scholarly journals On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom

2002 ◽  
Vol 9 (2) ◽  
pp. 233-262 ◽  
Author(s):  
Luigi Chierchia ◽  
Luca Biasco
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Two Degrees Of Freedom ◽  
The Stability ◽  
Degenerate Hamiltonian Systems

scholarly journals Topological obstacles to the realizability of integrable Hamiltonian systems by billiards

2019 ◽  
Vol 488 (5) ◽  
pp. 471-475
Author(s):  
V. V. Vedyushkina ◽  
A. T. Fomenko
Keyword(s):  
Magnetic Field ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Integrable Hamiltonian Systems ◽  
Two Degrees Of Freedom ◽  
Degenerate Hamiltonian Systems ◽  
Liouville Equivalence

We introduce the following classes of integrable billiards: elementary billiards, topological, books, with potential, magnetic field, geodesic billiards. These classes are used to test the A.T. Fomenko conjecture about the realizability up to Liouville equivalence by billiards of integrable non-degenerate Hamiltonian systems with two degrees of freedom. In the class of book billiards found topological obstacles to realizability.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

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