scholarly journals On Stability of Periodic Solutions of Lienard Type Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Zijian Yin ◽  
Hongbin Chen
Keyword(s):  
Asymptotic Behavior ◽  
Periodic Solutions ◽  
Exponential Decay ◽  
Floquet Theory ◽  
Linear Growth ◽  
Liénard Equation ◽  
Damping Term ◽  
Restoring Force ◽  
Stable Periodic ◽  
The Stability

We use the Floquet theory to analyze the stability of periodic solutions of Lienard type equations under the asymptotic linear growth of restoring force in this paper. We find that the existence and the stability of periodic solutions are determined primarily by asymptotic behavior of damping term. For special type of Lienard equation, the uniqueness and stability of periodic solutions are obtained. Furthermore, the sharp rate of exponential decay of the stable periodic solutions is determined under suitable conditions imposed on restoring force.

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.

An Experimental Study of Subcritical Bifurcations in Milling

2007 ◽  
Author(s):  
Anupam Radhakrishnan ◽  
Ben T. Edes ◽  
Brian P. Mann
Keyword(s):  
Periodic Solutions ◽  
Cutting Speed ◽  
Stable Solution ◽  
Stability Diagram ◽  
Depth Of Cut ◽  
Work Piece ◽  
Cutting Condition ◽  
Multiple Attractors ◽  
Stable Periodic ◽  
The Stability

The focus of the current paper is an experimental investigation of subcritical bifurcations in milling. Several researchers have reported results indicating that the secondary Hopf bifurcations of turning processes are subcritical. However, fewer results are available for milling — especially results that provide any substantial experimental evidence. Here, the experimental cutting tests were performed on a relatively long aluminum workpiece. The atypical length of the work piece will be used so that the depth of cut may be slowly increased or decreased during the cutting process. This provides visual evidence of hysteresis in the bifurcation diagram and the existence of multiple stable periodic solutions. The importance of this exploratory experimental effort is that multiple attractors may co-exist (stable periodic solutions) for the same cutting speed and depth of cut, but only one of these solutions would be the chatter free case. An important outcome from these results is that a small perturbation in the desirable stable solution near the borders of the stability diagram could result in a jump to the unstable cutting condition.

Effects of a Magnetic Modulation on the Stability of a Magnetic Liquid Layer Heated From Above

2001 ◽  
Vol 123 (3) ◽  
pp. 428-433 ◽  
Author(s):  
Saı¨d Aniss ◽  
Mohamed Belhaq ◽  
Mohamed Souhar
Keyword(s):  
Floquet Theory ◽  
Fluid Layer ◽  
Mathieu Equation ◽  
Magnetic Liquid ◽  
Damping Term ◽  
Onset Of Convection ◽  
First Order ◽  
Gravitational Forces ◽  
Sinusoidal Magnetic Field ◽  
The Stability

The effect of a time-sinusoidal magnetic field on the onset of convection in a horizontal magnetic fluid layer heated from above and bounded by isothermal non magnetic boundaries is investigated. The analysis is restricted to static and linear laws of magnetization. A first order Galerkin method is performed to reduce the governing linear system to the Mathieu equation with damping term. Therefore, the Floquet theory is used to determine the convective threshold for the free-free and rigid-rigid cases. With an appropriate choice of the ratio of the magnetic and gravitational forces, we show the possibility to produce a competition between the harmonic and subharmonic modes at the onset of convection.

scholarly journals Double Delayed Feedback Control of a Nonlinear Finance System

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zhichao Jiang ◽  
Yanfen Guo ◽  
Tongqian Zhang
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Chaotic Behavior ◽  
Threshold Value ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Finance System ◽  
Stable Periodic ◽  
The Stability ◽  
Manifold Theory

In this paper, a class of chaotic finance system with double delayed feedback control is investigated. Firstly, the stability of equilibrium and the existence of periodic solutions are discussed when delays change and cross some threshold value. Then the properties of the branching periodic solutions are given by using center manifold theory. Further, we give an example and numerical simulation, which implies that chaotic behavior can be transformed into a stable equilibrium or a stable periodic solution. Also, we give the local sensitivity analysis of parameters on equilibrium.

