We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to obtain rigorous conclusions on stability or instability. A complete study of the equilibrium stability in the above degenerate case is performed, giving sufficient conditions for instability and stability in the sense of Lyapunov. The above conditions are expressed in the form of inequalities with respect to the coefficients of the Hamiltonian function, normalized up to sixth-degree terms inclusive.

For the $4$-body problem there is the following conjecture: Given arbitrary positive masses, the planar $4$-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the $(\ell+2)$-body problem with $\ell \geqslant 3$. In particular, we prove that the symmetric $(2n+1)$-body problem with masses $m_1=\ldots=m_{2n-1}=1$ and $m_{2n}=m_{2n+1}=m$ sufficiently small has at least two classes of convex central configuration when $n=2$, five when $n=3$, and four when $n=4$. We conjecture that the $(2n+1)$-body problem has at least $n $ classes of convex central configurations for $n>4$ and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric $(2n+2)$-body problem with masses $m_1=\ldots=m_{2n}=1$ and $m_{2n+1}=m_{2n+2}=m$ sufficiently small has at least three classes of convex central configuration when $n=3$, two when $n=4$, and three when $n=5$. We also conjecture that the $(2n+2)$-body problem has at least $[(n+1)/2]$ classes of convex central configurations for $n>5$ and we give some numerical evidences that the conjecture can be true.

Nonlinear differential equations associated with the second Painlevé equation are considered. Transformations for solutions of the singular manifold equation are presented. It is shown that rational solutions of the singular manifold equation are determined by means of the Yablonskii – Vorob’ev polynomials. It is demonstrated that rational solutions for some differential equations are also expressed via the Yablonskii – Vorob’ev polynomials.

Intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories of the system from the chaotic region into the laminar one. The main results on chaotic intermittency strongly depend on the RPD. Recently a generalized power law RPD has been observed in a wide class of 1D maps. Noise has an impact on the intermittency phenomena and the generalized RPD introduces a novel scenario because it is affected by the noise. An analytical approach was introduced to estimate the noisy RPD (NRPD). In this work we investigate the noisy RPD in two cases: an experimental continuous system, by means of a Poincar´e map associated to it, and a numerical map chosen close to the experimental one. In the experimental map we use the internal noise of the circuit, whereas in the numerical map we introduce the noise in the usual way.
We have compared both noisy dynamics and found important differences between them, concerning the propagation of the noise effect from the maximum of the map (where the power law is generated) into the laminar region.
To mimic the numerical map by the experiment, we introduced an external noise during a short window of time, obtaining similar results to the ones obtained in the internal natural noise case. We found that our methodology developed to study the noise intermittency can be used to investigate which class of noise is present in continuous systems.

We present some necessary conditions for quasi-homogeneous differential systems to be completely integrable via Kovalevskaya exponents. Then, as an application, we give a new link between the weak-Painlevé property and the algebraical integrability for polynomial differential systems. Additionally, we also formulate stronger theorems in terms of Kovalevskaya exponents for homogeneous Newton systems, a special class of quasi-homogeneous systems, which gives its necessary conditions for B-integrability and complete integrability. A consequence is that the nonrational Kovalevskaya exponents imply the nonexistence of Darboux first integrals for two-dimensional natural homogeneous polynomial Hamiltonian systems, which relates the singularity structure to the Darboux theory of integrability.

A generalization of the Suslov problem with changing parameters is considered. The physical interpretation is a Chaplygin sleigh moving on a sphere. The problem is reduced to the study of a two-dimensional system describing the evolution of the angular velocity of a body. The system without viscous friction and the system with viscous friction are considered. Poincaré maps are constructed, attractors and noncompact attracting trajectories are found. The presence of noncompact trajectories in the Poincaré map suggests that acceleration is possible in this nonholonomic system. In the case of a system with viscous friction, a chart of dynamical regimes and a bifurcation tree are constructed to analyze the transition to chaos. The classical scenario of transition to chaos through a cascade of period doubling is shown, which may indicate attractors of Feigenbaum type.