List of authors > Trifonov Konstantin

Hamiltonian systems arise as mathematical mo dels in many branches of physics, chemistry, engineering. Such systems as their study shows have usually a rather complicated structure that leads to great difficulties in their examination. Therefore one of the fruitful metho d of their investigation is the study of the orbit behavior near some specific structures which can b e distinguished by simple conditions. The study of a system near
a homo clinic orbits or contours made up of several hetero clinic orbits and equilibria or p erio dic orbits is undoubtedly one of such problem. We study the dynamics of an analytic reversible Hamiltonian system XH with two
degrees of freedom assuming the system has a heteroclinic contour involving a symmetric saddle-center equilibrium p (its eigenvalues are nonzero numbers ±iω, ±λ, ω, λ ∈ R), an orientable symmetric saddle periodic orbit γ lying in the same level of Hamiltonian H = H(p) and two nonsymmetric heteroclinic orbits Γ1, Γ2 joining p with γ and interchanged by the involution L, Γ2 = L(Γ1). The reversible involution L is supposed to have a
smooth two-dimensional set Fix(L) of its fixed points. Such a system are met in generic one-parameter families of reversible Hamiltonian systems. Saddle periodic orbit γ belongs to a 1-parameter family γc of saddle periodic orbits in all close levels H = c forming a symplectic cylinder. Reversible Hamiltonian systems possessing the above mentioned contour can be of two different types in dependence on how the involution acts locally near a saddle-center. Our results demonstrate the existence in such a system:
• countable set of transverse 1-round homoclinic orbits to γ and related to them non-uniformly hyperbolic subsets;
• appearance for c > 0 of two transverse heteroclinic contours involving γc, a small Lyapunov periodic orbit lc near p and four heteroclinic orbits Γ ±1 and Γ ± 2 = L(Γ ±1 )
and related with them uniform hyperbolic subsets;
• a finite set of transverse 1-round homoclinic orbits to γc for | c | close to H(p) and uniformly hyperbolic sets related with them;
• a countable set of values c n < 0 accumulating at c = 0 such that on the level H = c_n the system has a tangent homoclinic orbit to γcn and bifurcations nearby orbits related to this tangency;
• countable sets of saddle and elliptic periodic orbits.
Some other bifurcation phenomena will be discussed when generic one parameter reversible unfoldings of such a system are considered. 

This work was supported by the Russian Foundation of Basic Research under the grant 18-29-10081.