Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays

We study two identical FitzHugh-Nagumo oscillators which are coupled with one or two different time delays. If only a single-delay coupling is used, the length of the delay determines whether the synchronization manifold is transversally stable or unstable, exhibiting mixed-mode or chaotic oscillations in which the small amplitude oscillations are always in phase but the large amplitude oscillations are in phase or out of phase, respectively. For two delays we find an intricate dynamics which comprises an irregular alteration of small amplitude oscillations, in-phase and out-of-phase large amplitude oscillations, also called extreme events. This transient chaotic dynamics is sandwiched between a bubbling transition and a blowout bifurcation.

Figure 1

Schematic representation of couplings between the two FHN units.

Figure 2

Bifurcation diagram showing the position of the fixed points for varying coupling strengths $M$. The solid circles represent stable fixed points and the small dots represent their unstable counterparts. In addition to the unstable origin, the plot shows the creation of two pairs of unstable fixed points via a fold bifurcation $\mathbf{F}$ at $M\approx 0.0058$. One of these pairs stabilizes due to a Hopf bifurcation $\mathbf{H}$ at $M\approx 0.0105$ (denoted by $\mathbf{H}$). The region in gray denotes the default parameter range chosen for our analysis. For this plot the FHN units using are connected by a single-delay with $\tau =80$.

Figure 3

Various representations of the synchronized long-term dynamics of two FHN units coupled with a single delay when the synchronization manifold is stable. Top-left panel: Time evolution of the $x$ coordinates of the oscillators. Top-right panel: Trajectories of the two oscillators in phase space. Bottom panel: Trajectory in a three-dimensional representation. The plane is the synchronization manifold. Color code: Small amplitude oscillations in black; synchronous large amplitude oscillations in cyan (light gray). Parameters: $M=0.01$, $\tau =80$.

Figure 4

Bifurcation diagram illustrating the period-adding cascade observed with increasing coupling strength $M$ and fixed time delay $\tau =80$. On the vertical axis the maxima corresponding to the oscillations are plotted. The upper panel shows the complete diagram, while the lower panel shows a closeup view of the regions corresponding to small amplitude oscillations. The gray regions in the lower panel correspond to the chaotic dynamics which intersperse the mixed-mode oscillations.

Figure 5

Various representations of the long-term dynamics of two FHN units coupled with a single delay when the synchronization manifold is unstable. Top-left panel: Time evolution of the $x$ coordinates of the oscillators. Top-right panel: Trajectories of the two oscillators in phase space. Bottom panel: Trajectory in a three-dimensional representation. The plane is the synchronization manifold. Color code: Small amplitude oscillations in black; asynchronous large amplitude oscillations in magenta (dark gray). Parameters: $M=0.01$, $\tau =70$.

Figure 6

Closeup view of typical time series showing in-phase (left panel) and out-of-phase (right panel) events. Color code: Small amplitude oscillations in black; synchronous large amplitude oscillations in cyan (light gray); asynchronous large amplitude oscillations in magenta (dark gray). Parameters: $M=0.01$, $\tau =80$ for the left panel; $M=0.01$, $\tau =70$ for the right panel.

Figure 7

Bifurcation diagram outlining the changes in the structure of the invariant sets on the synchronization manifold with varying coupling strength $M_2$. The vertical axis corresponds to the maxima of the oscillations. Upper panel: Complete diagram. Lower panel: Closeup view of the region corresponding to small amplitude oscillations. Fixed parameters: ${\tau }_1=80$, $M_1=0.005$, $\tau =70$.

Figure 8

Various representations of the long-term dynamics of two FHN units coupled with two delays. Top-left panel: Time evolution of the $x$ coordinates of the oscillators. Top-right panel: Trajectories of the two oscillators in the phase space. Bottom panel: Trajectory in a three-dimensional representation. The plane is the synchronization manifold. Color code: Small amplitude oscillations in black; synchronous large amplitude oscillations in cyan (light gray); asynchronous large amplitude oscillations in magenta (dark gray). Parameters: $M_1=0.005$, ${\tau }_1=80$; $M_2=0.0053$, ${\tau }_2=70$.

Figure 9

Loss of phase synchrony much before each “out-of-phase” extreme event. Top panel: Typical time series. Middle panel: Repeated approach to the synchronization manifold. Bottom panel: Sine of differences in phases of the oscillators. Color code: Small amplitude oscillations in black; synchronous large amplitude oscillations in cyan (light gray); asynchronous large amplitude oscillations in magenta (dark gray). Parameters: $M_1=0.005$, $M_2=0.0053$, ${\tau }_1=80$, and ${\tau }_2=70$.

Figure 10

Sketch representing the mechanisms of transition between the three major dynamical regimes obtained by varying $M_2$. For all three panels in the diagram, the synchronization manifold $\mathbf{S}$ of the system is represented by the dotted horizontal line. The transversally unstable limit cycle responsible for the ejection of trajectories away from the synchronization manifold is represented by a solid red (dark gray) dot on the synchronization manifold. All other nontrivial invariant subsets on the manifold $\mathbf{S}$ are represented by a thick black line which is solid (dashed) if the invariant subset is stable (unstable). Note that the periodic orbits which are transversally unstable are in fact embedded in the chaotic attractor but shown outside it for better visibility. Chaotic attractors and chaotic saddles which are not on the synchronization manifold are represented by the gray region. Attracting and repelling directions are represented by straight colored arrows and the curved or wiggly black arrows represent the trajectories in phase space.

Figure 11

Top panel shows part of a typical time series of two FHN units coupled using two delays exhibiting extreme events. Bottom panel shows the histogram of the interevent intervals $t_{\text{IEI}}$ for the system. The observation time for the entire simulation is $2\times {10}^9$ time units in which a total of 194 924 events were observed. The dashed line is a multiple of $exp\left(-r{t}_{\text{IEI}}\right)$ with $r=1.15\times {10}^{-4}$. Parameters: ${\tau }_1=80$, $M_1=0.01$, ${\tau }_2=65$, and $M_2=0.00255$.

Figure 12

Variance of interevent intervals for trajectories starting on the synchronization manifold for varying parameters ${\tau }_2$ and $M_2$. The white region corresponds to small amplitude oscillations and the brownish (dark gray) region corresponds to the formation of extreme events with very large interevent intervals.