Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem

Figure 1.  (a)-(b) Time series and phase portrait of a $ 1^5 $ MMO pattern observed in system (5) for $ \beta_1 = 0.35 $, $ h = 1.65 $ and the other parameter values as in (6)

Figure 2.  (a) The critical manifold $ \mathcal{M} = {\Pi}\cup S $. The folded node is marked by a cyan dot. (b) Reduced dynamics of system (11) zoomed near the folded node singularity (cyan) projected on the manifold $ S $. The black curves are trajectories of system (11). Also shown are the ordinary singularity (black dot), weak eigendirection (green), the strong eigendirection (red) and the singular funnel. Here $ h = 0.8 $ and the other parameter values are as in (6)

Figure 3.  One-parameter bifurcation of the desingularized system (12) as $ h $ is varied and the other parameter values are as in (6). The $ y $-axis represents the maximum of $ x $. The black curves are ordinary singularities (namely, the coexistent equilibria and the boundary equilibria $ E_{xy} $ (horizontal line) and $ E_{xz} $) and the red curve represents the folded singularities. These curves intersect at FSN II points (green dots). The stable parts of a branch are shown by solid curves, while the unstable parts by the dashed curves. The folded singularities exist in the positive octant for $ h>0.5958 $

Figure 4.  A one-parameter bifurcation diagram in $ h $ for $ \beta_1 = 0.25 $ and other parameter values as in (6). The red (black) represent stable (unstable) branches of equilibria, where $ E^* $ denotes the coexistent equilibrium, $ E_{xy} $ and $ E_{xz} $ are the boundary equilibria on the $ xy $ and $ xz $ planes respectively. The green (blue) circles represent stable (unstable) limit cycles. HB: Hopf bifurcation, PD: period doubling, SNL: saddle node bifurcation of limit cycles. See the next two figures for a detailed view of the inset. The plot was generated in XPPAUT [14]

Figure 5.  A one-parameter bifurcation structure of MMOs corresponding to the inset in Figure 4. Here $ x_{\max} $ is the maximum value of $ x $

Figure 6.  Numerical continuation of periodic orbits of system (5) as $ h $ is varied. Here, $ T $ represents the period. Green (blue) circles represent stable (unstable) orbits. The bottom curve shows the evolution of the family of orbits $ \Gamma_h $, and the curves above are continuations of the successive sequence of families of period-doubled Hopf orbits (see text for details). $ \text{PD}_1 $: $ 1 $st period-doubled bifurcation, $ \text{PD}_2 $: $ 2 $nd period-doubled bifurcation, SNL: saddle-node bifurcation of limit cycles. All parameter values are as in Figure 4

Figure 7.  (a) Phase portrait of a chaotic MMO orbit. (b) Corresponding time series in $ x $. (c) The trajectory rotates around the weak eigendirection (red) while getting pushed towards the equilibrium (magenta) along the stable eigendirection (green) of the equilibrium. The folded node is marked by a cyan dot. Here $ \beta_1 = 0.25 $, $ h = 0.8 $ and the other parameter values are as in (6)

Figure 8.  (a)-(b) Phase space and time series of a MMO orbit of signature $ 1^{11} $ for $ h = 0.633 $, $ \beta_1 = 0.1977 $ and the other parameter values as in (6). The SAOs rotate around the weak eigendirection (in red) and are induced by the rotational properties of the weak canard (in blue). The folded node is marked by a cyan dot

Figure 9.  (a) A chaotic MMO orbit rotating around the weak eigendirection (red) after leaving the equilibrium (magenta) along its unstable eigendirection. The folded node is shown in cyan. (b) Corresponding time series in $ x $. Note the amplitudes and epochs of small oscillations between two large oscillations vary by orders of magnitude. Here $ \beta_1 = 0.1825 $, $ h = 0.2648 $ and the other parameter values are as in (7)

Figure 10.  Time series in $ x $ demonstrating long transient dynamics in the form of MMOs past supercritical Hopf bifurcation as $ h $ is varied. The initial condition chosen is $ (0.01, 0.01, 0.12) $. Note the duration of the transients varies with $ h $. Here $ \beta_1 = 0.25 $ and the other parameter values are as in (6)

Figure 11.  Transient basins of attraction for $ h = 0.785 $ restricted to the plane $ x = 0.3428 $ computed over the interval $ [0, N] $. Blue dots lie in the basin of short transients and the red dots in the basin of long transients. Here $ \beta_1 = 0.25 $ and the other parameter values are as in (6). (a) $ N = 2000 $. (b) $ N = 5000 $

