On the appearance of horseshoe chaos in a nonlinear hysteretic systems with negative stiffness

The problem of inhibition of horseshoe chaos in a nonlinear hysteretic systems using negative stiffness is investigated in this paper. The Bouc–Wen model is used to describe the force produced by both the purely hysteretic and linear elastic springs. The analytical investigation of the Hamiltonian shows that the appearance of separatrix in the system is directly related to the parameters of the hysteretic forces. This means that the transverse intersection between the perturbed and unperturbed separatrix can be controlled according to the shape parameters of the hysteretic model.

Complex Surfaces and Coverings

This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally intersecting divisors. Applying this to linear arrangements in the complex projective plane, the chapter first blows up the projective plane at non-transverse intersection points, that is, at those points of the arrangement where more than two lines intersect. These points are called singular points of the arrangement. This gives rise to a complex surface and transversely intersecting divisors that contain the proper transforms of the original lines. The chapter also introduces the divisor class group, their intersection numbers, and the canonical divisor class. Finally, it describes the Chern numbers of a complex surface in order to define the proportionality deviation of a complex surface and to study its behavior with respect to finite covers.

Diffusion along transition chains of invariant tori and Aubry–Mather sets

We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry–Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry–Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

Quasi-steady linked vortices with chaotic streamlines

This paper describes the motion and the flow geometry of two or more linked ring vortices in an otherwise quiescent, ideal fluid. The vortices are thin tubes of near-circular shape which lie on the surface of an immaterial torus of small aspect ratio. Since the vortices are assumed to be identical and evenly distributed on any meridional section of the torus, the flow evolution depends only on the number of vortices ($n$) and the torus aspect ratio (${r}_{1} / {r}_{0} $, where ${r}_{0} $ is the centreline radius and ${r}_{1} $ is the cross-section radius). Numerical simulations based on the Biot–Savart law showed that a small number of vortices ($n= 2, 3$) coiled on a thin torus (${r}_{1} / {r}_{0} \leq 0. 16$) progressed along and rotated around the symmetry axis of the torus in an almost uniform manner while each vortex approximately preserved its shape. In the comoving frame the velocity field always possesses two stagnation points. The transverse intersection, along $2n$ streamlines, of the stream tube emanating from the front stagnation point and the stream tube ending at the rear stagnation point creates a three-dimensional chaotic tangle. It was found that the volume of the chaotic region increases with increasing torus aspect ratio and decreasing number of vortices.

SIMPLE VECTOR FIELDS WITH COMPLEX BEHAVIOR

We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R4 and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

Nearly real fronts in a Ginzburg–Landau equation

SynopsisSubcritical fronts are shown to exist in a quintic version of the well-known complex Ginzburg–Landau equation, which has a subcritical pitchfork as well as a supercritical saddle-node bifurcation. The fronts connect a finite amplitude plane wave state to a stable zero solution. The unstable manifold at finite amplitude and stable manifold of vanishing amplitude solutions are shown to intersect transversely on an invariant zero-wavenumber manifold with parameters set to be real. By the persistence of transverse intersection, frontal connections exist for a continuum of nearly real fronts parametrised by appropriate variables that exhibit some interesting changes in dimension.

On transversality

The notion of transversality has proved of immense value in differential topology. The Thom transversality lemma and its many variants show that transversality is a dense,and often open, property. In one parameter families the occurrence of non-transversality is inevitable; for example one cannot pull two linked curves in ℝ3 apart without a non transverse intersection. The aim of this note is to prove the following. In any generic family of mappings each map in the family fails to satisfy some fixed transversality conditions at worst at isolated points, and even at these points in rather special sorts of way. So, returning to the above example, given two space curves C1 and C2 without a (necessarily non-transverse) intersection we expect, in any genericisotopy of C2, that it will meet C1 if at all, at isolated points In particular generically we do not expect C1 and C2, any time, to have an arc in common