BIFURCATION STRUCTURES IN A MODEL OF A CO2 LASER WITH A FAST SATURABLE ABSORBER

2011 ◽  
Vol 21 (01) ◽  
pp. 305-322 ◽  
Author(s):  
EUSEBIUS J. DOEDEL ◽  
BART E. OLDEMAN ◽  
CARLOS L. PANDO L.
Keyword(s):  
Periodic Solutions ◽  
Saturable Absorber ◽  
Main Parameter ◽  
Solution Structure ◽  
Q Switching ◽  
Stable Periodic ◽  
The Stability ◽  
The One ◽  
Bifurcation Structures ◽  
Incoherent Pump

We consider a new mathematical model of a CO 2 laser with a fast saturable absorber. The system exhibits isolas of periodic solutions, along with their bifurcations, as the main parameter, the incoherent pump of the laser, is changed. The characteristic feature of these lasers is their spiking behavior; the spikes, passive Q-switching pulses, correspond to stable periodic solutions on the isolas. We also study the changes in the solution structure as a second parameter varies, namely, the one that is responsible for the extent of nonlinear losses. In particular, we determine what happens to the isolas and to the stability properties of the periodic solutions along them.

On the Stability of Forced Radial Oscillations of an Experimental Spring Pendulum

1993 ◽  
Author(s):  
Philip V. Bayly ◽  
Lawrence N. Virgin
Keyword(s):  
Fixed Point ◽  
Limit Cycle ◽  
Internal Resonance ◽  
Floquet Theory ◽  
Degree Of Freedom ◽  
Restoring Force ◽  
Linear Spring ◽  
Basic Ideas ◽  
The Stability ◽  
Elastic Pendulum

Abstract The elastic pendulum is a 2-degree-of-freedom, nonlinear device in which the pendulum bob may slide up and down the pendulum arm subject to the restoring force of a linear spring. In this study, radial motion (motion along the arm) is excited directly. Responses to this excitation include purely radial oscillations as well as swinging motion due to a 2:1 internal resonance. Changes in the behavior of the nonlinear spring pendulum occur when, under the control of a parameter, radial oscillations become unstable and are replaced by radial plus swinging motion. This bifurcation is explored analytically, numerically and experimentally, using the basic ideas of Floquet theory. Poincaré sampling is used to reduce the problem of describing the stability of a limit cycle to the easier task of defining the stability of the fixed point of a Poincaré map.

HOPF BIFURCATION AND STABILITY OF PERIODIC SOLUTIONS IN THE DELAYED LIÉNARD EQUATION

2008 ◽  
Vol 18 (10) ◽  
pp. 3147-3157 ◽  
Author(s):  
GUANG-PING HU ◽  
WAN-TONG LI ◽  
XIANG-PING YAN
Keyword(s):  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Liénard Equation ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Trivial Equilibrium ◽  
Functional Differential ◽  
The Stability ◽  
Lienard Equation ◽  
Zero Equilibrium

In this paper, the classical Liénard equation with a discrete delay is considered. Under the assumption that the classical Liénard equation without delay has a unique stable trivial equilibrium, we consider the effect of the delay on the stability of zero equilibrium. It is found that the increase of delay not only can change the stability of zero equilibrium but can also lead to the occurrence of periodic solutions near the zero equilibrium. Furthermore, the stability of bifurcated periodic solutions is investigated by applying the normal form theory and center manifold reduction for functional differential equations. Finally, in order to verify these theoretical conclusions, some numerical simulations are given.

Stability of periodic solutions near a collision of eigenvalues of opposite signature

1991 ◽  
Vol 109 (2) ◽  
pp. 375-403 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Periodic Orbit ◽  
Periodic Solutions ◽  
Floquet Theory ◽  
Stability Index ◽  
Surprising Result ◽  
Action Functional ◽  
Stability And Instability ◽  
The Stability ◽  
Pure Imaginary Eigenvalues ◽  
Stability Results

AbstractSome general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = ∮ p dq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI / dw, where w is the frequency of the periodic orbit. A stability index based on dI / dw is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: – 1 resonance) are obtained.

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