Figure 12.  (a) A trajectory exhibiting MMOs while approaching $ \Gamma_{h} $ (blue). The trajectory tightly wraps around the weak eigendirection $ v_w $ (in red) and approaches $ E_1^* $ (marked as a magenta dot), inducing SAOs in the MMOs. The folded node singularity is marked as the cyan dot and the stable eigendirection of $ E_1^* $ in green. (b) Corresponding time series in $ x $. (c) A zoomed view of the SAOs between two LAOs. Note the amount of time spent near the folded node and the equilibrium. (d) Time series of $ \Gamma_{h} $, the periodic stable attractor. Here, $ \beta_1 = 0.25 $, $ h = 0.785 $ and the other parameter values are as in (6)

Figure 13.  The Poincaré section $ \Sigma_h $ drawn transverse to the manifold $ S $. Also, shown are the strong and weak eigendirections $ v_s $ and $ v_w $ respectively, along with folded node singularity in cyan. All the parameter values are as in Figure 11

Figure 14.  (a) $ (x, z) $ coordinates of intersection of a trajectory starting at $ (0.01, 0.01, 0.12) $ with the cross-section $ \Sigma_h: = 2.94x+ 1.3y -1.52z = 0.588 $ such that $ dx/ds<0 $. (b) A zoomed view of the region $ a_2 $ near the folded node

Figure 15.  (a)-(b) Phase portrait and time series of a trajectory starting out at $ (0.27795, 0.4252, 0.0598) $ exhibits relaxation oscillations as transient dynamics and eventually asymptotes to $ E_{xz} = (0.357, 0, 0.615) $ for $ \beta_1 = 0.1923 $, $ h = 0.2649 $ and other parameter values as in (7)

Figure 16.  A two-parameter bifurcation diagram in $ (h, \beta_1) $ space. The other parameter values are as in (6). FH: fold-Hopf bifurcation (marked by black dot), SN: saddle-node bifurcation (red curve), H: Hopf bifurcation (blue curve and the vertical line), FSN II bifurcation (purple curve), PD: period-doubling bifurcation (yellow curve), BR: branch curve (cyan). The equilibria lying to the right of BR are biologically feasible. The plot was generated in XPPAUT [14]

Figure 17.  (a)-(b) Phase space and time series of a trajectory for $ h = 0.615 $. The trajectory approaches $ E_1^* $ along $ W^s(E_1^*) $ and spirals out along $ W^u(E_1^*) $ as it eventually settles down to the periodic orbit (red), which emerges through a Hopf bifurcation. Here $ \beta_1 = 0.1977 $ and the other parameter values are as in (6)

Figure 18.  Bifurcation diagram in $ h $ with $ \beta_1 = 0.35 $ and other parameter values as in (6). Here $ x_{max} $ is the maximum value of $ x $

Figure 19.  A two-parameter bifurcation diagram in $ (h, \beta_1) $ space. The other parameter values are as in (7). FH: fold-Hopf bifurcation (zero-pair), SN: saddle-node bifurcation of equilibria (red curve), H: Hopf bifurcation (blue curve), TR: torus bifurcation (magenta curve), BR: branch curve (cyan curve), GH: generalized Hopf bifurcation. Region 1: a unique coexistent equilibrium exists, which is unstable, Region 2i: existence of two coexistent equilibria, both unstable, Region 2ii: two coexistent equilibria exist, one of which is locally asymptotically stable, Region 3: a unique coexistent equilibrium exists, which is locally asymptotically stable. The plot was generated in XPPAUT [14]

Figure 20.  (a)-(b) Transient temporal dynamics of system (5) asymptotically approaching to the equilibrium $ E_{xz} = (0.5225, 0, 0.536) $. Here $ \beta_1 = 0.3001 $, $ h = 0.4767 $ and the other parameter values are as in (7). The chosen values lie in region 2i in Figure 19

Figure 21.  Bistable dynamics exhibited by system (5) at $ \beta_1 = 0.1923 $ and $ h = 0.2648 $. The system either approaches a periodic orbit (red) born out of torus bifurcation in (a) or to the equilibrium $ E_{xz} = (0.357, 0, 0.615) $ in (b). The other parameter values are as in (7). The initial conditions in (a) are $ (0.2995, 0.1175, 0.4154) $ and $ (0.34279, 0.104, 0.346) $ in (b)