The existence of periodic solutions for the third-order differential equation ̇̈ x+ ω2ẋ = μF(x,ẋ, ẍ) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bif...

A transversally driven isotropic ferromagnet being under the influence of a static external and an uniaxial internal anisotropy field is studied. We consider the dissipative Landau-Lifshitz equation as the fundamental equation of motion and treat it in 1 + 1 dimensions. The stability of the spatially homogeneous magnetizations against inhomogeneous perturbations is analyzed. Subsequently the dy...

The Hamiltonian Hopf bifurcation is described by part of a swallowtail surface. In this short note it is shown that this swallowtail is actually a non-transversally unfolded cusp singularity, and that the swallowtail surface is obtained by folding a cusp along a fold line that is tangent to the cusp and moving along the cusp.

We present the results of a theoretical investigation of droplets bouncing on a vertically vibrating fluid bath. An integro-differential equation describing the horizontal motion of the drop is developed by approximating the drop as a continuous moving source of standing waves. Our model indicates that, as the forcing acceleration is increased, the bouncing state destabilizes into steady horizo...

We present an algorithm for computing the direction of pitchfork bifurcation for two-point boundary value problems. The formula is rather involved, but its computational evaluation is quite feasible. As an application, we obtain a multiplicity result.

Gutzwiller's trace formula for the semiclassical density of states in a chaotic system diverges near bifurcations of periodic orbits, where it must be replaced with uniform approximations. It is well known that, when applying these approximations, complex predecessors of orbits created in the bifurcation (" ghost orbits ") can produce pronounced signatures in the semiclassical spectra in the vi...

We consider a nonlinear model for time-periodic oscillations of a suspension bridge. Under some additional restrictive assumptions we describe our model by a standard bifurcation scheme which allows us to use global bifurcation theorems and make some new conclusions.

A class of parameter dependent differential equations with one delay is considered. A decomposition of the parameter space into domains where the corresponding characteristic equation has a constant number of zeros with positive real part is provided. The local stability analysis of the zero solution and the computation of all Hopf bifurcation points with respect to the delay is given.

We investigate Hopf bifurcations in a delayed Nicholson’s blowflies equation of neutral type, derived from the Gurtin–MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criteri...

A well-known model due to J.-M. Lasry and P.L. Lions that presents the evolution of prices in a market as the evolution of a free boundary in a diffusion equation is suggested to be modified in order to show instabilities for some values of the parameters. This loss of stability is associated to the appearance of new types of solutions, namely periodic solutions, due to a Hopf bifurcation and r...