The multiplicity problem of limit cycles arising from a weak focus is addressed. The proposed methodology is a combination of the frequency domain method to handle some degenerate Hopf bifurcations with the powerful tools of the singularity theory. The frequency domain approach uses the harmonic balance method to study the existence of periodic solutions. On the other hand, the singularity theory provides the conditions and formulas for the classification problem of the unfolding of the singularity in terms of the distinguished and auxiliary parameters. A classical example introduced by Bautin is shown in which the multiplicity of limit cycles is recovered by using this type of hybrid methodology and standard software in the continuation of periodic solutions (LOCBIF and XPPAUT). For small amplitude limit cycles, the proposed methodology gives accurate results.

Weed species present high competitive capacity, rapid adaptability and herbicide resistance, hindering their effective control across worldwide cropping regions. Since field-conducted experiments are very time-consuming and usually expensive, mathematical population-based models are valuable tools to test and develop long-term weed management programs. Within this context, the objective of this paper is to formalize analytically the possible seed bank dynamics of the Lolium rigidum, subjected to different control strategies. The first focus is on studying in detail the effects of integrating constant actions, promoting more environmentally and economically sustainable scenarios. From the same perspective, an alternative to applying time-variant programs is introduced. The proposed control guarantees that the weed population is sufficiently small or, alternatively, is kept below a given economic threshold level in a ten-year planning horizon. Furthermore, an optimization criterion is adopted for distributing necessary efficiency into diverse integrated options. Numerical simulations are included to illustrate the analytical findings.

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