Asymptotically Stable Periodic Solutions in One Problem of Atmospheric Diffusion of Impurities: Asymptotics, Existence, and Uniqueness

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system. This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.

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Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops

Neural Computation ◽

10.1162/neco.2007.19.8.2124 ◽

2007 ◽

Vol 19 (8) ◽

pp. 2124-2148 ◽

Cited By ~ 32

Author(s):

Jianfu Ma ◽

Jianhong Wu

Keyword(s):

Periodic Solutions ◽

Asymptotically Stable ◽

Spiking Neuron ◽

Phase Resetting ◽

Neuron Models ◽

Absolute Refractory Period ◽

Integrate And Fire ◽

Stable Periodic ◽

Spiking Neuron Models ◽

Integrate And Fire Neuron

We consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop, when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate-and-fire neuron model. We show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations. We observe that the number of fixed points of the so-called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns. We demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns.

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Existence of asymptotically stable periodic solutions of a Rayleigh type equation

Nonlinear Analysis ◽

10.1016/j.na.2009.01.009 ◽

2009 ◽

Vol 71 (5-6) ◽

pp. 1728-1735 ◽

Cited By ~ 7

Author(s):

Yong Wang ◽

Liang Zhang

Keyword(s):

Periodic Solutions ◽

Type Equation ◽

Asymptotically Stable ◽

Stable Periodic

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New Criterion that Guarantees Sufficient Conditions for Globally Asymptotically Stable Periodic Solutions of Non-Linear Differential Equations with Delay

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2019/v31i530126 ◽

2019 ◽

pp. 1-10

Author(s):

Ebiendele Peter

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Periodic Solutions ◽

Sufficient Conditions ◽

Linear Differential Equations ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Stable Periodic ◽

Linear Differential ◽

Equations With Delay

The objective of this paper is to investigate and give sufficient conditions that we guarantees globally asymptotically stable periodic solutions, of non-linear differential Equations with Delay of the form (1.1). The Razumikhin’s technique was improve upon to enhance better result’s hence equation (1.2), was studied along side with equation (1.1). Equation (1.2) is an integro-differential equations with delay kernel. Since the coefficients of (1.2) are periodic, it is re-written as equation (3.1), where a ,b, and c ≥ 0, and ω- periodic continuious function on R. G ≥ 0, is a normalized kernel from equation (1.2), which enable us to defined equation (3.1) as a fixed point. Since the defined operator B, for equation (3.1) are not empty, claim1 -1V enable us to used the fixed point theorem to investigate and established our defined properties. See, (Theorem 3.1, Lemma 3.1 and Theorem 3.2) and the Liapunov’s direct (second) method to prove our main results. See, (Theorem3.3, 3.4, and 3.5) which established the objective of this study.

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