Saddle-point bifurcation and onset of large-scale stochasticity in 1.5-degree-of-freedom Hamiltonian systems

To investigate the variability of ground motion characteristics (GMC) with the angle of seismic incidence (ASI) and the impact of seismic incident directionality on structural responses, first, a large-scale database of recorded ground motions was used to analyze the causes of GMC variability due to the seismic incident directionality effect (SIDE). Then a single-mass bi-degree-of-freedom system (SM-BDOF-S) with different types of symmetrical sections was selected to explore the influence mechanism of SIDE on the seismic responses. The results illustrated that the GMC has substantial variability with the ASI, which is independent of the earthquake source, propagation distance, and site condition, and exhibits complex random characteristics. Additionally, a classification method for ground motions is proposed based on this GMC variability to establish a criterion for selecting ground motions in seismic analysis considering the SIDE. Moreover, for an SM-BDOF-S, the response spectral plane is proposed to explain the transition behavior of spectral responses that are very similar among different stiffness ratios, but divergent for different types of ground motions. The influence of SIDE on structures is determined by their stiffness and stiffness ratio in the [Formula: see text]- and [Formula: see text]-directions, as well as the type of ground motion.

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Dynamics Redesign of Marine Structures

Journal of Ship Research ◽

10.5957/jsr.1985.29.4.285 ◽

1985 ◽

Vol 29 (04) ◽

pp. 285-295 ◽

Cited By ~ 1

Author(s):

Curtis J. Hoff ◽

Michael M. Bernitsas

Keyword(s):

Large Scale ◽

Structural Changes ◽

Dynamic Equilibrium ◽

Degree Of Freedom ◽

Attractive Alternative ◽

Equilibrium Equations ◽

Element Analysis ◽

Solution Methods ◽

Tower Model ◽

Energy Equations

The dynamic response of a marine structure depends upon the exciting forces and the modal characteristics of the structure. Excessive vibratory response requires reduction of the exciting loads or redesign of the structure or both. In this paper the general redesign problem is formulated. It applies to large-scale structures and allows for large structural changes. Solution of the redesign problem is achieved through perturbation methods which are an attractive alternative to traditional trial-and-error methods. Perturbation solution methods are based on dynamic equilibrium equations or energy equations or both. A new method based on the energy equations which enforces the mode orthogonality conditions is developed and evaluated against all existing methods. Two test cases, a 191-degree-of-freedom two-dimensional ship model and a 810-degree-of-freedom offshore light tower model are used to compare the methods numerically. It is shown that the method developed in this paper can produce, with a single finite element analysis of the baseline system, a structure which satisfies within acceptable limits all nonconflicting design objectives.

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Stability and motion in two degree of freedom hamiltonian systems for three-to-one commensurability

International Journal of Non-Linear Mechanics ◽

10.1016/0020-7462(71)90020-5 ◽

1971 ◽

Vol 6 (5) ◽

pp. 563-568 ◽

Cited By ~ 2

Author(s):

K.T. Alfriend

Keyword(s):

Hamiltonian Systems ◽

Degree Of Freedom ◽

Two Degree Of Freedom

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Fault-tolerant strategy and workspace of the subreflector parallel adjusting mechanism

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406219865289 ◽

2019 ◽

Vol 233 (18) ◽

pp. 6656-6667

Author(s):

Jiantao Yao ◽

Bo Han ◽

Yuchao Dou ◽

Yundou Xu ◽

Yongsheng Zhao

Keyword(s):

