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Equivalence of MTS and CMR methods on the normal form of Turing–Hopf bifurcation in delayed reaction–diffusion equations

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  • Ding, Yuting
  • Yu, Pei

Abstract

In the study of dynamics and bifurcations of delay differential systems, computing the normal form associated with a singular point is both unavoidable and often technically challenging. Developing efficient computational methods or algorithms for such systems is therefore of great importance for the analysis of real-world problems. The multiple time scales (MTS) method is frequently applied in physics and engineering, but it lacks mathematical rigorousness. In contrast, the center manifold reduction (CMR) method is founded on rigorous mathematical theory, yet it is difficult to apply to practical problems. In this paper, we establish the third-order equivalence of the MTS and CMR methods in computing the normal form of Turing–Hopf bifurcation in general delayed reaction–diffusion equations with advection-diffusion and nonlocal effects. This result provides a rigorous mathematical foundation for applying the simpler MTS method to bifurcation problems in delayed reaction–diffusion systems. An illustrative example is presented, comparing the normal forms obtained by the two approaches and verifying the theoretical results.

Suggested Citation

  • Ding, Yuting & Yu, Pei, 2026. "Equivalence of MTS and CMR methods on the normal form of Turing–Hopf bifurcation in delayed reaction–diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 202(P1).
    • Handle: RePEc:eee:chsofr:v:202:y:2026:i:p1:s0960077925014985
    DOI: 10.1016/j.chaos.2025.117485
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    1. Merchant, Sandra M. & Nagata, Wayne, 2011. "Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition," Theoretical Population Biology, Elsevier, vol. 80(4), pages 289-297.
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