1,203 research outputs found
Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems
We consider the nonlinear boundary value problem consisting of the equation
\tag{1} -u" = f(u) + h, \quad \text{a.e. on ,} where ,
together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm
1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where are
integers, ,
, and we suppose that We also suppose that is continuous, and We allow --- such a
nonlinearity is {\em jumping}.
Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad
\text{on ,} where , and for . The problem (2)-(3) is `positively-homogeneous'
and jumping. Regarding as fixed, values of for
which (2)-(3) has a non-trivial solution will be called {\em
half-eigenvalues}, while the corresponding solutions will be called {\em
half-eigenfunctions}.
We show that a sequence of half-eigenvalues exists, the corresponding
half-eigenfunctions having specified nodal properties, and we obtain certain
spectral and degree theoretic properties of the set of half-eigenvalues. These
properties lead to solvability and non-solvability results for the problem
(1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum'
of the problem, which we briefly describe. Equivalent solvability and
non-solvability results for (1)-(2) are obtained from either the
half-eigenvalue or the Fucik spectrum approach
Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples
Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time
First order systems of odes with nonlinear nonlocal boundary conditions
In this article, we prove an existence of solutions for a non-local boundary
value problem with nonlinearity in a nonlocal condition. Our method is based
upon the Mawhin's coincidence theory
Wave excitations of drifting two-dimensional electron gas under strong inelastic scattering
We have analyzed low-temperature behavior of two-dimensional electron gas in
polar heterostructures subjected to a high electric field. When the optical
phonon emission is the fastest relaxation process, we have found existence of
collective wave-like excitations of the electrons. These wave-like excitations
are periodic in time oscillations of the electrons in both real and momentum
spaces. The excitation spectra are of multi-branch character with considerable
spatial dispersion. There are one acoustic-type and a number of optical-type
branches of the spectra. Their small damping is caused by quasi-elastic
scattering of the electrons and formation of relevant space charge. Also there
exist waves with zero frequency and finite spatial periods - the standing
waves. The found excitations of the electron gas can be interpreted as
synchronous in time and real space manifestation of well-known
optical-phonon-transient-time-resonance. Estimates of parameters of the
excitations for two polar heterostructures, GaN/AlGaN and ZnO/MgZnO, have shown
that excitation frequencies are in THz-frequency range, while standing wave
periods are in sub-micrometer region.Comment: 26 pages and 6 figure
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