Numerical simulations by Gao, Wang and Lü (Chem. Phys. Lett. 665 (2016) 10) showed that a weak periodic perturbation, confined to a small disk of radius 15 near the tip of a rotating spiral wave in a two-variable excitable reaction–diffusion model (a reduction of the Decroly–Goldbeter glycolytic model), splits spiral arms at distances of hundreds of spatial units from the tip when the forcing frequency matches the spiral's rotation frequency ω0≈0.143\omega_0 \approx 0.143; the breakup distance RR is minimal at resonance, decreases monotonically with forcing amplitude FF, and the initial phase ϕ\phi selects which sector of the arm breaks first. This observation was established in a single model at a single parameter set, with no reduced theory, no test in other excitable systems, and unresolved long-time behavior. The open problem is to bring this resonance-induced instability under scientific control: determine whether it is a general mechanism of spiral-wave breakup in excitable media or an artifact of one kinetic scheme, construct a stability-theoretic or reduced account that predicts the resonant amplification and the measured R(F,ω)R(F,\omega) dependences, settle the long-time fate of the new wave sources and the role of boundaries, establish whether the effect occurs in real chemical or biological excitable media including cardiac tissue, and determine whether detuning or phase control can predictably prevent or induce breakup.

Background

Rotating spiral waves are self-sustained patterns of excitation that occur in spatially extended excitable media — systems whose local dynamics rest at a stable steady state but respond to superthreshold perturbations with a stereotyped excursion before returning to rest. Canonical realizations are the Belousov–Zhabotinsky (BZ) oscillating chemical reaction, glycolytic reaction systems, aggregates of the slime mold Dictyostelium, and cardiac muscle, where rotating electrical spirals underlie life-threatening arrhythmias. In a reaction–diffusion description, two fields (e.g. activator α\alpha and substrate β\beta) diffuse and react; the wave tip (the free end where excitation and recovery fronts meet) traces a circle or meanders around a compact core, and the whole wave rotates at an angular frequency ω0\omega_0. Ordered spiral waves can lose stability by several known routes: the Eckhaus instability, a long-wavelength modulational instability of phase waves (relevant in oscillatory rather than strictly excitable media); line instabilities driven by spatially extended periodic forcing along a line; meandering instabilities in which the tip trajectory quasiperiodically meanders; and Doppler instabilities, where a rapidly meandering tip emits compressed waves that break near the tip itself. A separate, well-developed literature concerns the response of spiral waves to weak external forcing: periodic modulation of an excitability parameter at or near the spiral's own frequency produces resonant drift — the spiral core translates steadily along a line or circle — studied experimentally and numerically in light-forced BZ media by Schrader, Braune and Engel (Phys. Rev. E 52, 98 (1995)), analyzed theoretically for periodic parametric forcing by Mantel and Barkley (Phys. Rev. E 54, 4791 (1996)) and by Aranson, Kessler and Mitkov (Physica D 85, 142 (1995)), and formalized in the response-function method, which computes drift velocities as overlap integrals of the forcing with adjoint eigenmodes of the spiral (Biktasheva, Barkley, Biktashev and Foulkes, Phys. Rev. E 81, 066202 (2010)). In all of these results the observable is motion of the tip; none addresses the amplification of outgoing wave crests. Gao, Wang and Lü (Chem. Phys. Lett. 665 (2016) 10) numerically studied the two-variable reduction of the Decroly–Goldbeter model for glycolytic oscillations (Decroly and Goldbeter, Proc. Natl. Acad. Sci. USA 79, 6917 (1982)), with kinetics ∂tα=q1σΦ−σΨ+Dα∇2α+χ\partial_t\alpha = q_1\sigma\Phi - \sigma\Psi + D_\alpha\nabla^2\alpha + \chi, ∂tβ=q2σΨ−ksβ+Dβ∇2β+χ\partial_t\beta = q_2\sigma\Psi - k_s\beta + D_\beta\nabla^2\beta + \chi, where Φ=Aα2/(L1+Bα2)\Phi = A\alpha^2/(L_1+B\alpha^2) and Ψ=αβ2/L2\Psi = \alpha\beta^2/L_2, on a no-flux disk of radius 1000, with the localized additive periodic forcing χ=Fcos⁡(ωt+ϕ)\chi = F\cos(\omega t+\phi) applied identically to both equations inside a disk of radius 15 near the spiral tip — a region much smaller than the spiral wavelength. They found that at F=0.01F = 0.01 and ω=ω0≈0.143\omega = \omega_0 \approx 0.143 the amplitudes of outgoing wave crests undergo growing oscillations that split the spiral arms transversely at distances of order R≈300R \approx 300 from the tip, creating new stable-looking wave sources (point defects), while the original tip persists; constant (DC) and single-pulse perturbations of the same region produce no breakup, the breakup distance RR is minimal at ω=ω0\omega = \omega_0 and grows with detuning ∣ω0−ω∣|\omega_0-\omega|, RR decreases monotonically with FF (with the original tip destroyed for F>0.02F > 0.02), and the initial phase ϕ\phi selects the angular sector that breaks first. The authors classify this as a resonance-driven oscillatory instability distinct from the Eckhaus, line, and Doppler mechanisms. The study was confined to one kinetic scheme and one parameter set; the peak-amplification evidence was qualitative; the possibility of frequency locking under forcing was not checked; and the long-time fate of the created wave sources — a non-turbulent set of stable new tips versus eventual defect-mediated turbulence — was left open.

