Origin of bursting $p\mathrm{H}$ oscillations in an enzyme model reaction system

The transition from simple periodic to bursting behavior in a three-dimensional model system of the hemin–hydrogen-peroxide–sulfite $p\mathrm{H}$ oscillator is investigated. A two-parameter continuation in the flow rate and the hemin decay rate is performed to identify the region of complex dynamics. The bursting oscillations emerge subsequent to a cascade of period-doubling bifurcations and the formation of a chaotic attractor in parameter space where they are found to be organized in periodic-chaotic progressions. This suggests that the bursting oscillations are not associated with phase-locked states on a two-torus. The bursting behavior is classified by a bifurcation analysis using the intrinsic slow-fast structure of the dynamics. In particular, we find a slowly varying quasispecies (i.e., a linear combination of two species) which acts as an “internal” or quasistatic bifurcation parameter for the remaining two-dimensional subsystem. A systematic two-parameter continuation in the internal parameter and one of the external bifurcation parameters reveals a transition in the bursting mechanism from sub-Hopf/fold-cycle to fold/sub-Hopf type. In addition, the slow-fast analysis provides an explanation for the origin of quasiperiodic behavior in the hemin system, even though the underlying mechanism might be of more general importance.

Figure 1

Two-parameter bifurcation diagram in the flow rate $k_0$ and the hemin decay rate $k_8$. Bursting oscillations are stable in region 3, which is bounded by the period-doubling curve (PD) and the saddle-node curve ${\mathrm{SN}}_1$. The dotted line at $k_8=2.5$ marks the parameter path along which the codimension-1 bifurcation diagram in Fig. 2 has been calculated. Symbols denote ${\mathrm{SH}}_i$, curves of subcritical Hopf bifurcations (dashed line); H, curve of supercritical Hopf bifurcations (solid line); ${\mathrm{SN}}_i$, curves of saddle-node bifurcations of fixed points (solid line); PD, curve of period-doubling bifurcations (dash-dotted line); codimension-2 points: ${\mathrm{GH}}_i$, generalized Hopf bifurcations (open triangle); CP, cusp (solid triangle; see also the inset).

Figure 2

One-parameter bifurcation diagram along the line $k_8=2.5$ (cf. Fig. 1). Simple periodic oscillations (solid circles) emerge via a subcritical Hopf bifurcation (inset, SH) followed by a saddle-node bifurcation of periodic orbits (inset, open triangle, SNP) and an (inverse) Neimark-Sacker bifurcation (inset, solid square, NS). Between SNP and SH, the only stable attractor is a fixed point, while a torus is stable between SH and NS. Mixed-mode oscillations are observed beyond the period-doubling (PD) bifurcations where the primary limit cycle (open circles) is unstable (see text for details). The oscillatory region extends up to the saddle-node bifurcation ${\mathrm{SN}}_1$, where a homoclinic bifurcation occurs (see also Fig. 6). For the oscillatory states, the minimum and maximum amplitudes of the oscillation are plotted.

Figure 3

Deformation of the two-torus as the flow rate $k_0$ is decreased from $k_0=1.65189$ (a),(c) to $k_0=1.6518$ (b),(d): (a),(b) represent projections onto the $p\text{−}y$ plane while (c),(d) depict the corresponding time series. Close to the Neimark-Sacker point NS in the inset of Fig. 2, the torus looks smooth (a). At a slightly decreased value of the flow rate, the “inner part” of the torus rapidly shrinks to a linelike manifold along which the trajectory approaches the stable manifold of the saddle point $S$ (b).

Figure 4

Bifurcation scenario beyond the period-doubling bifurcation PD in Fig. 2: period-2 (a) at $k_0=2.52$, period-4 (b) at $k_0=2.5252$, and two subsequent chaotic orbits (c),(d) at $k_0=2.529$ and $k_0=2.53$, respectively, are shown. The chaotic trajectory in (c) performs only large amplitude oscillations while that one in (d) makes irregular excursions to the neighborhood of the saddle point (open triangle). The associated Poincaré map of the chaotic orbit in (c) is shown in the inset. It exhibits the shape of an inverse tent map with a cuspoid tip (see text for details). The inset in (d) shows the same chaotic orbit as in (d) but in a $y\text{−}z$ projection where it becomes apparent that the chaotic attractor is contained in a thin layer in phase space.

Figure 5

The ${11}^{20}$ MMO at $k_0=2.545$ is shown in a $y\text{−}x$ projection (a) and in a $p\text{−}y$ projection (b) from which the unfolding of the mixed-mode state along the $p$ direction becomes apparent. The corresponding time series is presented in (c) while the dashed rectangular region in (c) is magnified in (d) showing the small amplitude oscillations.

Figure 6

Large relaxational oscillations at $k_0=3.8$ close to a homoclinic orbit (a). The corresponding trajectory in phase space is shown in (b). ${\mathrm{SN}}_1$ marks the location where a saddle-node bifurcation will occur at $k_0=3.858$ on the formerly periodic solution. This yields a saddle-node homoclinic bifurcation and causes the oscillations to cease.

