Solitary death in coupled limit cycle oscillators with higher-order interactions

Letter

Solitary death in coupled limit cycle oscillators with higher-order interactions

Subhasanket Dutta, Umesh Kumar Verma, and Sarika Jalan

Phys. Rev. E 108, L062201 – Published 18 December 2023

Abstract

Coupled limit cycle oscillators with pairwise interactions are known to depict phase transitions from an oscillatory state to amplitude or oscillation death. This Research Letter introduces a scheme to incorporate higher-order interactions which cannot be decomposed into pairwise interactions and investigates the dynamical evolution of Stuart-Landau oscillators under the impression of such a coupling. We discover an oscillator death state through a first-order (explosive) phase transition in which a single, coupling-dependent stable death state away from the origin exists in isolation without being accompanied by any other stable state usually existing for pairwise couplings. We call such a state a solitary death state. Contrary to widespread subcritical Hopf bifurcation, here we report homoclinic bifurcation as an origin of the explosive death state. Moreover, this explosive transition to the death state is preceded by a surge in amplitude and followed by a revival of the oscillations. The analytical value of the critical coupling strength for the solitary death state agrees with the simulation results. Finally, we point out the resemblance of the results with different dynamical states associated with epileptic seizures.

Received 26 May 2023
Accepted 16 November 2023

DOI:https://doi.org/10.1103/PhysRevE.108.L062201

Physics Subject Headings (PhySH)

Bifurcations Coupled oscillators Dynamics of networks Synchronization transition

Collective dynamics

Nonlinear Dynamics

Authors & Affiliations

Subhasanket Dutta, Umesh Kumar Verma, and Sarika Jalan ^*

Complex Systems Lab, Department of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore-453552, India

^*sarika@iiti.ac.in

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Issue

Vol. 108, Iss. 6 — December 2023

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Images

Figure 1
(a) $A, R$ vs $ɛ$ and (b)–(d) time series of globally coupled identical SL oscillators [Eq. (1)]: a synchronized state with enhanced oscillation (EO; $ɛ = 2.3$ ) (b), solitary death (SD; $ɛ = 3.4$ ) (c), and a revival of oscillations (RO) state with a toroid ( $ɛ = 7.5$ ) (d). Red diamonds (circles) represent $A$ in the forward (backward) direction, and green diamonds (circles) represent $R$ in the forward (backward) direction. Other parameters are $N = 1000, ω = 4.0$ , and $x_{i} (0), y_{i} (0) \in [0, 1] \forall i$ .
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Figure 2
Coordinate space plot $y_{3}$ vs $x_{3}$ at various coupling strengths depicting the disappearance of the stable limit cycle via homoclinic bifurcation for SL oscillators [Eq. (1)]. (a) $ɛ = 2.0$ ; existence of the limit cycle. (b) $ɛ = 2.51$ ; birth of a pair of stable and saddle fixed points. (c) $ɛ = 3.14$ ; increase in the amplitude of the limit cycle and approaching the saddle point. (d) $ɛ = 3.18$ ; disappearance of the limit cycle through homoclinic bifurcation. Other parameters are $N = 3$ and $ω = 4.0$ . Solid red and black circles represent the stable fixed point and the saddle fixed point, respectively.
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Figure 3
Bifurcation diagram plotted using xppaut [27] for $ω = 4.0$ and $N = 3$ . The stable oscillatory state is depicted by solid green circles, while the unstable oscillatory state is depicted by open blue circles. A stable steady state is represented by the red solid curve, while an unstable steady state is represented by the black dashed curve.
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Figure 4
Phase diagram in the parameter space $(ɛ, ω)$ . The different states are as follows: EO, enhancement of oscillations; HA, hysteresis; SD, solitary death; RO, revival of oscillations. The blue dashed lines are obtained from analytical calculations [Eq. (2)]. The other parameters are $x_{i} (0), y_{i} (0) \in [0, 1] \forall i$ and $N = 1000$ .
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Figure 5
Basin of attraction for $N = 3$ and $ω = 4.0$ in Eq. (1). (a) Synchronized state at $ɛ = 1.0$ , (b) hysteresis at $ɛ = 2.7$ , (c) solitary death at $ɛ = 4.0$ , and (d) RO state at $ɛ = 7.0$ . The different states are as follows: EO, enhancement of oscillations; PL, phase locked; SD, solitary death; RO, revival of oscillations.
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Figure 6
(a) $A$ vs $ɛ$ for globally coupled identical SL oscillators having pairwise interactions as well [Eq. (4)] for $ɛ_{p} = 0.1, ɛ_{p} = 0.5$ , and $ɛ_{p} = 1.0$ ; $ω = 4.0$ and $N = 1000$ . (b) $A$ vs $ɛ$ for globally coupled nonidentical SL oscillators [Eq. (5)] for $ɛ_{p} = 0.0, ɛ_{p} = 1.0$ , and $ɛ_{p} = 2.0$ ; $ω \in [4, 5]$ and $N = 1000$ . Diamonds, $A$ in the forward direction; circles, $A$ in the backward direction.
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