Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors

2017 ◽  
Vol 27 (14) ◽  
pp. 1750225 ◽  
Author(s):  
Atanasiu Stefan Demian ◽  
Stephen Wiggins

The purpose of this paper is to apply Lagrangian Descriptors, a concept used to describe phase space structure, to autonomous Hamiltonian systems with two degrees of freedom in order to detect periodic solutions. We propose a method for Hamiltonian systems with saddle-center equilibrium and apply this approach to the classical Hénon–Heiles system. The method was successful in locating the unstable Lyapunov orbits in phase space.

1998 ◽  
Vol 5 (2) ◽  
pp. 69-74 ◽  
Author(s):  
M. G. Brown

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.


1989 ◽  
Vol 135 (3) ◽  
pp. 347-356 ◽  
Author(s):  
S.C. Farantos ◽  
M. Founargiotakis ◽  
C. Polymilis

1995 ◽  
Vol 05 (02) ◽  
pp. 545-549 ◽  
Author(s):  
C. GROTTA RAGAZZO

We show that the equation [Formula: see text], x ∈ (0, π), α < -1, which models transversal nonlinear vibrations of a buckled beam, has invariant four-dimensional manifolds of solutions containing periodic orbits with transversal homoclinic orbits to them. The basic tool used in the proof is a theorem concerning two degrees of freedom Hamiltonian systems with saddle-center loops.


2020 ◽  
Vol 501 (1) ◽  
pp. 1511-1519
Author(s):  
Junjie Luo ◽  
Weipeng Lin ◽  
Lili Yang

ABSTRACT Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.


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