Problem Statement

Establish and characterize, beyond the original single-model simulation, the resonance-induced instability of spiral waves produced by weak periodic forcing localized to a sub-wavelength region near the spiral tip. Concretely, for the excitable reaction–diffusion system studied by Gao, Wang and Lü (2016) — the two-variable Decroly–Goldbeter reduction with localized additive forcing χ=Fcos⁡(ωt+ϕ)\chi = F\cos(\omega t+\phi) for r≤15r \le 15 near the tip, with the kinetic parameter values, the placement of the forcing disk relative to the tip, and the measurement convention for the breakup distance RR as specified in the source paper — and for other excitable models and parameter regimes: (1) determine whether the resonant crest amplification and distant transverse arm breakup at forcing frequency ω≈ω0\omega \approx \omega_0 is a general mechanism of spiral-wave breakup in excitable media, and whether the measured dependence of the breakup distance RR on forcing amplitude FF and frequency detuning ∣Δω∣=∣ω0−ω∣|\Delta\omega| = |\omega_0-\omega| is quantitatively reproduced and the initial phase ϕ\phi continues to select the angular sector of the arm that breaks first, including the fate of the instability in the presence of frequency locking between the forcing and the spiral; (2) construct a reduced theoretical description or linear/nonlinear stability analysis that predicts the resonance condition and the amplification of outgoing wave-crest oscillations, and accounts for the observed R(F)R(F) and R(∣Δω∣)R(|\Delta\omega|) behavior; (3) determine how domain geometry, boundary conditions, and finite-size effects modify the breakup thresholds, and settle the long-time fate of the created point defects — stable new spiral tips versus eventual defect-mediated turbulence; (4) establish whether the same instability can be realized experimentally in chemical or biological excitable media (including cardiac tissue) and identify the parameter regimes required; and (5) determine whether detuning, phase selection, or local amplitude modulation can predictably prevent or induce breakup. Questions (1)–(3) characterize whether localized resonant forcing is a genuine and general route to spiral-wave breakup; questions (4)–(5), concerning experimental realization and control, build on this characterization.

