Determination of the stability boundaries for the hamiltonian systems with periodic coefficients

We consider the hamiltonian system of linear differential equations with periodic coefficients. Using the infinite determinant method based on the existence of periodic solutions on the boundaries between the domains of stability and instability in the parameter space we have developed the algorithm for analytical computation of the stability boundaries. The algorithm has been realized for the second and the fourth order hamiltonian systems arising in the restricted many‐body problems. The stability boundaries have been found in the form of powers series, accurate to the sixth order in a small parameter. All the computations are done with the computer algebra system Mathematica. Nagrinejama Hamiltono tiesiniu diferencialiniu lygčiu su periodiniais koeficientais sistema. Remiantis tuo, kad parametru erdveje stabilumo ir nestabilumo sritis skiriančioje sienoje egzistuoja periodinis sprendinys, sukurtas analitinis minetos sienos apskaičiavimo algoritmas. Algoritmas realizuotas antros ir ketvirtos eiles Hamiltono sistemoms, kylančioms nagrinejant apribotu keleto kūnu uždavinius. Stabilumo srities siena randama laipsnines eilutes pavidalu mažojo parametro šešto laipsnio tikslumu. Skaičiavimai atlikti skaičiavimo algebros paketo Mathematica pagalba. First Published Online: 14 Oct 201

Computing the stability boundaries for the lagrange triangular solutions in the elliptic restricted three‐body problem

An algorithm is proposed for analytical computing the stability boundaries of the Lagrange triangular solutions in the elliptic restricted three‐body problem. It is based on the infinite determinant method. The algorithm has been implemented by using the computer algebra system Mathematica and the stability boundaries have been determined in the form of power series with respect to a small parameter with accuracy up to the 10th order. First Published Online: 14 Oct 201

On the Stability of the Homographic Polygon Configuration in the Many-Body Problem

In this paper the stability of a new class of exact symmetrical solutions in the Newtonian gravitational (n+1)-body problem is studied. This class of solution follows from a suitable geometric distribution of the (n+1)-bodies, and initial conditions, so that the solution is represented geometrically by an oscillating regular polygon with n sides rotating non-uniformly about its center. The body having a mass m_0 is at the center of the polygon, while nbodies having the same mass m are at the vertices of the polygon and move about the central body in identical elliptic orbits. It is proved that for n=2 and for regular polygons 3<= n<= 6 each corresponding solution is unstable for any value of the central mass m_0. For n=>7 the solution is linearly stable and the eccentricity of the particles’ orbits if both \mu = m_0 / m > 141.477 and the eccentricity of the particles' orbits e is sufficiently smal

On the stability of the Hill's equation with damping

We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude \epsilon with accuracy O(\epsilon^2) and verify conditions for its stability. In the first order approximation in \epsilon, we prove that the resonance domain exists only if the excitation frequency \Omega is sufficiently close to the double natural frequency of the pendulum; the corresponding equation of the stability boundary is obtained

Simulation of quantum circuits with Mathematica

Компьютерная математика и компьютерная механик