Fault Tolerance ◽

Large Scale ◽

Fault Tolerant ◽

Degree Of Freedom ◽

Proper Function ◽

Maintenance Cost ◽

Mechanical Equipment ◽

Identification Method ◽

Boundary Identification ◽

Workspace Boundary

Parallel mechanism has been widely used in large-scale and heavy-duty attitude adjustment equipment. In order to improve the work reliability of the subreflector parallel adjusting mechanism, a fault-tolerant strategy based on the redundant degree-of-freedom and a workspace boundary identification method are proposed in this paper, which can realize the proper function of the subreflector parallel adjusting mechanism when it has a driven fault. The configuration and parameters of the parallel adjusting mechanism are introduced firstly, then the degrees-of-freedom of the parallel adjusting mechanism is calculated when it has a driven fault, and the principle of the fault-tolerant strategy based on the redundant degree-of-freedom is deduced in detail. Next, the method to solve the workspace boundary identification problem for the parallel adjusting mechanism in fault tolerance conditions is proposed, the maximum and minimum workspaces of the parallel adjusting mechanism at fault tolerance conditions in different frequency bands are analyzed. The results showed that the workspace calculated by the fault-tolerant strategy in the fault condition can completely meet the needs of the subreflector, where this method can also be applied to other parallel mechanisms. Lastly, an experiment is conducted to verify the correctness and effectiveness of the fault-tolerant strategy, in which the results showed that the fault-tolerant strategy can effectively improve the work reliability of the parallel adjusting mechanism. The fault-tolerant strategy and workspace boundary identification method can make the subreflector parallel adjusting mechanism work normally when it has a driven fault, which can significantly improve the work reliability and work efficiency and the maintenance cost can also be reduced. The fault-tolerant strategy and workspace boundary identification method can also be well applied to the research and development for this kind of parallel mechanical equipment.

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Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems

Science Advances ◽

10.1126/sciadv.aav2372 ◽

2019 ◽

Vol 5 (4) ◽

pp. eaav2372 ◽

Cited By ~ 30

Author(s):

Hayato Goto ◽

Kosuke Tatsumura ◽

Alexander R. Dixon

Keyword(s):

Combinatorial Optimization ◽

Hamiltonian Systems ◽

Large Scale ◽

Optimization Problems ◽

Approximate Solutions ◽

Combinatorial Optimization Problems ◽

Bifurcation Phenomena ◽

Field Programmable ◽

Max Cut Problem ◽

Adiabatic Optimization

Combinatorial optimization problems are ubiquitous but difficult to solve. Hardware devices for these problems have recently been developed by various approaches, including quantum computers. Inspired by recently proposed quantum adiabatic optimization using a nonlinear oscillator network, we propose a new optimization algorithm simulating adiabatic evolutions of classical nonlinear Hamiltonian systems exhibiting bifurcation phenomena, which we call simulated bifurcation (SB). SB is based on adiabatic and chaotic (ergodic) evolutions of nonlinear Hamiltonian systems. SB is also suitable for parallel computing because of its simultaneous updating. Implementing SB with a field-programmable gate array, we demonstrate that the SB machine can obtain good approximate solutions of an all-to-all connected 2000-node MAX-CUT problem in 0.5 ms, which is about 10 times faster than a state-of-the-art laser-based machine called a coherent Ising machine. SB will accelerate large-scale combinatorial optimization harnessing digital computer technologies and also offer a new application of computational and mathematical physics.

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

Nonlinear Processes in Geophysics ◽

10.5194/npg-5-69-1998 ◽

1998 ◽

Vol 5 (2) ◽

pp. 69-74 ◽

Cited By ~ 11

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.

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On the geometry of transport in phase space I. Transport in k-degree-of-freedom Hamiltonian systems, 2≤k<∞

Physica D Nonlinear Phenomena ◽

10.1016/0167-2789(90)90159-m ◽

1990 ◽

Vol 44 (3) ◽

pp. 471-501 ◽

Cited By ~ 76

Author(s):

Stephen Wiggins

Keyword(s):

Phase Space ◽

Hamiltonian Systems ◽

Degree Of Freedom

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CHAOTIC DYNAMICS OF N-DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406015672 ◽

2006 ◽

Vol 16 (06) ◽

pp. 1777-1793 ◽

Cited By ~ 30

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.

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