scholarly journals The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition

2018 ◽  
Vol 264 (2) ◽  
pp. 604-623 ◽  
Author(s):  
Nina Xue ◽  
Xiong Li
Keyword(s):  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Diophantine Condition

scholarly journals Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

1998 ◽  
Vol 5 (2) ◽  
pp. 69-74 ◽  
Author(s):  
M. G. Brown
Keyword(s):  
Phase Space ◽  
Time Dependence ◽  
Hamiltonian Systems ◽  
Fluid Flows ◽  
Space Structure ◽  
Degree Of Freedom ◽  
Periodic Functions ◽  
Phase Space Structure ◽  
Fractal Properties ◽  
Incompressible Fluid Flows

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.

scholarly journals CHAOTIC DYNAMICS OF N-DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

2006 ◽  
Vol 16 (06) ◽  
pp. 1777-1793 ◽  
Author(s):  
CHRIS ANTONOPOULOS ◽  
TASSOS BOUNTIS ◽  
CHARALAMPOS SKOKOS
Keyword(s):  
Lyapunov Exponents ◽  
Hamiltonian Systems ◽  
Finite Size ◽  
Period Doubling ◽  
Power Laws ◽  
Degree Of Freedom ◽  
Einstein Condensation ◽  
Global Dynamics ◽  
Local And Global Dynamics ◽  
Very High

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi–Pasta–Ulam (FPU) model and a discretized version of the nonlinear Schrödinger equation related to Bose–Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents Li, i = 1,…, N - 1, exhibit a transition between two power laws, Li ∝ EBk, Bk > 0, k = 1, 2, occurring at the same value of E. The destabilization energy Ec per particle goes to zero as N → ∞ following a simple power-law, Ec/N ∝ N-α, with α being 1 or 2 for the cases we studied. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Furthermore, the IPM orbit of the BEC Hamiltonian never destabilizes, having finite-size islands around it, even for very high N and E. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov–Sinai entropies per particle h KS /N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

scholarly journals Periodic solutions and their stability for some perturbed Hamiltonian systems

2020 ◽  
Vol 18 (01) ◽  
pp. 2150013
Author(s):  
Juan L. G. Guirao ◽  
Jaume Llibre ◽  
Juan A. Vera ◽  
Bruce A. Wade
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Averaging Theory ◽  
Differential Systems ◽  
Type Of Stability

We deal with non-autonomous Hamiltonian systems of one degree of freedom. For such differential systems, we compute analytically some of their periodic solutions, together with their type of stability. The tool for proving these results is the averaging theory of dynamical systems. We present some applications of these results.

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