Floquet Theory and Stability of Periodic Solutions of Renewal Equations

2020 ◽  
Author(s):  
Dimitri Breda ◽  
Davide Liessi
Keyword(s):  
Periodic Solutions ◽  
Floquet Theory ◽  
Renewal Equations

Periodic Solutions by Picard's Approximations

1968 ◽  
Vol 11 (5) ◽  
pp. 743-745 ◽  
Author(s):  
T.A. Burton
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
General Framework ◽  
Floquet Theory ◽  
Linear Differential Equations ◽  
All Solutions ◽  
Private Correspondence ◽  
Linear Differential ◽  
Image Position

In [1] Demidovic considered a system of linear differential equationswith A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.

A NOTE ON PERIODIC SOLUTIONS OF A FORCED LIÉNARD-TYPE EQUATION

2010 ◽  
Vol 51 (3) ◽  
pp. 350-368 ◽  
Author(s):  
LIE-HUI ZHANG ◽  
YONG WANG
Keyword(s):  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Stability Theory ◽  
Floquet Theory ◽  
Type Equation ◽  
Continuation Theorem ◽  
Liapunov Stability ◽  
Analysis Techniques ◽  
Liénard Type Equation ◽  
Mawhin Continuation Theorem

AbstractCriteria for guaranteeing the existence, uniqueness and asymptotic stability (in the sense of Liapunov) of periodic solutions of a forced Liénard-type equation under certain assumptions are presented. These criteria are obtained by application of the Manásevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. An example is provided to demonstrate the applicability of our results.

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.

scholarly journals Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Hunki Baek ◽  
Younghae Do
Keyword(s):  
Numerical Simulations ◽  
Periodic Solutions ◽  
Small Amplitude ◽  
Floquet Theory ◽  
Dynamical Behavior ◽  
Seasonal Effects ◽  
Predator Prey ◽  
Impulsive Equation ◽  
Prey System ◽  
Impulsive Perturbations

We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations.

Periodic solutions and stability analysis of electrodynamic tethers for 13 degree international geomagnetic reference field

2018 ◽  
Vol 233 (11) ◽  
pp. 4115-4128
Author(s):  
Yu-wei Yang ◽  
Hong Cai
Keyword(s):  
Periodic Solutions ◽  
Dynamic Equation ◽  
Geomagnetic Field ◽  
Floquet Theory ◽  
Failure Time ◽  
Reference Field ◽  
International Geomagnetic Reference Field ◽  
Orbital Inclination ◽  
International Geomagnetic Reference ◽  
Free Parameters

The geomagnetic field plays a crucial role in the operation of the electrodynamic tether system in the space. Using the 13 degree International Geomagnetic Reference Field to model the geomagnetic field, the libration dynamic equation of the system is established. The nonlinear libration dynamic equation of the system is a typical chaotic system. In the dynamic equation, there are four free parameters: the orbital inclination, the orbital eccentricity, the altitude of the perigee and the electrodynamic parameter. After adding dummy control terms into the dynamic system, the extended time delayed autosynchronization control method is employed to calculate the periodic solutions of the dynamic equation. The shapes and amplitudes of the periodic libration motions change with the free parameters regularity. Based on the Floquet theory, the stabilities of the periodic solutions are analyzed. The orbit time during which the initial periodic libration turns to the rotation motion is defined as the failure time. It is used to measure the instability of the periodic solution and validate the results from the Floquet theory. Simulations show that the results from the failure time are consistent with the results from the Floquet theory. For all parameters, the periodic solutions are all unstable. In addition, the relationships between the instabilities of the periodic solutions and the free parameters are obtained.

scholarly journals Dynamics of Numerics of Nonautonomous Equations with Periodic Solutions: Introducing the Numerical Floquet Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Melusi Khumalo
Keyword(s):  
Numerical Methods ◽  
Periodic Solutions ◽  
Floquet Theory ◽  
Poincaré Map ◽  
Collocation Methods ◽  
Discrete Version ◽  
Qualitative Behaviour ◽  
Nonautonomous Systems ◽  
Nonautonomous Equations ◽  
Simple Systems

Nonautonomous systems with periodic solutions are encountered frequently in applications. In this paper, we will consider simple systems whose solutions are periodic with a known period. Their transformation under linearized collocation methods is investigated, using a technique called stroboscopic sampling, a discrete version of the well-known Poincaré map. It is shown that there is an inextricable relationship between AN stability (or BN stability) of the numerical methods and the correct qualitative behaviour of solutions.

A PERTURBATION-INCREMENTAL METHOD FOR DELAY DIFFERENTIAL EQUATIONS

2006 ◽  
Vol 16 (09) ◽  
pp. 2529-2544 ◽  
Author(s):  
K. W. CHUNG ◽  
C. L. CHAN ◽  
J. XU
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Floquet Theory ◽  
Period Doubling ◽  
Present Scheme ◽  
Pi Method ◽  
Incremental Method ◽  
Delay Differential ◽  
Branch Switching

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of periodic solutions of nonlinear systems of delay differential equations (DDEs). Periodic solutions can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at a period-doubling bifurcation is made simple by the present scheme as a parameter is simply increased from zero to a small positive value so that a solution on the new branch is obtained. Subsequent continuation of an emanating branch is also discussed. The advantage of the PI method lies in its simplicity and ease of implementation.

STABILITY OF T-PERIODIC SOLUTION ON THE EXTENDED SIMPLIFIED BRUSSELATOR MODEL

2008 ◽  
Vol 01 (01) ◽  
pp. 19-27 ◽  
Author(s):  
JIN-GUO LIAN ◽  
HONG-KUN ZHANG
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Floquet Theory ◽  
Degree Theory ◽  
Reaction Diffusion ◽  
Natural World ◽  
Autocatalytic Reaction ◽  
Periodic Patterns ◽  
Invariant Region ◽  
Brusselator Model

Brusselator model is a very typical autocatalytic reaction diffusion system. The bifurcation of steady-states of Brusselator model can be used to explain spot patterns of certain animals such as leopard and jaguar. Periodic patterns can be found throughout whole natural world, so it is very interesting to study patterns generated by the bifurcation of periodic solutions in extended Brusselator (EB) model, which extends Brusselator to T-periodic coefficients. In this paper, we study extended simplified Brusselator (ESB) model, which is EB model without diffusion terms. We find a unique T-periodic solution x0(t) in the strictly positively invariant region [Formula: see text] and prove its stability. This result establishes a foundation to study the bifurcation of EB model from x0(t). We also develop techniques of using degree theory and Floquet theory to analyze existence, uniqueness and stability of a periodic solution.

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.

scholarly journals EXISTENCE AND STABILITY OF PERIODIC SOLUTIONS OF A RAYLEIGH TYPE EQUATION

2009 ◽  
Vol 79 (3) ◽  
pp. 377-390 ◽  
Author(s):  
YONG WANG ◽  
XIAN-ZHI DAI
Keyword(s):  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Floquet Theory ◽  
Type Equation ◽  
Lyapunov Stability Theory ◽  
Analysis Techniques ◽  
Existence And Stability ◽  
Image Position ◽  
Mawhin Continuation Theorem ◽  
Lyapunov Sense

AbstractIn this work, we shall be concerned with the following forced Rayleigh type equation: Under certain assumptions, some criteria for guaranteeing the existence, uniqueness and asymptotic stability (in the Lyapunov sense) of periodic solutions of this equation are presented by applying the Manásevich–Mawhin continuation theorem, Floquet theory, Lyapunov stability theory and some analysis techniques. Moreover, an example is provided to demonstrate the applications of our results.

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