# American Institute of Mathematical Sciences

March  2008, 1(1): 15-25. doi: 10.3934/dcdss.2008.1.15

## Counting uniformly attracting solutions of nonautonomous differential equations

 1 Department of Mathematics and Statistics, University of Canterbury, Christchurch

Received  September 2006 Revised  August 2007 Published  December 2007

Bounded uniform attractors and repellors are the natural nonautonomous analogues of autonomous stable and unstable equilibria. Unlike for equilibria, it is generally a difficult dynamical task to determine the number of uniformly attracting or repelling solutions for a given nonautonomous equation, even if the latter exhibits strong structural properties such as e.g. polynomial growth in space or periodicity in time. The present note highlights this aspect by proving that the number of uniform attractors is locally finite for several classes of equations, and by providing examples for which this number can be any $N\in \N$. These results and examples extend and complement recent work on nonautonomous differential equations.
Citation: Arno Berger. Counting uniformly attracting solutions of nonautonomous differential equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 15-25. doi: 10.3934/dcdss.2008.1.15
 [1] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure and Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [2] Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663 [3] Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 [4] Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013 [5] Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589 [6] Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899 [7] Mustapha Yebdri. Existence of $\mathcal{D}-$pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036 [8] Antonio Garijo, Xavier Jarque. The secant map applied to a real polynomial with multiple roots. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6783-6794. doi: 10.3934/dcds.2020133 [9] Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 [10] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215 [11] Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190 [12] Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 [13] Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727 [14] David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96 [15] Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065 [16] Qiyuan Wei, Liwei Zhang. An accelerated differential equation system for generalized equations. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021195 [17] Victor Kozyakin. Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587 [18] Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259 [19] M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129 [20] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

2020 Impact Factor: 2.425