Quasinormal Forms Research Articles

Impact of quadratic non-linearity on the dynamics of periodic solutions of a wave equation

For the non-linear telegraph equation with homogeneous Dirichlet or Neumann conditions at the end-points of a finite interval the question of the existence and the stability of time-periodic solutions bifurcating from the zero equilibrium state is considered. The dynamics of these solutions under a change of the diffusion coefficient (that is, the coefficient of the second derivative with respect to the space variable) is investigated. For the Dirichlet boundary conditions it is shown that this dynamics substantially depends on the presence - or the absence - of quadratic terms in the non-linearity. More precisely, it is shown that a quadratic non-linearity results in the occurrence, under an unbounded decrease of diffusion, of an infinite sequence of bifurcations of each periodic solution. En route, the related issue of the limits of applicability of Yu.S. Kolesov's method of quasinormal forms to the construction of self-oscillations in singularly perturbed hyperbolic boundary value problems is studied.

Justification of the method of quasinormal forms for Hutchinson's equation with a small diffusion coefficient

Although we consider only the concrete problem indicated in the title (the proofs of general theorems would be bulky), our arguments can be adapted for a wide class of singularly perturbed systems of reaction-diffusion type with time-delay in the ordinary part.

Perturbation theory and the energy spectrum of normal Fermi systems

This paper is a study in a possible steady-state perturbation theory for arbitrary excited multiply degenerate states of a normal Fermi system in the statistical limit. An operator technique has been developed to transform the relevant Hamiltonian into an operator that is a function of occupation number operators only. The perturbation theory equations have been proved to be solvable in the space of quasinormal form operators and their formal solutions have been derived. It is shown that the operator series of perturbation theory can be transformed to a linked cluster expansion with the nonphysical powers of volume eliminated. Elimination of nonphysical terms is effected without use of the diagrams technique.

The attractor problem for nonlinear wave equations in plane domains

We consider a model example of a quasilinear wave equation in the unit square with zero boundary conditions and use the method of quasinormal forms to prove that there are quite a few dichotomic cycles and tori bifurcating from zero equilibrium. A conjecture concerning the attractor structure is presented.

Parametric oscillations of a singularly perturbed telegraph equation with a pendulum non-linearity

The solution of the problem in the title is reduced to an analysis of the question of the number of and stability of equilibrium states of the quasi-normal form of the boundary-value problem under consideration. A mechanism is revealed for the origin of its so-called simple equilibrium states. It is shown that as the coefficient of elasticity decreases, the number of such states increases, and that those of them with the most complex spatial structure are stable.

Bifurcation of spatially inhomogeneous auto-oscillations from a non-local homogeneous cycle in systems with small diffusion

The problem stated in the title is of interest both for theory and applications (e.g., in mathematical ecology). To solve this problem, the authors use the method of quasi-normal forms for constructing auto-oscillations in parabolic systems with small diffusion. Bibliography: 19 titles.

Asymptotics and stability of non-linear parametric oscillations of a singularly perturbed telegraph equation

The solution of the problem indicated in the title is based on the method of quasi-normal forms developed previously by the author for the construction of stationary regimes of parabolic systems with small diffusion. As in the parabolic case, it relies on the theory of exponential dichotomy of solutions and on an algorithm in the study of stability of linear differential equations whose coefficients are nearly independent of the time.

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionut=eDuxx+(A+eA1)u+F(u),u ∈ ℝ2. These equations (quasinormal forms) describe the behaviour of solutions of the original equation fore → 0.

Method of quasinormal forms in the problem of steady-state conditions for parabolic systems with small diffusion

Method of quasinormal forms in the problem of steady-state conditions for parabolic systems with small diffusion

Randwertaufgaben halbzahliger charakteristik für eltiptische systeme

This paper deals with boundary value problems of elliptic systems in the plane using W. Wendland's quite general quasinormal form. The usual periodicity condition for boundary data with respect to circulations around the boundary is weakened preserving local continuity of the boundary data and the Lopatinski condition which is satisfied everywhere on the boundary. This yields a half-integer for the characteristic (index) of the boundary condition instead of an integer in the classical case. Beyond that no other values of the characteristic are possible. Using a statement contained in R. Bott's periodicity theorem one can show that, assuming the general validity of the Lopatinski condition, two boundary conditions (with the same system dimension n) are continuously deformable into each other (i.e. homotopically equivalent) if and only if they have the same characteristic. W. Wendland's proof that the boundary value problem is semi-Fredholm remains valid for all boundary data which are periodic in the above...