On a certain type of functional differential equations
On a codimension 3 bifurcation of plane vector fields with symmetry
On gradient at infinity of semialgebraic functions
Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number such that |x|·|∇f| and are separated at infinity. If c is a regular value and , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.
On numerical solution of a mildly nonlinear turning point problem
On stability and bifurcation of solutions of an SEIR epidemic model with vertical transmission.
On the critical periods of Liénard systems with cubic restoring forces.
On the direction of pitchfork bifurcation.
On the exact multiplicity of solutions for boundary-value problems via computing the direction of bifurcations.
On the existence of canard solutions
We study the existence of global canard surfaces for a wide class of real singular perturbation problems. These surfaces define families of solutions which remain near the slow curve as the singular parameter goes to zero.
On the existence of chaotic behaviour of diffeomorphisms
For several specific mappings we show their chaotic behaviour by detecting the existence of their transversal homoclinic points. Our approach has an analytical feature based on the method of Lyapunov-Schmidt.
On the number of zeros of Melnikov functions
We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order of the Melnikov function. The generic case was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.