American Institute of Mathematical Sciences

Advanced
October  2006, 14(4): 721-736. doi: 10.3934/dcds.2006.14.721

Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof

A. Aschwanden 1, , A. Schulze-Halberg 1, and  D. Stoffer 1,

1. 

Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland, Switzerland, Switzerland

Received  October 2004 Revised  July 2005 Published  January 2006

We study the delay equation $\dot{x}(t)=-\mu x(t)+f(x(t-1))$ with $\mu>0$ and a nonmonotone $C^1$-function $f$ obeying $x f(x)>0$ (positive feedback) outside a small neighbourhood of zero. By means of a computer-assisted method we prove the existence of asymptotically orbitally stable periodic solutions. The main idea behind our proof is the reduction of the infinite-dimensional dynamics to a finite-dimensional map. In particular, for two classes of nonlinearities $f$ we construct two types of solutions, the dynamics of which is reduced to a one- and a two-dimensional map, respectively.
Keywords: Stable periodic solution, computer-assisted proof., interval arithmetic.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721
[1]

Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164
[2]

Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045
[3]

István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003
[4]

Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95
[5]

Bao Qing Hu, Song Wang. A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers. Journal of Industrial & Management Optimization, 2006, 2 (4) : 351-371. doi: 10.3934/jimo.2006.2.351
[6]

Xavier Cabré. A new proof of the boundedness results for stable solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7249-7264. doi: 10.3934/dcds.2019302
[7]

Song Wang. Numerical solution of an obstacle problem with interval coefficients. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019030
[8]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080
[9]

Carl. T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin. Finding a stable solution of a system of nonlinear equations arising from dynamic systems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 497-521. doi: 10.3934/jimo.2011.7.497
[10]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015
[11]

E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207
[12]

Josep M. Olm, Xavier Ros-Oton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 197-215. doi: 10.3934/dcdsb.2011.15.197
[13]

Raoul-Martin Memmesheimer, Marc Timme. Stable and unstable periodic orbits in complex networks of spiking neurons with delays. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1555-1588. doi: 10.3934/dcds.2010.28.1555
[14]

Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157
[15]

Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177
[16]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[17]

Maicon Sônego. Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5981-5988. doi: 10.3934/dcdsb.2019116
[18]

Tanja Eisner, Rainer Nagel. Arithmetic progressions -- an operator theoretic view. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 657-667. doi: 10.3934/dcdss.2013.6.657
[19]

Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186
[20]

Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences