane associated with $ W^u_+(p) $, with previously shown bifurcation curves and $ \bf{HT_{[i, j]}} $ for $ \bf{i, j} = 1, ..., 4 $ of fold bifurcations of homoclinic orbits to $ \Gamma_o $; see also the enlargement in panel (a2). For chaotic attractors at green squares see Fig. 3, at red and blue squares see Fig. 20. Panelsk (b) and (c) show $ W^u_+(p) $ at the green triangles in panel (a2) at $ (f, \delta) = (2.6, -4.89) $ and $ (f, \delta) = (2.6, -4.91) $, respectively. Panelsk (d1) and (d2) show the four transversal homoclinic orbits to $ \Gamma_o $, labeled $ \gamma_1 $ to $ \gamma_4 $, that exist simulaneously for $ (f, \delta) = (2.65, -4.985) $, the yellow square in panel (a2)
Fig. 24(a2) near the point, and panel (a2) shows the same data after a coordinate change to straightened parameters $ \widehat{f} $ and $ \widehat{\delta} $, where the curves $ \bf{Hep} $ and $ \bf{F} $ are the coordinate axes. The grey curve, the continuation of $ \bf{HT_{[3, 4]}} $ oscillates into the point $ \bf{CF} $, which generates further pairs of curves $ \bf{HT_{[i, i+1]}} $, $ \bf{i} = 2, 4, 6 $ and $ \bf{HT_{[i, i+3]}} $, $ \bf{i} = 1, 3, 5 $ (fuchsia) of fold bifurcations of homoclinic orbits to $ \Gamma_o $. Panelsk (b1) and (b2) show the continuation in $ f $ for $ \delta = -4.985 $ of the two transversal homoclinic orbits $ \gamma_3 $ and $ \gamma_4 $ from Fig. 24(d2), where the time $ T_{out} $ is the time that the homoclinic orbit spends outside a tubular neighborhood of $ \Gamma_o $. Panelsk (c1) and (c2) show the two respective homoclinic orbits with large $ T_{out} $ as they approach the bifurcation $ \bf{Hep} $">Figure 25.  Bifurcation diagram near the codimension-two point $ \bf{CF} $ in the $ (f, \delta) $-plane. Panel (a1) is an enlargement of Fig. 24(a2) near the point, and panel (a2) shows the same data after a coordinate change to straightened parameters $ \widehat{f} $ and $ \widehat{\delta} $, where the curves $ \bf{Hep} $ and $ \bf{F} $ are the coordinate axes. The grey curve, the continuation of $ \bf{HT_{[3, 4]}} $ oscillates into the point $ \bf{CF} $, which generates further pairs of curves $ \bf{HT_{[i, i+1]}} $, $ \bf{i} = 2, 4, 6 $ and $ \bf{HT_{[i, i+3]}} $, $ \bf{i} = 1, 3, 5 $ (fuchsia) of fold bifurcations of homoclinic orbits to $ \Gamma_o $. Panelsk (b1) and (b2) show the continuation in $ f $ for $ \delta = -4.985 $ of the two transversal homoclinic orbits $ \gamma_3 $ and $ \gamma_4 $ from Fig. 24(d2), where the time $ T_{out} $ is the time that the homoclinic orbit spends outside a tubular neighborhood of $ \Gamma_o $. Panelsk (c1) and (c2) show the two respective homoclinic orbits with large $ T_{out} $ as they approach the bifurcation $ \bf{Hep} $
Figure 26.  Bifurcation diagrams in the $ (f, \delta) $-plane of system (1) showcasing the curves that emanate from the Bykov T-point $ \bf{T} $ in panel (a), and the ones that emanate from the flip bifurcation $ \bf{Fl} $ in panel (b1). Shilnikov bifurcations may be encountered only in the blue highlighted region in panel (a). The regions where $ W^u_+(p) $ converges to dominant chaotic behavior are highlighted in panel (b1) and the enlargement panel (b2), where coloring distinguishes different types of chaotic attractors
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