scholarly journals On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in the Case of Two Identical Integer or Half-Integer Frequencies

2021 ◽  
Vol 17 (1) ◽  
pp. 77-102
Author(s):  
O. V. Kholostova ◽  
Keyword(s):  
Hamiltonian System ◽  
Normal Form ◽  
Small Parameter ◽  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Periodic Motions ◽  
Half Integer ◽  
Stability And Instability ◽  
Trivial Equilibrium ◽  
Two Parameters

This paper examines the motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small; when it has zero value, the system is autonomous. Consideration is given to a set of values of the other two parameters for which, in the autonomous case, two frequencies of small oscillations of the linearized equations of perturbed motion are identical and are integer or half-integer numbers (the case of multiple parametric resonance). It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form in terms through degree four in perturbations and up to various degrees in a small parameter (systems of first, second and third approximations). The structure of the regions of stability and instability of trivial equilibrium is investigated, and solutions are obtained to the problems of the existence, number, as well as (linear and nonlinear) stability of the system’s periodic motions analytic in fractional or integer powers of the small parameter. For some cases, conditionally periodic motions of the system are described. As an application, resonant periodic motions of a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their stability is solved.

scholarly journals On the motions of a near-autonomous Hamiltonian system in the cases of two zero frequencies

2020 ◽  
Vol 30 (4) ◽  
pp. 672-695
Author(s):  
O.V. Kholostova
Keyword(s):  
Hamiltonian System ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Coefficient Matrix ◽  
Periodic Motions ◽  
Dimensional Parameter ◽  
Stability And Instability ◽  
Autonomous Case ◽  
Trivial Equilibrium ◽  
Regions Of Stability

We consider the motion of a near-autonomous, time-periodic two-degree-of- freedom Hamiltonian system in the vicinity of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small, and when it is zero, the system is autonomous. Suppose that in the autonomous case for a set of two other parameters, both frequencies of small linear oscillations of the system in the vicinity of the equilibrium are equal to zero, and the rank of the coefficient matrix of the linearized equations of perturbed motion is three, two, or one. We study the structure of the regions of stability and instability of the trivial equilibrium of the system in the vicinity of the resonant point of a three-dimensional parameter space, as well as the existence, number and stability (in a linear approximation) of periodic motions of the system that are analytic in integer or fractional powers of the small parameter. As an application, periodic motions of a dynamically symmetric satellite (solid) with respect to the center of mass are obtained in the vicinity of its stationary rotation (cylindrical precession) in a weakly elliptical orbit in the case of two zero frequencies under study, and their instability is proved.

Elastically Mounted Double Aerodynamic Pendulum

2019 ◽  
Vol 19 (05) ◽  
pp. 1941007 ◽  
Author(s):  
Yury D. Selyutskiy ◽  
Andrei P. Holub ◽  
Marat Z. Dosaev
Keyword(s):  
Experimental Data ◽  
Numerical Simulation ◽  
Degrees Of Freedom ◽  
State University ◽  
Periodic Motions ◽  
Aeroelastic System ◽  
Subsonic Wind Tunnel ◽  
Trivial Equilibrium ◽  
Rotational Degrees Of Freedom ◽  
Moscow State University

Elastically mounted double aerodynamic pendulum is an aeroelastic system with two rotational degrees of freedom. A wing is attached to the second link of the pendulum. It is shown that it is possible to select values of parameters in such a way as to make the trivial equilibrium (where both links of the pendulum are stretched along the flow) unstable. Numerical simulation of behavior of the system in such situations is performed, and arising limit cycles are studied. Experimental investigation of such aerodynamic pendulum is performed in the subsonic wind tunnel of the Institute of Mechanics of Lomonosov Moscow State University. Characteristics of periodic motions are registered for different values of parameters of the system. It is shown that experimental data are in qualitative agreement with results of numerical simulation.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

2005 ◽  
Vol 1 (2) ◽  
pp. 135-142 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Hamiltonian System ◽  
Two Degree Of Freedom

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a two-degree-of-freedom (2-DOF) nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the incremental energy approach. The Poincaré mapping surfaces of chaotic motions for this specific nonlinear Hamiltonian system are illustrated. The chaotic and quasi-periodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian system. The resonant-periodic motions for such a system are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

2005 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Periodic Motion ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a 2-DOF nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the energy incremental energy approach. The Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The chaotic and quasiperiodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian systems. The resonant-periodic motions are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Symmetric Homoclinic Orbits at the Periodic Hamiltonian Hopf Bifurcation

2014 ◽  
Vol 24 (08) ◽  
pp. 1440006 ◽  
Author(s):  
Lev Lerman ◽  
Anna Markova
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Normal Form ◽  
Periodic Orbit ◽  
Degrees Of Freedom ◽  
Homoclinic Orbits ◽  
Bifurcation Parameter ◽  
Hamiltonian Hopf Bifurcation ◽  
Generic Family ◽  
Three Degrees Of Freedom

We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori.

Nonlinear oscillations of an elastic two-degrees-of-freedom pendulum

2007 ◽  
Vol 53 (1-2) ◽  
pp. 19-30 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Alexander G. Petrov
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Two Degrees Of Freedom
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