Computer-Assisted Methods for Analyzing Periodic Orbits in Vibrating Gravitational Billiards

2021 ◽  
Vol 31 (08) ◽  
pp. 2130021
Author(s):  
Kevin E. M. Church ◽  
Clément Fortin
Keyword(s):  
Numerical Methods ◽  
Periodic Orbits ◽  
Upper Bound ◽  
Simplified Model ◽  
Computer Assisted ◽  
Critical Amplitude ◽  
Numerical Investigations ◽  
Model Based ◽  
Physical Intuition ◽  
Pitchfork Bifurcations

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair [Formula: see text], there is a constant [Formula: see text] for which periodic orbits consisting of [Formula: see text] impacts per period [Formula: see text] cannot be sustained for amplitudes of oscillation below [Formula: see text]. We compute a verified upper bound for the conjectured critical amplitude for [Formula: see text] using our rigorous pseudo-arclength continuation.

scholarly journals On the Effects of Computer-assisted Teaching on Learning Results Based on Blended Learning Method

2019 ◽  
Vol 14 (01) ◽  
pp. 58 ◽  
Author(s):  
Xinwen Bi ◽  
Xiaodan Shi
Keyword(s):  
Blended Learning ◽  
Computer Network ◽  
Rapid Development ◽  
Learning Model ◽  
Sample Survey ◽  
Survey Method ◽  
Computer Assisted ◽  
Learning Method ◽  
Model Based ◽  
Teachers And Students

With the rapid development of the computer network technology, the blended learning method based on computer-assisted teaching has also been booming. This paper analyzes the connotations of blended learning and the architecture and main functions of the Moodle system by using research methods such as litera-ture analysis, development and research method and sample survey method and then establishes a blended learning model based on the Moodle platform. Then, with the course of C Programming Language as an example, this paper gives de-tailed design of the process and implementation plan of the blended learning model based on the Moodle platform. At last, this paper investigates and analyzes the teaching effects among teachers and students participating in the learning ac-tivity, and the results show that the blended learning model based on the Moodle platform is helpful to improving the teaching effects and can serve as a reference for the innovation of teaching models and methods.

scholarly journals Numerical investigations to design a novel model based on the fifth order system of Emden–Fowler equations

2020 ◽  
Vol 10 (5) ◽  
pp. 333-342 ◽  
Author(s):  
Zulqurnain Sabir ◽  
Mehmet Giyas Sakar ◽  
Manshuk Yeskindirova ◽  
Onur Saldir
Keyword(s):  
Order System ◽  
Numerical Investigations ◽  
Model Based ◽  
Fifth Order ◽  
Novel Model

The Modeling of the Real Building Object by Using the Model of a Two-Layer Beam of Variable Rigidity on an Elastic Basis

2012 ◽  
Vol 204-208 ◽  
pp. 3596-3599 ◽  
Author(s):  
Andreev Vladimir Igorevich ◽  
Barmenkova Elena Vjacheslavovna
Keyword(s):  
Numerical Methods ◽  
Lower Layer ◽  
System Structure ◽  
Simplified Model ◽  
Sandwich Beams ◽  
Variable Rigidity ◽  
The Real ◽  
Methods Of Calculation ◽  
Elastic Basis ◽  
Real Building

In the present paper is given the calculation of the real building object by using the model of a two-layer beam of variable rigidity on an elastic basis. The lower layer of a two-layer beam simulates the foundation, and the upper - the structure, at the same time is considered the weight of each layer. The characteristics of the upper layer change on length. To solve this problem were used an analytical and numerical methods of calculation. On the basis of the calculations can make the following conclusion: for the calculation of system «structure-foundation-basis» for the stage of pre-proposals it is advisable to apply the simplified model as sandwich beams and plates on elastic basis.

A Higher-Order Period Function and Its Application

2015 ◽  
Vol 25 (10) ◽  
pp. 1550140 ◽  
Author(s):  
Linping Peng ◽  
Lianghaolong Lu ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Critical Period ◽  
Higher Order ◽  
Quadratic System ◽  
Critical Periods ◽  
Explicit Formulas ◽  
Period Function ◽  
Isochronous System ◽  
Isochronous Centers

This paper derives explicit formulas of the q th period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

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