1,721,017 research outputs found
Maslov index for Hamiltonian systems
No full textThe aim of this article is to give an explicit formula for computing the Maslov index of the fundamental solutions of linear autonomous Hamiltonian systems in terms of the Conley-Zehnder index and the map time one flow
A K-theoretical invariant and bifurcation for a parameterized family of functionals
No full textAbstractLet F:={fx:x∈X} be a family of functionals defined on a Hilbert manifold E˜ and smoothly parameterized by a compact connected orientable n-dimensional manifold X, and let σ:X→E˜ be a smooth section of critical points of F. The aim of this paper is to give a sufficient topological condition on the parameter space X which detects bifurcation of critical points for F from the trivial branch. Finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi-Riemannian manifolds
Morse theory for a fourth order elliptic equation with exponential nonlinearity
No full textGiven a Hilbert space H, an interval Λ⊂(0,+∞) and a map K∈C2(H,R) whose gradient is a compact mapping, the authors consider a family of functionals of the form
I(λ,u)=12⟨u,u⟩−λK(u),(λ,u)∈Λ×H.
Based on a recently proven deformation lemma, they show a Poincaré-Hopf-type theorem which they use, together with precise homological properties of the formal set of barycenters, in order to establish a direct and geometrically clear degree counting formula for the following fourth-order nonlinear scalar field equation on bounded, smooth domains of R4:
⎧⎩⎨Δ2u=τh(x)eu∫Ωh(x)eudxu=Δu=0in Ω,on ∂Ω,(1)
where h∈C2,α(Ω) is a positive function and τ>0. More specifically, they re-prove the following result proved earlier by other authors using blow-up estimates:
Theorem 2. For τ∈(64kπ2,64(k+1)π2) and k∈N, the Leray-Schauder degree dτ of (1) is given by
dτ=(k−χ(Ω)k),
where χ(Ω) denotes the Euler characteristic of the domain Ω.
In particular, if χ(Ω)⩽0 and τ≠64kπ2, then problem (1) has a solution
On the dihedral n-body problem.
No full textConsider 4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral
group. By adding a homogeneous potential (which recovers the gravitational Newtonian potential), one finds a special n-body problem with
three degrees of freedom, which is a kind of generalization of the Devaney
isosceles problem, in which all orbits have zero angular momentum. In the
paper we find all the central configurations and we compute the dimension of
the stable/unstable manifolds
A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation
No full textAbstractWe obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct.Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point
A mathematical model of flavescence dorée epidemiology
No full textFlavescence dorée (FD) is a disease of grapevine transmitted by an insect vector, Scaphoideus titanus Ball.
At present, no prophylaxis exists, so mandatory control procedures (e.g. removal of infected plants, and
insecticidal sprays to avoid transmission) are in place in Italy and other European countries. We propose
a model of the epidemiology of FD by taking into account the different aspects involved into the transmis-
sion process (acquisition of the disease, latency and expression of symptoms, recovery rate, removal and
replacement of infected plants, insecticidal treatments, and the effect of hotbeds). The model was con-
structed as a system of first order nonlinear ODEs in four compartment variables. A bifurcation analysis
shows that, in the absence of hotbeds, the state of healthy vineyard is stable, if removal and replacement
of infected plants is implemented. In the presence of hotbeds, depending on the grapevine density, we
find either a single family of equilibria in which the health of the vineyard gradually deteriorates for pro-
gressively more severe hotbeds, or multiple equilibria that give rise to sudden transitions from a nearly
healthy vineyard to a highly deteriorated one when the severity of the hotbeds crosses a critical value.
These results show the long-term risks in planting new vineyards in environmental situations where
strong hotbeds of FD are present or may arise in the surroundings
- …
