Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Bifurcations analysis of turbulent energy cascade

This note studies the mechanism of turbulent energy cascade through an opportune bifurcations analysis of the Navier--Stokes equations, and furnishes explanations on the more significant characteristics of the turbulence. A statistical bifurcations property of the Navier--Stokes equations in fully developed turbulence is proposed, and a spatial representation of the bifurcations is presented, which is based on a proper definition of the fixed points of the velocity field. The analysis first shows that the local deformation can be much more rapid than the fluid state variables, then explains the mechanism of energy cascade through the aforementioned property of the bifurcations, and gives reasonable argumentation of the fact that the bifurcations cascade can be expressed in terms of length scales. Furthermore, the study analyzes the characteristic length scales at the transition through global properties of the bifurcations, and estimates the order of magnitude of the critical Taylor--scale Reynolds number and the number of bifurcations at the onset of turbulence.Comment: 14 pages, 5 figures, available online Annals of Physics, 201

Imperfect Homoclinic Bifurcations

Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.Comment: 8 pages, 11 figures, submitted to PR

The level of occlusion of included bark affects the strength of bifurcations in hazel (Corylus avellana L.)

Bark-included junctions in trees are considered a defect as the bark weakens the union between the branches. To more accurately assess this weakening effect, 241 bifurcations from young specimens of hazel (Corylus avellana L.), of which 106 had bark inclusions, were harvested and subjected to rupture tests. Three-point bending of the smaller branches acted as a benchmark for the relative strength of the bifurcations. Bifurcations with included bark failed at higher displacements, and their modulus of rupture was 24% lower than normally formed bifurcations, while stepwise regression showed that the best predictors of strength in these bark-included bifurcations were the diameter ratio and width of the bark inclusion, which explained 16.6% and 8.1% of the variability, respectively. Cup-shaped, bark-included bifurcations where included bark was partially occluded by xylem were found, on average, to be 36% stronger than those, where included bark was situated at the bifurcation apex. These findings show that there are significant gradations in the strength of bark-included bifurcations in juvenile hazel trees that relate directly to the level of occlusion of the bark into the bifurcation. It therefore may be possible to assess the extent of the defect that a bark-included bifurcation represents in a tree by assessing the relative level of occlusion of the included bark

Uniform approximations for non-generic bifurcation scenatios including bifurcations of ghost orbits

Gutzwiller's trace formula allows interpreting 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 studying an example taken from the Diamagnetic Kepler Problem, viz. the period-quadrupling of the balloon-orbit, we demonstrate that these ghost orbits themselves can undergo bifurcations, giving rise to non-generic complicated bifurcation scenarios. We extend classical normal form theory so as to yield analytic descriptions of both bifurcations of real orbits and ghost orbit bifurcations. We then show how the normal form serves to obtain a uniform approximation taking the ghost orbit bifurcation into account. We find that the ghost bifurcation produces signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits does.Comment: 56 pages, 21 figure, LaTeX2e using amsmath, amssymb, epsfig, and rotating packages. To be published in Annals of Physic

Automated measurements of retinal bifurcations

This paper presents an analysis of the bifurcations of retinal vessels. The angles and relative diameters of blood vessels in 230 bifurcations were measured using a new automated procedure, and used to calculate the values of several features with known theoretical properties. The measurements are compared with predictions from theoretical models, and with manual measurements. The automated measurements agree with the theoretical prediction measurements with slightly different bias. The automated method can measure a large number of retinal bifurcations very rapidly, and may be useful in correlating bifurcation geometry with clinical conditions

Small aspect ratio Taylor-Couette flow: onset of a very-low-frequency three-torus state

The nonlinear dynamics of Taylor-Couette flow in a small aspect ratio annulus (where the length of the cylinders is half of the annular gap between them) is investigated by numerically solving the full three-dimensional Navier-Stokes equations. The system is invariant to arbitrary rotations about the annulus axis and to a reflection about the annulus half-height, so that the symmetry group is SO(2)×Z2. In this paper, we systematically investigate primary and subsequent bifurcations of the basic state, concentrating on a parameter regime where the basic state becomes unstable via Hopf bifurcations. We derive the four distinct cases for the symmetries of the bifurcated orbit, and numerically find two of these. In the parameter regime considered, we also locate the codimension-two double Hopf bifurcation where these two Hopf bifurcations coincide. Secondary Hopf bifurcations (Neimark-Sacker bifurcations), leading to modulated rotating waves, are subsequently found and a saddle-node-infinite-period bifurcation between a stable (node) and an unstable (saddle) modulated rotating wave is located, which gives rise to a very-low-frequency three-torus. This paper provides the computed example of such a state, along with a comprehensive bifurcation sequence leading to its onset.Postprint (published version