Application of the Characteristic Bisection Method for locating and computing periodic orbits in molecular systems

Vrahatis, Michael N.; Perdiou, A. E.; Kalantonis, V. S.; Perdios, E. A.; Papadakis, K. E.; Prosmiti, Rita; Farantos, Stavros C.

doi:10.1016/s0010-4655(01)00190-4

Cited by 19 publications

(14 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The finding of POs and their continuation in energy has been amply described in previous applications. 11,35,36 The C/B diagram of HOBr, which is very similar to the one for HOCl, 8 is shown in Fig. 2͑c͒.…”

Section: Analysis Of the Classical Phase Spacementioning

confidence: 54%

The bound state spectrum of HOBr up to the dissociation limit: Evolution of saddle-node bifurcations

Azzam¹,

Schinke²,

Farantos

et al. 2003

The Journal of Chemical Physics

View full text Add to dashboard Cite

Based on an accurate potential energy surface ͓J. Chem. Phys. 113, 4598 ͑2000͔͒ we calculated ca. 700 bound state energies and wave functions of nonrotating HOBr using the filter-diagonalization method. Similar to HOCl, a 1:2 anharmonic resonance between the HOBr bending and the OBr stretching mode determines the general structure of the level spectrum. One of the results of this resonance is a saddle-node bifurcation at which a new class of states ͑''dissociation states''͒ comes into existence, which advance along the HO-Br dissociation path. Because the resonance condition at low energies is better fulfilled for HOBr, the bifurcation occurs at considerably lower energies than for HOCl. The results of the quantum mechanical calculations are interpreted in terms of classical periodic orbits ͑continuation/bifurcation diagram͒ and a semiclassical analysis based on a spectroscopic Hamiltonian, which is fitted to the exact energy levels and also taking into account the wave functions.

show abstract

Section: Analysis Of the Classical Phase Spacementioning

confidence: 54%

The bound state spectrum of HOBr up to the dissociation limit: Evolution of saddle-node bifurcations

Azzam¹,

Schinke²,

Farantos

et al. 2003

The Journal of Chemical Physics

View full text Add to dashboard Cite

show abstract

“…Furthermore, fixed points of contractive, or nonexpanding (L = 1), functions appear in numerous fields including economics, game theory (especially ergodic games), meromorphic functions, nonlinear differential equations and dynamical systems. In the study of dynamical systems with two degrees of freedom, such fixed point (periodic orbits) problems model conservative or dissipative systems depending on whether the mapping is area-preserving or area-contracting, respectively (see [19][20][21][22][23][24][25][26][27]). …”

Section: Fixed Points and Their Computationmentioning

confidence: 99%

Existence and computation of short-run equilibria in economic geography

Pavlidis

Vrahatis

Mossay

2007

Applied Mathematics and Computation

Self Cite

View full text Add to dashboard Cite

The new economic geography literature provides a general equilibrium framework that explains the emergence of economic agglomerations as a trade-off between increasing returns at the firm level and transportation costs related to the shipment of goods. The existence and uniqueness of short-run equilibria of this model has been shown for the case of two regions. The proposed approach employs the differential evolution algorithm to obtain estimates of the Lipschitz constant and the infinity norm of the function along the boundary and utilizes these values to investigate the existence of solutions of a function, and the computational burden of computing the topological degree of this function. This approach is employed to investigate the existence of short-run equilibria for more than two regions using fixed point and topological degree theory, as well as, the differential evolution algorithm. Irrespective of parameter settings the criteria from topological degree theory suggest that the model can have equilibria. The differential evolution algorithm identified such equilibria and for none of the parameter settings that were considered more than one equilibria were detected. The experimental results obtained also indicate that the computation of such equilibria has an exponential worst case lower bound complexity, as the model yields a function that is neither contractive, nor nonexpanding. Finally, the computation of the topological degree to identify the number of equilibria also has a very high computational cost.

show abstract

“…The restricted three-body problem is the most celebrated problem of Celestial Mechanics (see, among others, Szebehely [1], Marsden and Ross [2], and references therein) and due to its interdisciplinary applications it has also attracted the interest of many researchers from different fields of science (see, e.g., Prosmiti et al [3], Vrahatis et al [4], Sano [5]). During the last century, several modifications of this classical problem have been introduced in order to make it more relevant and applicable to certain systems of Dynamical Astronomy.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium Points and Related Periodic Motions in the Restricted Three-Body Problem with Angular Velocity and Radiation Effects

Perdios

Kalantonis

Perdiou

et al. 2015

Advances in Astronomy

View full text Add to dashboard Cite

The paper deals with a modification of the restricted three-body problem in which the angular velocity variation is considered in the case where the primaries are sources of radiation. In particular, the existence and stability of its equilibrium points in the plane of motion of the primaries are studied. We find that this problem admits the well-known five planar equilibria of the classical problem with the difference that the corresponding collinear points may be stable depending on the parameters of the problem. For all planar equilibria, sufficient parametric conditions for their stability have been established which are used for the numerical determination of the stability regions in various parametric planes. Also, for certain values of the parameters of the problem for which the equilibrium points are stable, the short and long period families have been computed. To do so, semianalytical expressions have been found for the determination of appropriate initial conditions. Special attention has been given to the continuation of the long period family, in the case of the classical restricted three-body problem, where we show numerically that periodic orbits of the short period family, which are bifurcation points with the long period family, are connected through the characteristic curve of the long period family.

show abstract

Application of the Characteristic Bisection Method for locating and computing periodic orbits in molecular systems

Cited by 19 publications

References 33 publications

The bound state spectrum of HOBr up to the dissociation limit: Evolution of saddle-node bifurcations

The bound state spectrum of HOBr up to the dissociation limit: Evolution of saddle-node bifurcations

Existence and computation of short-run equilibria in economic geography

Equilibrium Points and Related Periodic Motions in the Restricted Three-Body Problem with Angular Velocity and Radiation Effects

Contact Info

Product

Resources

About