References

  1. Jian Gao, Qun Wang, Huaping Lü, 'A new type of instability of spiral waves induced by resonance effects', Chemical Physics Letters 665 (2016) 10–13. DOI: 10.1016/j.cplett.2016.10.049, https://doi.org/10.1016/j.cplett.2016.10.049 (the source paper)
  2. A. Schrader, M. Braune, H. Engel, 'Dynamics of spiral waves in excitable media subjected to external periodic forcing', Physical Review E 52 (1995) 98. DOI: 10.1103/PhysRevE.52.98, https://doi.org/10.1103/PhysRevE.52.98
  3. Rolf-Martin Mantel, Dwight Barkley, 'Periodic forcing of spiral waves in excitable media', Physical Review E 54 (1996) 4791. DOI: 10.1103/PhysRevE.54.4791, https://doi.org/10.1103/PhysRevE.54.4791
  4. Igor Aranson, David Kessler, Igor Mitkov, 'Drift of spiral waves in excitable media', Physica D 85 (1995) 142–155. DOI: 10.1016/0167-2789(95)00113-I, https://doi.org/10.1016/0167-2789(95)00113-i
  5. I. V. Biktasheva, D. Barkley, V. N. Biktashev, A. J. Foulkes, 'Computation of the drift velocity of spiral waves using response functions', Physical Review E 81 (2010) 066202. DOI: 10.1103/PhysRevE.81.066202, https://doi.org/10.1103/PhysRevE.81.066202
  6. O. Decroly, A. Goldbeter, 'Birhythmicity, chaos, and other patterns of temporal self-organization in a multiply regulated biochemical system', Proceedings of the National Academy of Sciences USA 79 (1982) 6917. DOI: 10.1073/pnas.79.22.6917, https://doi.org/10.1073/pnas.79.22.6917
  7. Jian Gao, Changgui Gu, Huijie Yang, Tongfeng Weng, 'Excited state of spiral waves in oscillatory reaction-diffusion systems caused by a pulse', Physical Review E 101 (2020) 042219. DOI: 10.1103/PhysRevE.101.042219, https://doi.org/10.1103/PhysRevE.101.042219
  8. Jian Gao, Qun Wang, Huaping Lü, 'Super-spiral structures of bi-stable spiral waves and a new instability of spiral waves', Chemical Physics Letters 685 (2017) 205–209. DOI: 10.1016/j.cplett.2017.07.061, https://doi.org/10.1016/j.cplett.2017.07.061 (same authors, a different instability)
  9. J. Schlesner, V. S. Zykov, H. Brandtstädter, I. Gerdes, H. Engel, 'Efficient control of spiral wave location in an excitable medium with localized heterogeneities', New Journal of Physics 10 (2008) 015003. DOI: 10.1088/1367-2630/10/1/015003, https://doi.org/10.1088/1367-2630/10/1/015003
  10. Yu Deng, Bao Yu Liu, Tong Wu, Yan Yan Shangguan, Jun Ma, Jun Tang, 'Parametric wave induces straight drift of spiral waves in excitable medium', EPL (Europhysics Letters) 119 (2017) 58002. DOI: 10.1209/0295-5075/119/58002, https://doi.org/10.1209/0295-5075/119/58002
  11. Jinming Luo, Teng-Chao Li, Hong Zhang, 'Resonant drift of synchronized spiral waves in excitable media', Physical Review E 101 (2020) 032205. DOI: 10.1103/PhysRevE.101.032205, https://doi.org/10.1103/PhysRevE.101.032205
  12. Debei Pan, Mingyu Lu, 'Resonance-induced drift and mutual annihilation of counter-rotating spiral waves in excitable media', Physics Letters A 593 (2026) 132051. DOI: 10.1016/j.physleta.2026.132051, https://doi.org/10.1016/j.physleta.2026.132051
  13. Wu Ning-Jie, Li Bing-Wei, Ying He-Ping, 'Effects of Periodic Forcing Amplitude on the Spiral Wave Resonance Drift', Chinese Physics Letters 23 (2006) 2030–2033. DOI: 10.1088/0256-307X/23/8/019, https://doi.org/10.1088/0256-307x/23/8/019
  14. E. Boccia, S. Luther, U. Parlitz, 'Modelling far field pacing for terminating spiral waves pinned to ischaemic heterogeneities in cardiac tissue', Philosophical Transactions of the Royal Society A 375 (2017) 20160289. DOI: 10.1098/rsta.2016.0289, https://doi.org/10.1098/rsta.2016.0289