We construct a global bifurcation diagram of the plane differential system ẋ = x(1− x)− xy/(a+ x), ẏ = y(δ − βy/x), x(t) > 0, y(t) > 0, a > 0, δ > 0, β > 0, which describes the predator-prey interaction. In [1] was considered the following planar differential systems which describes a mathematical predator-prey model ẋ = rx(1 − x K )− mx b+x y, ẏ = sy(1− y hx ). (1) Here x(t) > 0, y(t) > 0 — th...

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle-node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obt...

This paper illustrates transcritical and Hopf bifurcations in a realistic ac/dc power system model. These bifurcations are thoroughly analyzed based on bifurcation theory to understand and characterize their e ect on the dynamic behavior of the system, particularly on voltage stability. A new technique to trace transcritical bifurcations diagrams is proposed, based on the generic characteristic...

Macroscopic systems with dissipation and time-modulated injection of energy, parametrically driven systems, can self-organize into localized states and/or patterns. We investigate a pulse that travels over a one-dimensional pattern in parametrically driven systems. Based on a minimal prototype model, we show that the pulses emerge through a subcritical Andronov-Hopf bifurcation of the underlyin...

Consider a polynomial Liénard system depending on three parameters a, b, c and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii). If a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the equilibrium. We investigate analytically this bifurcation in dependence on the parameters b and c and establish the existenc...

Bosch Vivancos, Chossat and Melbourne showed that two types of steadystate bifurcations are possible from trivial states when Euclidean equivariant systems are restricted to a planar lattice—scalar and pseudoscalar—and began the study of pseudoscalar bifurcations. The scalar bifurcations have been well studied since they appear in planar reaction–diffusion systems and in plane layer convection ...

We recall three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifurcations, and analyze these bifurcations in presence of certain degeneracy conditions, when the above mentioned theorems are not applied. The occurrence of such degenerate bifurcations is particularly important in pi...

Using the singularity theory of scalar functions, we derive a classification of sliding bifurcations in piecewise-smooth flows. These are global bifurcations which occur when distinguished orbits become tangent to surfaces of discontinuity, called switching manifolds. The key idea of the paper is to attribute sliding bifurcations to singularities in the manifold’s projection along the flow, nam...