Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

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

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} $

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 $

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)

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)

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)

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) $

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

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

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

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 $

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

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)

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} $

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)

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)

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