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July  2022, 27(7): 4023-4075. doi: 10.3934/dcdsb.2021217

## 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 Published  July 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (7) : 4023-4075. doi: 10.3934/dcdsb.2021217
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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. 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Parkins, Superradiant switching, quantum hysteresis, and oscillations in a generalized dicke model, Phys. Rev. A, 102 (2020), 063702. [55] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Texts in Applied Mathematics, Springer-Verlag, New York, 2003. [56] S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics. Springer, Cham, 2014. doi: 10.1007/978-3-319-06343-0. [57] T. Xing, R. Barrio and A. Shilnikov, Symbolic quest into homoclinic chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 1440004.  doi: 10.1142/S0218127414400045. [58] W. Zhang, B. Krauskopf and V. Kirk, How to find a codimension-one heteroclinic cycle between two periodic orbits, Discrete Contin. Dyn. Syst., 32 (2012), 2825-2851.  doi: 10.3934/dcds.2012.32.2825.

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, Lyapunov–Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Differential Equations, 249 (2010), 305-348.  doi: 10.1016/j.jde.2010.04.007. [52] L. P. Shilnikov, A case of the existence of a denumerable set of periodic motions, Dokl. Akad. Nauk SSSR, 160 (1965), 558-561. [53] K. C. Stitely, A. Giraldo, B. Krauskopf and S. Parkins, Nonlinear semiclassical dynamics of the unbalanced, open dicke model, Phys. Rev. Research, 2 (2020), 033131.  doi: 10.1103/PhysRevResearch.2.033131. [54] K. C. Stitely, S. J. Masson, A. Giraldo, B. Krauskopf and S. 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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
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 $f \approx 2.5900$
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
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 $\bf{Fl}$
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)
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 $S$
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)
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)
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)
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)
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 $(f, \delta) = (3.2, -6.25)$; $(3.05, -6.1)$; $(2.93, -5.9)$
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
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
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
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
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)
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)
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 $\Gamma_o$
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
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)
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)
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 $\bf{H}$
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)
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)
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 $\bf{Hep}$
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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