Bifurcation Results for a Class of Perturbed Fredholm Maps

Abstract We prove a global bifurcation result for an equation of the type , where is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset of , are and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions with , whose closure contains a trivial solution . The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called -Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi

A degree theory for a class of perturbed Fredholm maps

We define a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between infinite-dimensional real Banach spaces. Our notion extends the degree introduced by Nussbaum for locally -contractive perturbations of the identity, as well as the recent degree for locally compact perturbations of Fredholm maps of index zero defined by the first and third authors

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.Comment: arXiv admin note: text overlap with arXiv:2006.1553

On the existence of forced oscillations of retarded functional motion equations on a class of topologically nontrivial manifolds

Using a topological approach, based on the fixed point index theory for locally compact maps on metric ANRs, we prove the existence of forced oscillations for retarded functional motion problems constrained on compact manifolds with nontrivial Euler–Poincar´e characteristic, provided that the frictional coefficient is nonzero. We do not know if an analogous result holds true in the frictionless case

A degree associated to linear eigenvalue problems in Hilbert spaces and applications to nonlinear spectral theory

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated in a recent article. Our result (the ex conjecture) is applied to prove a Rabinowitz type global continuation property of the solutions to a perturbed motion equation containing an air resistance frictional force

Global branches of periodic solutions for forced delay differential equations on compact manifolds

AbstractWe prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ⩾0. The approach is based on the fixed point index theory for maps on ANRs

Global continuation of periodic solutions for retarded functional differential equations on manifolds

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Sensorless Control with Switching Frequency Square Wave Voltage Injection for SPMSM with Low Rotor Magnetic Anisotropy

High-frequency signal injection sensorless algorithms are widely studied and used for rotor angle estimation in PMSM at low speed or standstill. One of the main drawbacks of such methods is the acoustic noise connected to the voltage injection. In order to minimize this problem, it is advisable to increase the frequency of the injected signal. Thus, many studies focus on square-wave injection at the switching frequency, which is the maximum theoretical frequency. Since these methods exploit the rotor magnetic anisotropy, it is relatively easy to use them in interior PMSMs, where the rotor anisotropy is high. On the contrary, it is hard to exploit them in surface PMSMs, which have an almost symmetric rotor, although a low rotor magnetic anisotropy is still present. In this paper, a sensorless algorithm with switching frequency squarewave injection is developed for surface PMSMs. To increase the signal-to-noise ratio, current oversampling is exploited. The benefits of such a technique are demonstrated with experimental results on a 2 Nm SPMSM