Hamiltonian systems with widely separated frequencies

Tuwankotta, J.M.; Verhulst, Ferdinand

doi:10.1088/0951-7715/16/2/319

Cited by 15 publications

(15 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Two of the frequencies will be near √ 2, one will be √ 2a, the associated modes will be called the optical group (x 1 , x 2 ) and the acoustical group (x 3 ). System (18) is an example of a system with widely separated frequencies, see Tuwankotta and Verhulst (2003) and further references there. Following the analysis in Tuwankotta and Verhulst (2003) we apply normalization considering x 3 as slowly varying.…”

Section: Four Alternating Masses a Summarymentioning

confidence: 99%

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Bruggeman

Verhulst

2018

J Nonlinear Sci

View full text Add to dashboard Cite

An important aspect of understanding FPU chains is the existence of invariant manifolds (called "bushes") in FPU chains. We will focus on the classical periodic FPU chain and on the FPU chain with alternating masses where we show that also in the alternating case nested manifolds (related to bushes) exist. The use of symmetries leads to the emergence of systems of n particles as invariant manifolds of systems with a multiple of n particles. This analysis is followed by examples of existence and stability of special invariant manifolds and phase-space dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles. It turns out that in the alternating case the dynamics is strongly affected by the choice of the alternating mass m. Normal form calculations help to identify quasi-trapping regions leading to delay of recurrence. The results suggest that equipartition of energy near stable equilibrium is improbable.

show abstract

Section: Four Alternating Masses a Summarymentioning

confidence: 99%

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Bruggeman

Verhulst

2018

J Nonlinear Sci

View full text Add to dashboard Cite

show abstract

“…This fact leads to small denominators or coupling terms in the corresponding averaged equations or normal forms. 2,14,18,20,[26][27][28][29]35 Since other modal frequencies are not rationally commensurate or have significant time scale separation, we do not expect strong resonance among them. Extensive simulations confirmed our expectation.…”

Section: Introductionmentioning

confidence: 93%

“…The success of Eisenhower and Mezić, 12 Du Toit et al, 8 Nayfeh et al, 26,28 and other researchers 18,23,35 shows that the method of modal truncation is a viable method if it is used with care. Our understanding of the nearly 0:1 resonance in our DNA model and the results of our extensive numerical simulation have convinced us that two mode truncation is adequate for our analytical study of the phenomenon of structured activations.…”

Section: B Two Mode Truncation Is Adequatementioning

confidence: 99%

“…Hence, we believe that nearly 0:1 resonance should be the main focus of our study. Nayfeh et al, 26,28 Haller, 18 Feng and Liew, 14 Tuwankotta and Verhulst, 35 Langford and Zhan, 20 and Broer et al 2 have studied such a degenerate resonance. Both Langford and Zhan and Broer et al used the method of normal form.…”

Section: A Nearly 0:1 Resonance and Partial Averagingmentioning

confidence: 99%

“…By applying the method of partial averaging 1,12,14,18,26,28,35 for nearly 0:1 resonance, we obtained the averaged equations for the reduced models of this chain of a) Author to whom correspondence should be addressed. Electronic mail: koon@cds.caltech.edu.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Control of a model of DNA division via parametric resonance

Koon

Owhadi

Tao

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendula in a Morse potential. While the (possibly parametrically excited) system has a large number of degrees of freedom and a large number of intrinsic time scales, global and slow variables can be identified by (1) first reducing its dynamic to two modes exchanging energy between each other and (2) averaging the dynamic of the reduced system with respect to the phase of the fastest mode. Surprisingly, the global and slow dynamic of the system remains Hamiltonian (despite the parametric excitation) and the study of its associated effective potential shows how parametric excitation can turn the unstable open state into a stable one. Numerical experiments support the accuracy of the time-averaged reduced Hamiltonian in capturing the global and slow dynamic of the full system. In this paper, we study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While DNA macro-molecules are robust to noise, our study shows that they are sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electromagnetic fields for triggering and controlling the division. The suggested method of control is supported not only by the observation that DNA vibrations induced by electricfields or microwave absorption are an experimental reality but also by the fact that electric field-induced molecular vibrations have already been used as a noninvasive cell transfection protocol. Our study also raises the question on whether enzymes are using the proposed mechanism to initiate the opening of DNA strands. This question is to put into correspondence with increasing theoretical and experimental evidence that low-frequency vibrations do exist and play significant biological functions in proteins, DNA molecules, and other bio-macromolecules.

show abstract

Dynamics of a Chain with Four Particles, Alternating Masses and Nearest-Neighbor Interaction

Bruggeman¹,

Verhulst²

2017

Springer Proceedings in Physics

View full text Add to dashboard Cite

We formulate the periodic FPU problem with four alternating masses which is the simplest nontrivial version. The analysis involves normal form calculations to second order producing integrable normal forms with three timescales. In the case of large alternating mass the system is an example of dynamics with widely separated frequencies and three timescales. The presence of approximate integrals and the stability characteristics of the periodic solutions lead to weak interaction of the modes of the system.

show abstract

Hamiltonian systems with widely separated frequencies

Cited by 15 publications

References 23 publications

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Control of a model of DNA division via parametric resonance

Dynamics of a Chain with Four Particles, Alternating Masses and Nearest-Neighbor Interaction

Contact Info

Product

Resources

About