scholarly journals Geometric analysis of a two-body problem with quick loss of mass

2021 ◽  
Author(s):  
Zhouqian Miao ◽  
Nikola Popović ◽  
Thomas Zacharis
Keyword(s):  
Singular Perturbation ◽  
Vector Fields ◽  
Body Problem ◽  
Blow Up ◽  
Geometric Approach ◽  
Resonance Phenomenon ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Two Body Problem ◽  
Loss Of Mass

AbstractWe consider a two-body problem with quick loss of mass which was formulated by Verhulst (Verhulst in J Inst Math Appl 18: 87–98, 1976). The corresponding dynamical system is singularly perturbed due to the presence of a small parameter in the governing equations which corresponds to the reciprocal of the initial rate of loss of mass, resulting in a boundary layer in the asymptotics. Here, we showcase a geometric approach which allows us to derive asymptotic expansions for the solutions of that problem via a combination of geometric singular perturbation theory (Fenichel in J Differ Equ 31: 53–98, 1979) and the desingularization technique known as “blow-up” (Dumortier, in: Bifurcations and Periodic Orbits of Vector Fields, Springer, Dordrecht, 1993). In particular, we justify the unexpected dependence of those expansions on fractional powers of the singular perturbation parameter; moreover, we show that the occurrence of logarithmic (“switchback”) terms therein is due to a resonance phenomenon that arises in one of the coordinate charts after blow-up.

scholarly journals Symmetries in central-force problems

2003 ◽  
pp. 47-52 ◽  
Author(s):  
V. Mioc ◽  
M. Barbosu
Keyword(s):  
Vector Fields ◽  
Body Problem ◽  
Blow Up ◽  
Central Force ◽  
Polar Coordinates ◽  
Global Flow ◽  
Two Body Problem ◽  
Concrete Problem ◽  
Central Force Problems ◽  
Force Problem

The two-body problem in central fields (reducible to a central-force problem) models a lot of concrete astronomical situations. The corresponding vector fields (in Cartesian and polar coordinates, extended via collision-blow-up and infinity-blow-up transformations) exhibit nice symmetries that form eight-element Abelian groups endowed with an idempotent structure. All these groups are isomorphic, which is not a trivial result, given the different structures of the corresponding phase spaces. Each of these groups contains seven four-element subgroups isomorphic to Klein?s group. These symmetries are of much help in understanding various characteristics of the global flow of the general problem or of a concrete problem at hand, and are essential in searching for periodic orbits.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

Firing patterns of the CA1 pyramidal neuron with geometric singular perturbation: a model study

2020 ◽  
Vol 34 (32) ◽  
pp. 2050316
Author(s):  
Yaru Liu ◽  
Shenquan Liu
Keyword(s):  
Singular Perturbation ◽  
Pyramidal Neurons ◽  
Sodium Current ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
External Stimulation ◽  
Model Study ◽  
Geometric Singular Perturbation ◽  
Current Activation ◽  
Ca1 Pyramidal Neuron

An investigation of CA1 pyramidal model is an important issue for applications, which is intimately related to the composition of ions in the extracellular environment and external stimulation. In this paper, it is demonstrated that the effects of different electrophysiological parameters such as muscarinic-sensitive potassium current activation variable and sustained sodium current inactivation variable on the firing sequence of model by numerical simulations. Furthermore, the paper also discusses that the temperature affects the firing of the CA1 model from direct current (DC) and alternating current (AC) stimuli. It is found that the model exhibits excellent spiking and bursting patterns, even chaotic patterns occur. Meanwhile, generalized mixed oscillations emerge in the model. Additionally, the firing modes are depicted by providing the response curve (RC), inter-spike interval curve (ISI), phase diagram curve (PDC) and the number of spikes per burst curve (NC). Mathematically, the paper elaborates the results which are presented to obtain two lower dimensional subsystems, which govern the fast and slow dynamics for giving insight into the dynamic behaviors of the full 5D system based on the geometric singular perturbation theory (GSPT). Particularly, we analyse the phase diagrams of the CA1 model to understand the properties better. The present results may contribute to further understand the information processing of the CA1 pyramidal neurons.

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

2020 ◽  
pp. 2150003
Author(s):  
Ali Atabaigi
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Limit Cycles ◽  
Heteroclinic Orbits ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Generalist Predator ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Geometric Singular Perturbation

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

Analysis of the Interaction between Modes by Normal Forms and Singular Perturbation Theory

2012 ◽  
Vol 157-158 ◽  
pp. 381-385
Author(s):  
Jing Zhang ◽  
Ting Ting Jiang
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Power System ◽  
Voltage Stability ◽  
Vector Fields ◽  
Normal Forms ◽  
Interaction Analysis ◽  
System Structure ◽  
System Stability ◽  
Singular Perturbation Theory

Due to the nonlinear characteristics of modern power systems, strong interaction between system angle stability and voltage stability will present. However, because of extreme complexity, these two kinds of stability are often investigated separately and the interactions between them are normally neglected in traditional studies. In this paper, a new method based on the theory of normal forms of vector fields and singular perturbation theory is introduced to deal with the power system structure preserving model for system stability modes interaction analysis. The relationship between angle and voltage stability is discovered. The method is validated with a sample power system.

Sign in / Sign up

Export Citation Format

Share Document