scholarly journals Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

2019 ◽  
Vol 21 (07) ◽  
pp. 1850064 ◽  
Author(s):  
Guido Gentile ◽  
Alessandro Mazzoccoli ◽  
Faenia Vaia
Keyword(s):  
Potential Energy ◽  
Finite Dimension ◽  
Dissipative Systems ◽  
Diophantine Condition ◽  
Frequency Vector ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Term ◽  
Degeneracy Condition ◽  
The One

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence of dissipation, so that the forcing term is — up to the sign — the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular, we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exists provided the dissipation is strong enough. This extends results previously available in the literature in the one-dimensional case.

scholarly journals Quasi-periodic motions in strongly dissipative forced systems

2009 ◽  
Vol 30 (5) ◽  
pp. 1457-1469 ◽  
Author(s):  
GUIDO GENTILE
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Mechanical Force ◽  
Frequency Vector ◽  
Periodic Motions ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Term

AbstractWe consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.

Integration of one-dimensional equations of motions

2020 ◽  
pp. 79-90
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Potential Energy ◽  
Field Change ◽  
One Dimensional ◽  
System A ◽  
Small Correction ◽  
The Law ◽  
The One ◽  
The Given ◽  
Equations Of Motions

If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.

scholarly journals Escaping Orbits Are also Rare in the Almost Periodic Fermi–Ulam Ping-Pong

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Henrik Schließauf
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Initial Condition ◽  
Almost Periodic ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Function ◽  
Ping Pong ◽  
The One ◽  
Escaping Orbits

AbstractWe study the one-dimensional Fermi–Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.

Integration of one-dimensional equations of motion

2020 ◽  
pp. 3-5
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Potential Energy ◽  
Equations Of Motion ◽  
Field Change ◽  
One Dimensional ◽  
System A ◽  
Small Correction ◽  
The Law ◽  
The One ◽  
The Given

If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.

scholarly journals Convergent series for quasi-periodically forced strongly dissipative systems

2014 ◽  
Vol 16 (03) ◽  
pp. 1350022 ◽  
Author(s):  
Livia Corsi ◽  
Roberto Feola ◽  
Guido Gentile
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solution ◽  
Trigonometric Polynomial ◽  
Convergent Series ◽  
Dissipative Systems ◽  
Diophantine Condition ◽  
Frequency Vector ◽  
First Derivative ◽  
Odd Order

We study the ordinary differential equation εẍ + ẋ + εg(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ ℝd. We show that if there exists c0∈ ℝ such that g(c0) equals the average of f and the first non-zero derivative of g at c0is of odd order 𝔫, then, for ε small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution close to c0, with the same frequency vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for 𝔫 = 1. We also point out that, if 𝔫 = 1 and the first derivative of g at c0is positive, then the quasi-periodic solution is locally unique and attractive.

THE PARTIAL POTENTIAL ENERGY SURFACE AND SCATTERING RESONANCE STATE OF THE STATE-TO-STATE REACTION Br + HBr(v) → BrH(v′) + Br

2006 ◽  
Vol 05 (spec01) ◽  
pp. 307-316 ◽  
Author(s):  
HUAYANG WANG ◽  
XIAOMIN SUN ◽  
DACHENG FENG ◽  
ZHENGTING CAI
Keyword(s):  
Potential Energy ◽  
Potential Energy Surface ◽  
Energy Surface ◽  
Minimum Energy ◽  
Resonance State ◽  
Minimum Energy Path ◽  
Resonance States ◽  
One Dimensional ◽  
Well Model ◽  
The One

In this paper, the partial potential energy surface (PPESs) of the Br + HBr and Br - + HBr systems including the minimum energy path and the vibrational potential curves were constructed at MP2/6-311++G** level, based on the conception and constructing approach of the PPESs previously proposed. These results obtained from the PPESs were compared with those from the high resolved threshold photodetachment spectrum of the BrHBr - anion measured by Neumark et al., J Phys Chem94, 1377–1388, 1990. On the basis of the PPESs, the scattering resonance states of the Br + HBr (v) → BrH (v′) + Br state-to-state reaction were studied and the satisfactory results were obtained. Subsequently, we calculated the width and lifetime of the resonance states in this reaction by the one-dimensional square potential well model, and obtained some results consistent to the experiments.

One-dimensional dislocations - III. Influence of the second harmonic term in the potential representation, on the properties of the model

1949 ◽  
Vol 200 (1060) ◽  
pp. 125-134 ◽  
Keyword(s):  
Potential Energy ◽  
Second Harmonic ◽  
Dislocation Model ◽  
Stability Conditions ◽  
One Dimensional ◽  
Harmonic Term ◽  
A Chain ◽  
The Stability ◽  
The One ◽  
Work Done

In the previous one-dimensional dislocation model, a single sinusoidal term was taken to represent the potential energy of the deposit as a function of its position on the substrate. In this model a more general representation of the potential, containing a second harmonic term as well, is used, and it is shown that the solution in this case is also expressible in terms of elliptic integrals. The displacements corresponding to a sequence of dislocations (or a single one) are calculated. The work done in generating a single dislocation by a force on a free end is derived and the stability conditions for such a chain determined. It turns out that the properties of single dislocations, especially as concerns their application to misfitting monolayers and oriented overgrowth, remain almost uninfluenced, unless the amplitude of the second harmonic term is so large as to produce a new minimum and provided the overall amplitude of the potential energy is taken to be constant. When the amplitude of the second harmonic term is large, so that the potential curve has a second minimum, a complete dislocation splits up into two halves which are the one-dimensional analogues of Shockley’s ‘half-dislocations’ in close-packed lattices. The equilibrium separation of the two halves, as well as the stability conditions for the existence of a single half, are determined.

Group theory of constrained reaction mechanisms

1985 ◽  
Vol 63 (7) ◽  
pp. 1972-1975
Author(s):  
Paul G. Mezey
Keyword(s):  
Group Theory ◽  
Potential Energy ◽  
Algebraic Structure ◽  
Reaction Mechanisms ◽  
One Dimensional ◽  
The Family ◽  
The One ◽  
Energy Hypersurface ◽  
Potential Energy Hypersurface

It is shown that the family of all possible reaction mechanisms on a given potential energy hypersurface, which reaction mechanisms are constrained by a localstabilitycriterion, has an algebraic structure. The fundamental relations among all such "λ-constrained" reaction mechanisms are described in terms of a group, the one dimensional homotopygroup π1(Y). In contrast with the energy-dependentgroup π1(F) of fundamentalreactionmechanisms, proposed earlier for the lowenergydomainsF of potential energy hypersurfaces, the new group π1(Y) is fully determined by the distributionofcatchmentregions, representingchemicalspecies on the potential energy hypersurface.

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

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