scholarly journals The stability of periodic solutions of discontinuous systems that intersect several surfaces of discontinuity

1998 ◽  
Vol 62 (5) ◽  
pp. 677-685 ◽  
Author(s):  
A.P. Ivanov
Keyword(s):  
Periodic Solutions ◽  
Discontinuous Systems ◽  
The Stability

On the Stability Properties of a Periodically Forced, Time-Varying Mass System

2009 ◽  
Author(s):  
W. T. van Horssen ◽  
O. V. Pischanskyy ◽  
J. L. A. Dubbeldam
Keyword(s):  
Periodic Solutions ◽  
Forced Vibrations ◽  
Time Varying ◽  
Mass System ◽  
Single Degree Of Freedom ◽  
Stability Properties ◽  
Single Degree ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Varying Mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Not only solutions of the oscillator equation will be constructed, but also the stability properties, and the existence of periodic solutions will be discussed.

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

scholarly journals Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kaushik Dehingia ◽  
Hemanta Kumar Sarmah ◽  
Yamen Alharbi ◽  
Kamyar Hosseini
Keyword(s):  
Time Delay ◽  
Immune Responses ◽  
Periodic Solutions ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Cancer Model ◽  
Delay Effect ◽  
Discrete Time Delay ◽  
The Stability ◽  
Vital Parameters

AbstractIn this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

Effects of Delay and Diffusion on the Dynamics of a Leslie–Gower Type Predator–Prey Model

2014 ◽  
Vol 24 (04) ◽  
pp. 1450043
Author(s):  
Jia-Fang Zhang ◽  
Xiang-Ping Yan
Keyword(s):  
Periodic Solutions ◽  
Positive Constant ◽  
Neumann Boundary ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Delayed System ◽  
The Stability ◽  
And Diffusion

In this paper, we consider the effects of time delay and space diffusion on the dynamics of a Leslie–Gower type predator–prey system. It is shown that under homogeneous Neumann boundary condition the occurrence of space diffusion does not affect the stability of the positive constant equilibrium of the system. However, we find that the incorporation of a discrete delay representing the gestation of prey species can not only destabilize the positive constant equilibrium of the system but can also cause a Hopf bifurcation at the positive constant equilibrium as it crosses some critical values. In particular, we prove that these Hopf bifurcations' periodic solutions are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as periodic solutions of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system will generate spatially nonhomogeneous periodic solutions. The results in this work demonstrate that diffusion plays an important role in deriving complex spatiotemporal dynamics.

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