scholarly journals Asymptotic behavior of periodic solutions in one-parameter families of Liénard equations

2020 ◽  
Vol 190 ◽  
pp. 111617 ◽  
Author(s):  
Pedro Toniol Cardin ◽  
Douglas Duarte Novaes
Keyword(s):  
Asymptotic Behavior ◽  
Periodic Solutions ◽  
Liénard Equations

ASYMPTOTIC BEHAVIOR OF SPATIALLY INHOMOGENEOUS BALANCE LAWS

2005 ◽  
Vol 02 (03) ◽  
pp. 645-672 ◽  
Author(s):  
JULIA EHRT ◽  
JÖRG HÄRTERICH
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Initial Data ◽  
Periodic Solutions ◽  
Balance Laws ◽  
Spatially Inhomogeneous ◽  
Periodic Initial Data ◽  
Left And Right ◽  
Longtime Behavior ◽  
Time Periodic

We study the longtime behavior of spatially inhomogeneous scalar balance laws with periodic initial data and a convex flux. Our main result states that for a large class of initial data the entropy solution will either converge uniformly to some steady state or to a discontinuous time-periodic solution. This extends results of Lyberopoulos, Sinestrari and Fan and Hale obtained in the spatially homogeneous case. The proof is based on the method of generalized characteristics together with ideas from dynamical systems theory. A major difficulty consists of finding the periodic solutions which determine the asymptotic behavior. To this end we introduce a new tool, the Rankine–Hugoniot vector field, which describes the motion of a (hypothetical) shock with certain prescribed left and right states. We then show the existence of periodic solutions of the Rankine–Hugoniot vector field and prove that the actual shock curves converge to these periodic solutions.

Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions

1988 ◽  
Vol 110 (3-4) ◽  
pp. 183-198 ◽  
Author(s):  
R. Iannacci ◽  
M.N. Nkashama ◽  
P. Omari ◽  
F. Zanolin
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Periodic Solutions ◽  
Homogeneous Problem ◽  
A Priori ◽  
Critical Values ◽  
Liénard Equations ◽  
Jumping Nonlinearities ◽  
Existence Of Periodic Solutions ◽  
Image Position

SynopsisThis paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equationThe key assumptions relate the asymptotic behaviour as x →± ∞of g(t; x)/x to the “critical values” of the positively 1-homogeneous problemNo condition on f, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

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