A general parametrically and externally excited mechanical system is considered. The main attention is focused on the dynamical properties of local bifurcations as well as global bifurcations including homoclinic and heteroclinic bifurcations. In particular, degenerate bifurcations of codimension 3 are studied in detail. The original mechanical system is first transformed to averaged equations ...

In this work we study a wide class of symmetric control systems that has the Chua’s circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-thre...

In this article, we analyse bifurcations from stationary stable spots to travelling spots in a planar three-component FitzHugh–Nagumo system that was proposed previously as a phenomenological model of gas-discharge systems. By combining formal analyses, center-manifold reductions, and detailed numerical continuation studies, we show that, in the parameter regime under consideration, the station...

We investigate the behaviour of a neural network model consisting of three neurons with delayed self and nearest-neighbour connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use ...

The detection of vascular bifurcations in retinal fundus images is important for finding signs of various cardiovascular diseases. We propose a novel method to detect such bifurcations. Our method is implemented in trainable filters that mimic the properties of shape-selective neurons in area V4 of visual cortex. Such a filter is configured by combining given channels of a bank of Gabor filters...

Network architecture can lead to robust synchrony in coupled systems and, surprisingly, to codimension one bifurcations from synchronous equilibria at which the associated Jacobian is nilpotent. We prove three theorems concerning nilpotent Hopf bifurcations from synchronous equilibria to periodic solutions, where the critical eigenvalues have algebraic multiplicity two and geometric multiplicit...

Gutzwiller's trace formula allows interpretation of the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex ``ghost orbits''. By...

Control systems and other dynamic systems may lose stability at critical points from which bifurcations take place. However, due to the existence of small disturbances or imperfections, the dynamic behaviour of a system-in particular, the stability conditions-may change drastically. This paper is concerned with the imperfection sensitivity of dynamic bifurcations (degenerate or generalized Hopf...

We investigate classical and semiclassical aspects of codimension-two bifurcations of periodic orbits in Hamiltonian systems. A classification of these bifurcations in autonomous systems with two degrees of freedom or time-periodic systems with one degree of freedom is presented. We derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurc...