We study small periodic solutions of a class of damped second-order boundary-value problems with local odd symmetry near a one-dimensional loss of invertibility in the constant Fourier mode. A mean–zero-mean decomposition gives a uniquely solvable range equation and an exact scalar Lyapunov–Schmidt equation. The reduced equation yields the forcing-dependent displacement of the critical curve, a supercritical pitchfork, and its cusp unfolding under a constant unfolding parameter. Its derivative is shown to be the exact scalar Schur complement of the full periodic linearization. A parallel reduction of the Floquet problem identifies the same derivative as the quantity governing the multiplier that crosses one. This common scalar structure gives a complete local classification of the number, bifurcation type, and stability of all sufficiently small periodic solutions.
This paper investigates Hopf bifurcations and the existence of limit cycles in the three-dimensional Atkinson-Allen competitive system. By employing center manifold theory and the successive function method, we first derive explicit analytical expressions for the first and second focal values of general three-dimensional autonomous systems, providing rigorous alternatives to numerical simulations. Applying these formulas to the Atkinson-Allen system, we prove that class 27 in the Zeeman classification exhibits a first-order fine focus, leading exclusively to a supercritical Hopf bifurcation and a unique stable limit cycle. This distinguishes the model from Lotka-Volterra and Ricker systems, where class 27 can admit more complex bifurcation structures. Furthermore, we resolve a pending problem regarding class 28 by constructing critical bifurcation conditions to prove the emergence of limit cycles through boundary perturbations. Finally, by calculating the second focal value, we identify a second-order fine focus in class 26 and rigorously establish the coexistence of two limit cycles. These results provide a complete bifurcation analysis for the Atkinson-Allen system and offer new insights into the periodic dynamics of competitive ecosystems.
In this paper, we establish the existence of positive radial solutions for an indefinite singular Neumann problem involving the mean curvature operator in Minkowski space via the Leray–Schauder degree theory. We present an example to illustrate our theory.
We analyze a class of Gause-type predator–prey models with logistic prey growth and a generalized non-monotonic functional response of the form h(x)=q x^m/x^n+a, n>m≥ 1. The model incorporates antipredator group-defense behavior and is shown to be orbitally topologically equivalent to a polynomial Kolmogorov system. We characterize the existence and stability of equilibria and derive explicit trace–determinant formulas at coexistence states. These formulas yield a parametrization of the genuine Hopf crossing curve, distinguish it from the neutral-saddle locus, and identify both the generic Bogdanov–Takens points and the codimension-three degenerate cusp cases determined by the exponents (m, n). Homoclinic phenomena are formulated through a conditional separatrix-splitting criterion, and numerical simulations for a representative case are used only as evidence for global connections and large-amplitude oscillations. Stability claims for bifurcating cycles are tied to the corresponding Lyapunov or saddle quantities.
This paper is concerned with bifurcations of mixed-mode oscillations (MMOs) in the FitzHugh–Rinzel (FHR) equation. Our attentions are given to reveal the underlying mechanisms responsible for the births of MMOs and relaxation oscillations (ROs) via geometric singular perturbation theory and canard theory. It is shown that the FHR model can exhibit MMOs with single small amplitude oscillations (SAOs) and double SAOs, and MMOs can transition to ROs under the suitable relative geometries as the control parameters varying. MMOs in this model can be generated via canard mechanisms associated with folded nodes and folded saddle-node (FSN) II singularities, Shilnikov-type singular Hopf bifurcation and delayed-Hopf bifurcation (DHB) respectively. Numerical simulations verify all the theoretical predictions.
