Center Configurations of Hamiltonian Cubic Systems

Bifurcations of Limit Cycles from Cubic Hamiltonian Systems with a Center and a Homoclinic Saddle-loop

It is proved in this paper that the maximum number of limit cycles of system { dx dt = y, dy dt = kx− (k + 1)x2 + x3 + (α + βx + γx2)y is equal to two in the finite plane, where k > 11+ √ 33 4 , 0 < | | 1, |α| + |β| + |γ| = 0. This is partial answer to the seventh question in [2], posed by Arnold.

The domination number of cubic Hamiltonian graphs

Let γ(G) denote the domination number of a graph, and let C be the set of all Hamiltonian cubic graphs. Let γ̄(n) = max {γ(G)| G ∈ C and |V (G)| = n} , and γ(n) = min {γ(G)| G ∈ C and |V (G)| = n} . Then, for n ≥ 4, n even, γ̄(n) = ⌊ n + 1 3 ⌋ and γ(n) = ⌊ n + 2 4 ⌋ .

Hamiltonian decompositions of prisms over cubic graphs

Cubic planar hamiltonian graphs of various types

Let U be the set of cubic planar hamiltonian graphs, A the set of graphs G in U such thatG− v is hamiltonian for any vertex v of G, B the set of graphs G in U such thatG− e is hamiltonian for any edge e of G, and C the set of graphs G in U such that there is a hamiltonian path between any two different vertices of G. With the inclusion and/or exclusion of the sets A,B, and C, U is divided into ...

Homoclinic Orbits in Saddle-center Reversible Hamiltonian Systems

We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u + au′′ − u + f(u, b) = 0 as a model. Here f is an analytic function and a, b real parameters. These equations are important in several physical situations such as solitons and in the existence of “finite energy” stationary states of partial differential equations. We red...