Spontaneous pattern formation and competition in a nonlinear ring cavity are studied. Complex patterns arising from static and Hopf bifurcations and Hopf–static interactions have been numerically observed and analysed. Among them are flower-like patterns, alternating rolls and oscillating hexagonal structures.

Abstract This paper proposes a virtual oscillator control method by reducing the order of amplitude term Andronov-Hopf dynamic process. The proposed retains its steady-state droop characteristic and power dispatchability in grid-connected mode, but with less nonlinearity. At same time, based on performance criterion inverter, detailed parameter selection principle controller is given, verified ...

A model of balancing upright on a unicycle has pitchfork and Hopf bifurcations, and describes riders from novice to expert with both good and not-so-good reactions.

In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark–Hopf (N–H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark–Hopf bifurcations. We shall see that the mechanism is not unique, and th...

We present experimental measurements near the onset of electroconvection (EC) of homeotropically aligned nematic liquid crystals Phase 5A and MBBA. A voltage of amplitude square root 2V0 and frequency f was applied. With increasing V0, EC occurred after the bend Freedericksz transition. We found supercritical bifurcations to EC that were either stationary bifurcations or Hopf bifurcations to tr...

In this paper, we analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at...

A reaction–diffusion–advection predator–prey model with Holling type-II predator functional response is considered. We show the stability/instability of positive steady state and existence a Hopf bifurcation when diffusion advection rates are large. Moreover, we that rate can affect not only occurrence bifurcations but also values bifurcations.

We consider iterated maps with a reflectional symmetry. Possible bifurcations in such systems include period-doubling bifurcations (within the symmetric subspace) and symmetry-breaking bifurcations. By using a second parameter, these bifurcations can be made to coincide at a mode interaction. By reformulating the period-doubling bifurcation as a symmetry-breaking bifurcation, two bifurcation eq...

Group theoretic means are employed to analyse the Hopf bifurcation on pattern forming systems with the periodicity of the face-centred (FCC) and bodycentred (BCC) cubic lattices. We find all C-axial subgroups of the normal form symmetry group by first extending the symmetry to a larger group. There are 15 such solutions for the FCC lattice, of which at least 12 can be stable for appropriate par...

We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons andmultiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to Wu and Bendixson’s criterion for high-dimensional ordinary differential equations due to Li and Muldowney. It is shown th...