We show that bifurcations of four-dimensional symplectic diffeomorphisms with a quadratic homoclinic tangency to a saddle periodic orbit with real multipliers produce 2-elliptic periodic orbits if the tangency is not partially hyperbolic. We show that a normal form for the rescaled first-return maps near such tangency is given by a four-dimensional symplectic Hénon-like map and study bifurcations of the first-return maps in generic two-parameter families.

The dynamics that necessarily coexists with a homoclinic orbit is captured by its dynamical core. In this work we characterize the dynamical core of a broad class of homoclinic orbits in the Smale horseshoe, specifically those with decorations of three types: maximal, P-lists and star decorations. For each of these families, we construct an explicit pruning region whose survival set — consisting of all symbolic sequences whose orbits avoid the region under the shift — coincides with the dynamical core. This provides a unified symbolic description of the forced dynamics and establishes a framework for computing dynamical invariants such as topological entropy.

This paper is a continuation of our previous work where we investigated the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable connected surfaces such that their nonwandering sets consist of one-dimensional basic sets (attractors and repellers). In that work, we showed that the dynamical properties of each diffeomorphism from a given class define a collection consisting of nonempty multisets of natural numbers (each such collection contains at least two multisets). These multisets are topological invariants of the diffeomorphism and uniquely determine the topology of the ambient surface. In this paper, we solve the problem of realization of diffeomorphisms from the class $\mathbb G(M^2)$ with respect to a given collection of multisets of natural numbers. We describe all possible collections of multisets from which one can construct a diffeomorphism from the class $\mathbb G(M^2)$, presenting a step-by-step algorithm of construction.

In this paper, we prove the theorem given in the title and a similar theorem for generalized pseudo-Anosov homeomorphisms of the Klein bottle with only one “needle” singularity. We reformulate both statements in terms of the invariant foliation singularity type and orientability of foliations. The method we use to prove the theorems implies using the construction of band surface. Every band surface is represented with its combinatorial description, called the configuration. Applying a series of Rauzy transformations to all possible configurations in the cases considered, we show that the necessary conditions imposed on the invariant foliations are violated whence the results follow.

An original effective method for constructing explicit solutions of integrable Davey – Stewartson type equations is proposed, based on the use of dressing chains. The main difficulty arising when using the symmetry approach in 3D is associated with nonlocal variables entering the equation. To solve the nonlocality problem, it is proposed to replace the infinite dressing chain with its finite-field reductions preserving the integrability property. The application of the method is illustrated by the DS I equation, for which a new class of explicit solutions is constructed that depend on two arbitrary functions. In this example, the dressing chain is replaced by a finite-field reduction of the Toda lattice corresponding to a simple Lie algebra $A_2$.

This paper establishes the existence of periodic response solutions for multidimensional nonlinear Schrödinger equations (NLS) with unbounded perturbations. We adapt the Craig – Wayne – Bourgain (CWB) method to this context. The central point is that the core separation lemma remains valid for sufficiently weak unbounded perturbations, which allows the CWB method to be successfully applied. This work extends the applicability of the CWB method to problems with unbounded perturbations.

The purpose of this study is to investigate a simple model for the half-center oscillator (HCO), which consists of two nonidentical phase oscillators ($\phi$-neurons) with mutual chemical excitatory couplings. In the absence of couplings, each element is in an excitable state. Using one- and two-parameter bifurcation analysis, we determine regions of stability of various synchronous temporal patterns typical for HCO and describe in detail bifurcation transitions between them. The proposed simple HCO model allows one to reproduce the main effects observed in biologically plausible HCO models, and can be easily implemented in locomotor units in neuromorphic robotics.

In this paper we investigate rhomboidal central configurations for the planar fivebody problem. These are configurations whose convex hulls are rhombuses, and one interior mass lies on a diagonal of the rhombus. Consider partition of the configuration space by orderings of mutual distances, our main theorem provides criteria for central configurations in every region of this partition. For some regions we obtain existence and uniqueness of central configurations, for some we obtain nonexistence, for others our criteria are necessary conditions on masses. Our proofs are elementary and completely analytic.

In this study, we analyze a planar mathematical pendulum whose suspension point oscillates vertically according to a harmonic law. The pendulum bob is electrically charged and positioned slightly above two electric charges of equal sign and intensity, which are equidistant from the suspension point and separated by a distance of $2d$. Here, $d$ denotes the distance from each charge to the orthogonal projection of the suspension point onto the horizontal line where the charges lie. We formulate the Hamiltonian structure of this mechanical system, identify two equilibrium points, and examine the system’s linear stability. The dynamics are governed by three dimensionless parameters: $\mu$ which relates to the electric charges; $\varepsilon$, associated with the amplitude of oscillation of the suspension point; and $\alpha$, determined by the frequency of the system. We then investigate the parametric stability of the equilibrium points. Finally, we present the boundary surfaces that separate regions of stability and instability in the parameter space. For specific values of $\mu$, we derive cross-sectional curves that delineate these regions, using results from the Krein – Gelfand – Lidskii theorem and the Deprit – Hori method.