Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case

2017 ◽  
Vol 22 (7) ◽  
pp. 880-892
Author(s):  
Rodrigo Gutierrez ◽  
Claudio Vidal
Keyword(s):  
Hamiltonian Systems ◽  
Equilibrium Points ◽  
Degenerate Case ◽  
Degree Of Freedom ◽  
Stability Of Equilibrium

scholarly journals Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 755
Author(s):  
Rebiha Benterki ◽  
Jaume LLibre
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Upper Bounds ◽  
Linear Hamiltonian Systems ◽  
Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

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