scholarly journals On the periodic orbits of perturbed Hooke Hamiltonian systems with three degrees of freedom

2012 ◽  
Vol 62 (5) ◽  
pp. 1054-1063 ◽  
Author(s):  
Jaume Llibre ◽  
Luis Fernando Mello
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Three Degrees Of Freedom

A class of C∞-integrable Hamiltonian systems

1989 ◽  
Vol 113 (3-4) ◽  
pp. 293-314 ◽  
Author(s):  
W. M. Oliva ◽  
M. S. A. C. Castilla
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
First Integrals ◽  
Integrable Hamiltonian Systems ◽  
Integral Of Motion ◽  
Toda System ◽  
Nonnegative Coefficients ◽  
Special Case ◽  
Three Degrees Of Freedom

SynopsisWe discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

Near-Integrability and Recurrence in FPU-Cells

2016 ◽  
Vol 26 (14) ◽  
pp. 1650230 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Hamiltonian Systems ◽  
Building Block ◽  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Chain Recurrence ◽  
Low Dimension ◽  
A Cell ◽  
Linearized System ◽  
A Chain ◽  
Three Degrees Of Freedom

In a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the [Formula: see text] resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in [Formula: see text] resonance.

Hamiltonian systems with three degrees of freedom, singular-point analysis, and chaotic behavior

1986 ◽  
Vol 33 (3) ◽  
pp. 2131-2133 ◽  
Author(s):  
W.-H. Steeb ◽  
J. A. Louw ◽  
P. G. L. Leach ◽  
F. M. Mahomed
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Point Analysis ◽  
Three Degrees Of Freedom
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