On the Stability of Equilibrium Paths Associated With Autonomous Systems

1981 ◽  
Vol 48 (1) ◽  
pp. 183-187 ◽  
Author(s):  
K. Huseyin
Keyword(s):  
Equilibrium Points ◽  
Autonomous Systems ◽  
Perturbation Approach ◽  
Stability Of Equilibrium ◽  
Symmetric Point ◽  
Divergence Point ◽  
Stability And Instability ◽  
Space Formulation ◽  
The Stability ◽  
Conditions Of Stability

The postcritical behavior and stability distribution on the equilibrium paths emanating from a divergence point associated with an autonomous system are studied within a state-space formulation. The analysis concerning the stability of equilibrium paths is based on the eigenvalues of the Jacobian evaluated at arbitrary equilibrium points in the vicinity of a critical point. Explicit conditions of stability and instability concerning the initial and postcritical paths are obtained through a perturbation approach. It is shown that at an asymmetric point of bifurcation an exchange of stabilities between two paths occurs in complete analogy with conservative systems. Similarly, a symmetric point of bifurcation involves a postcritical path which is totally stable (unstable) if the initial path is unstable (stable).

scholarly journals Stability of Nonlinear Fractional Neutral Differential Difference Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Kewei Liu ◽  
Wei Jiang
Keyword(s):  
Caputo Derivative ◽  
Neutral Systems ◽  
Difference Systems ◽  
Stability And Instability ◽  
The Stability ◽  
Conditions Of Stability ◽  
Fractional Neutral Systems

We study the stability of a class of nonlinear fractional neutral differential difference systems equipped with the Caputo derivative. We extend Lyapunov-Krasovskii theorem for the nonlinear fractional neutral systems. Conditions of stability and instability are obtained for the nonlinear fractional neutral systems.

Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

2021 ◽  
Vol 31 (11) ◽  
pp. 2130031
Author(s):  
José Alejandro Zepeda Ramírez ◽  
Martha Alvarez-Ramírez ◽  
Antonio García
Keyword(s):  
Nonlinear Stability ◽  
Mass Formula ◽  
Body Problem ◽  
Equilibrium Points ◽  
Constant Angular Velocity ◽  
Gravitational Attraction ◽  
Stability Of Equilibrium ◽  
Four Body Problem ◽  
Elliptic Equilibrium ◽  
The Stability

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

scholarly journals Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Xuerong Shi ◽  
Zuolei Wang
Keyword(s):  
Qualitative Analysis ◽  
Neuron Model ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
External Forcing ◽  
Coupling Coefficients ◽  
Initial Value ◽  
Firing Patterns ◽  
Stability Of Equilibrium ◽  
The Stability

A model is introduced by coupling two three-dimensional Hindmarsh-Rose models with the help of a nonsmooth memristor. The firing patterns dependent on the external forcing current are explored, which undergo a process from adding-period to chaos. The stability of equilibrium points of the considered model is investigated via qualitative analysis, from which it can be gained that the model has diversity in the number and stability of equilibrium points for different coupling coefficients. The coexistence of multiple firing patterns relative to initial values is revealed, which means that the referred model can appear various firing patterns with the change of the initial value. Multiple firing patterns of the addressed neuron model induced by different scales are uncovered, which suggests that the discussed model has a multiscale effect for the nonzero initial value.

scholarly journals Limit cycles in a Kolmogorov-type model

1990 ◽  
Vol 13 (3) ◽  
pp. 555-566 ◽  
Author(s):  
Xun-Cheng Huang
Keyword(s):  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Equilibrium Points ◽  
Type Model ◽  
Kolmogorov Type ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stability Of Equilibrium ◽  
Uniqueness Of Limit Cycles ◽  
The Stability

In this paper, a Kolmogorov-type model, which includes the Gause-type model (Kuang and Freedman, 1988), the general predator-prey model (Huang 1988, Huang and Merrill 1989), and many other specialized models, is studied. The stability of equilibrium points, the existence and uniqueness of limit cycles in the model are proved.

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