Figure 7

Slow-fast analysis at a fixed value of the flow rate $k_0=2.8$: The bifurcation diagram of the fast subsystem (6) is shown in (a). The dotted rectangular region is magnified in (b) together with the trajectory (medium solid line) of a $4^{19}$ state at $k_8=2.5$. The orbit wraps 4 times around the cylinderlike quasistationary manifold that is composed of stable limit cycle solutions and subsequently performs 19 small amplitude oscillations along the linelike quasistationary manifold which is magnified in (c). In (d) a three-dimensional view of the $4^{19}$ state is shown together with a projection onto the $x\text{−}y$ plane. Solid and dashed bold lines denote maxima and minima of a stable and an unstable limit cycle while solid and dashed lines denote stable and unstable fixed points of the fast subsystem.

Figure 8

Overlay of one- and two-parameter bifurcation diagrams for the fast subsystem (6): Curves I, II, and III show how the branch of stationary states of the fast subsystem changes as the flow rate $k_0$ is increased from 2.8 via 3.6 to 3.8. The codimension-1 bifurcations occurring along curves I, II, and III are found at the intersection points of these curves with the two-parameter continuation curves ${\mathrm{SH}}_i$ and ${\mathrm{SN}}_i$. Symbols denote ${\mathrm{SH}}_i$, curves of subcritical Hopf bifurcations (dashed line); H, curve of supercritical Hopf bifurcations (solid line); ${\mathrm{SN}}_i$, curves of saddle-node bifurcations of fixed points (dotted line); codimension-2 points: ${\mathrm{GH}}_i$, generalized Hopf bifurcations (open triangle); BT, Bogdanov-Takens (diamond); CP, cusp (solid triangle).

Figure 9

Two-parameter bifurcation diagram of the fast subsystem (6) using the slow variable $p$ and the flow rate $k_0$ as parameters. If $p$ sweeps back and forth between region 1 and 2 crossing the ${\mathrm{SNP}}_1$ curve, the dynamics of the whole system exhibits bursting behavior. In the neighborhood of the intersection point 3 a transition in the bursting mechanism occurs (see text and Fig. 10 for details). Symbols denote ${\mathrm{SH}}_i$, curves of subcritical Hopf bifurcations (dashed line); H, curve of supercritical Hopf bifurcations (solid line); ${\mathrm{SN}}_i$, curves of saddle-node bifurcations of fixed points (dotted line); ${\mathrm{SNP}}_1$, curve of saddle-node bifurcations of periodic orbits (dash-dotted line); codimension-2 points: ${\mathrm{GH}}_i$, generalized Hopf bifurcations (open triangle); BT, Bogdanov-Takens (diamond); CP, cusp (solid triangle).

Figure 10

Transition in the bursting behavior from sub-Hopf/fold-cycle at $k_0=3.6$ (a),(b) to fold/sub-Hopf type at $k_0=3.8$ (c),(d). (a),(c) show codimension-1 bifurcation diagrams of the fast subsystem together with trajectories calculated for $k_8=2.5$ while (b),(d) depict the corresponding time series. As the flow rate $k_0$ increases, the invariant cylinderlike manifold (formed by stable limit cycles of the fast subsystem) is destroyed as the location of the saddle-node bifurcation ${\mathrm{SN}}_1$ approaches the unstable limit cycle that is created in the subcritical Hopf bifurcation ${\mathrm{SH}}_1$ (a). Henceforth, the fast subsystem is bistable (c) and only relaxational oscillations are observed (d). Symbols denote ${\mathrm{SH}}_1$, subcritical Hopf bifurcation; ${\mathrm{SNP}}_1$, saddle-node bifurcation of periodic orbit; ${\mathrm{SN}}_i$, saddle-node bifurcation of fixed points; SHC, saddle homoclinic orbit; SNHC, saddle-node homoclinic orbit. Solid and dashed bold lines denote maxima and minima of a stable and an unstable limit cycle while solid and dashed lines denote stable and unstable fixed points of the fast subsystem.

Figure 11

Slow-fast analysis of behavior in the neighborhood of the Neimark-Sacker bifurcation at $k_0=1.6519$ (cf. Fig. 3): Close to the Neimark-Sacker point at $k_0=1.65189$, the trajectory basically sweeps back and forth the saddle-node bifurcation point SNP of the fast subsystem (a). As the flow rate $k_0$ decreases to $k_0=1.6518$, the phase flow on the torus changes: The trajectory approaches the stable manifold of the saddle point $S$ (open triangle) in damped oscillations as the saddle-node bifurcation point SNP is passed to the left (b). Along the two-dimensional unstable manifold of the saddle point $S$, the trajectory reaches the cylinderlike manifold (bold solid lines) that is composed of stable limit cycle solutions of the fast subsystem. As the trajectory slowly moves to the left, it performs large amplitude oscillations in the vicinity of the cylinderlike manifold until it again passes the saddle-node bifurcation point SNP.