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In our previous work [SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 626{671], we studied an interconnected bursting neuron model for insect locomotion, and its corresponding phase oscillator model, which at high speed can generate stable tripod gaits with three legs off the ground simultaneously in swing and at low speed can generate stable tetrapod gaits with two legs off the ground simultaneously in swing. However, at low speed several other stable locomotion patterns that are not typically observed as insect gaits may coexist. In the present paper, by adding heterogeneous external input to each oscillator, we modify the bursting neuron model so that its corresponding phase oscillator system produces only one stable gait at each speed, specifically a unique stable tetrapod gait at low speed, a unique stable tripod gait at high speed, and a unique branch of stable transition gaits connecting them. This suggests that control signals originating in the brain and central nervous system can modify gait patterns. [ABSTRACT FROM AUTHOR]

Litvak-Hinenzon et al. developed the phenomenological model to simulate the horizontal motion of particles in the atmosphere, and a series of their papers (see e.g., Litvak-Hinenzon et al. (Phys. D 164:213-250, 2002; Nonlinearity 15:1149-1177, 2002; SIAM J. Appl. Dyn. Syst. 3:525-573, 2004)) focused on studying the fate of parabolic resonance lower dimensional tori (as part of a quasi-periodic Hamiltonian pitchfork bifurcation (HPB) scenario in the unperturbed phenomenological model) under perturbations. However, the structural stability of quasi-periodic HPB involved therein has not been fully exposed theoretically (Litvak-Hinenzon et al. have only given some numerical explanations). Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), we consider a more general quasi-periodic HPB triggered by the Z 2 -invariant universal unfolding N b c k v = a y 2 2 − (λ + b I 1) x 2 2 + c x 4 4 with respect to Z 2 -equivariant BCKV-restricted morphisms of the planar singularity a 2 y 2 + c 4 x 4 (the coefficients a , b , c ≠ 0 , the I 1 is regarded as distinguished parameter with respect to the external parameter λ). We prove a KAM (Kolmogorov–Arnold–Moser) theorem concerning parabolic tori in such quasi-periodic HPB, by which Diophantine parabolic tori (and the whole corresponding Diophantine HPB scenario) survive non-integrable and Z 2 -invariant Hamiltonian perturbations, parametrized by pertinent large Cantor sets. In the context of Z 2 -symmetry, our results can be seen as rigorous proof of the structural stability problem of bifurcations of Floquet-tori triggered by the universal unfolding N b c k v with distinguished parameters which is proposed by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993). Ultimately, we similarly obtain the structural stability result of quasi-periodic HPB in the phenomenological model mentioned above, which can be utilized as a starting point for a deeper understanding of the various resonances and chaotic dynamics (in the gaps of the Cantor sets), just as Litvak-Hinenzon et al. did for normally parabolic tori undergoing a HPB. • We prove a KAM theorem in a quasi-periodic Hamiltonian pitchfork bifurcation. • We obtain the structural stability of a bifurcation in a phenomenological model. • We verify some of the previous numerical phenomena of Litvak-Hinenzon et al. [ABSTRACT FROM AUTHOR]