American Institute of Mathematical Sciences

Advanced
January  1999, 5(1): 61-82. doi: 10.3934/dcds.1999.5.61

Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$

Peter Dormayer 1, and  Anatoli F. Ivanov 2,

1. 

Mathematisches Institut der Universität Giessen, Arndtstrasse 2, 35392 Giessen, Germany
2. 

Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, United States

Received  September 1997 Revised  October 1998 Published  October 1998

We study special symmetric periodic solutions of the equation

$\dot x(t) =\alphaf(x(t), x(t-1))$

where $\alpha$ is a positive parameter and the nonlinearity $f$ satisfies the symmetry conditions $f(-u, v) = -f(u,-v) = f(u, v).$ We establish the existence and stability properties for such periodic solutions with small amplitude.

Keywords: Symmetric periodic solutions, floquet multipliers, bifurcations..
Mathematics Subject Classification: 34.
Citation: Peter Dormayer, Anatoli F. Ivanov. Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 61-82. doi: 10.3934/dcds.1999.5.61
[1]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047
[2]

Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573
[3]

Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795
[4]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[5]

Yuta Ishii. Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2965-3031. doi: 10.3934/cpaa.2020130
[6]

Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709
[7]

Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997
[8]

Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677
[9]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ Ⅱ: Discrete torus bifurcations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1847-1874. doi: 10.3934/cpaa.2020081
[10]

Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756
[11]

P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure & Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161
[12]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[13]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923
[14]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[15]

Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[16]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523
[17]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080
[18]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[19]

Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629
[20]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences