38,455 research outputs found
Resonances in a spring-pendulum: algorithms for equivariant singularity theory
A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.
Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping
Explicit and semi-explicit geometric integration schemes for dissipative
perturbations of Hamiltonian systems are analyzed. The dissipation is
characterized by a small parameter , and the schemes under study
preserve the symplectic structure in the case . In the case
the energy dissipation rate is shown to be asymptotically
correct by backward error analysis. Theoretical results on monotone decrease of
the modified Hamiltonian function for small enough step sizes are given.
Further, an analysis proving near conservation of relative equilibria for small
enough step sizes is conducted.
Numerical examples, verifying the analyses, are given for a planar pendulum
and an elastic 3--D pendulum. The results are superior in comparison with a
conventional explicit Runge-Kutta method of the same order
Energy-Entropy-Momentum integration of discrete thermo-visco-elastic dynamics.
A novel time integration scheme is presented for the numerical solution of the dynamics of discrete systems consisting of point masses and thermo-visco-elastic springs. Even considering fully coupled constitutive laws for the elements, the obtained solutions strictly preserve the two laws of thermo dynamics and the symmetries of the continuum evolution equations. Moreover, the unconditional control over the energy and the entropy growth have the effect of stabilizing the numerical solution, allowing the use of larger time steps than those suitable for comparable implicit algorithms. Proofs for these claims are provided in the article as well as numerical examples that illustrate the performance of the method
Metropolis Integration Schemes for Self-Adjoint Diffusions
We present explicit methods for simulating diffusions whose generator is
self-adjoint with respect to a known (but possibly not normalizable) density.
These methods exploit this property and combine an optimized Runge-Kutta
algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting
numerical integration scheme is shown to be weakly accurate at finite noise and
to gain higher order accuracy in the small noise limit. It also permits to
avoid computing explicitly certain terms in the equation, such as the
divergence of the mobility tensor, which can be tedious to calculate. Finally,
the scheme is shown to be ergodic with respect to the exact equilibrium
probability distribution of the diffusion when it exists. These results are
illustrated on several examples including a Brownian dynamics simulation of DNA
in a solvent. In this example, the proposed scheme is able to accurately
compute dynamics at time step sizes that are an order of magnitude (or more)
larger than those permitted with commonly used explicit predictor-corrector
schemes.Comment: 54 pages, 8 figures, To appear in MM
Time-delayed autosynchronous swarm control
In this paper a general Morse potential model of self-propelling particles is considered in the presence of a time-delayed term and a spring potential. It is shown that the emergent swarm behavior is dependent on the delay term and weights of the time-delayed function which can be set to induce a stationary swarm, a rotating swarm with uniform translation and a rotating swarm with a stationary center-of-mass. An analysis of the mean field equations shows that without a spring potential the motion of the center-of-mass is determined explicitly by a multi-valued function. For a non-zero spring potential the swarm converges to a vortex formation about a stationary center-of-mass, except at discrete bifurcation points where the center-of-mass will periodically trace an ellipse. The analytical results defining the behavior of the center-of-mass are shown to correspond with the numerical swarm simulations
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