### Application of the Lyapunov-Schmidt method to the problem of the branching of a cycle from a family of equilibria in a system with multicosymmetry.

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In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the $\mathrm{SO}\left(2\right)$ symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.

We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed riemannian manifolds and the latter is in addition of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is...

Symmetries of the control systems of the form ${u}_{t}=f(t,u,v)$, $u\in {\mathbb{R}}^{n}$, $v\in {\mathbb{R}}^{m}$ are studied. Some general results concerning point symmetries are obtained. Examples are provided.

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $\cdots \<{\lambda}_{k}^{-}\le {\lambda}_{k}^{+}\<{\lambda}_{k+1}^{-}\le \cdots $ of a Zakharov-Shabat operator is symmetric,i.e. ${\lambda}_{k}^{\pm}=-{\lambda}_{-k}^{\mp}$ for all $k$, if and only if the sequence ${\left({\gamma}_{k}\right)}_{k\in \mathbb{Z}}$ of gap lengths, ${\gamma}_{k}:={\lambda}_{k}^{+}-{\lambda}_{k}^{-}$, is symmetric with respect to $k=0$.