# American Institute of Mathematical Sciences

• Previous Article
Scattering results for Dirac Hartree-type equations with small initial data
• CPAA Home
• This Issue
• Next Article
Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $\mathbb{R} ^{3}$
July  2019, 18(4): 1695-1709. doi: 10.3934/cpaa.2019080

## Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium

 1 IMAS – CONICET, Universidad de Buenos Aires, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria - Pabellón I - (C1428EGA), Buenos Aires, Argentina 2 Universidad de Chile, Departamento de Matemáticas, Facultad de Ciencias, Casilla 653, Santiago, Chile

* Corresponding author

Received  May 2018 Revised  August 2018 Published  January 2019

Fund Project: The first author is supported by projects CONICET PIP 11220130100006CO and UBACyT 20020160100002BA.

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset \mathbb{R}^{N}$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the Euler characteristic of $\Omega$. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.

Citation: Pablo Amster, Mariel Paula Kuna, Gonzalo Robledo. Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1695-1709. doi: 10.3934/cpaa.2019080
##### References:

show all references

##### References:
 [1] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [2] Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013 [3] Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517 [4] Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861 [5] Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765 [6] 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 [7] Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059 [8] Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222 [9] Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827 [10] Shangbing Ai. Multiple positive periodic solutions for a delay host macroparasite model. Communications on Pure & Applied Analysis, 2004, 3 (2) : 175-182. doi: 10.3934/cpaa.2004.3.175 [11] Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801 [12] Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301 [13] Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264 [14] Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031 [15] Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105 [16] Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529 [17] Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157 [18] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 [19] Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure & Applied Analysis, 2007, 6 (1) : 103-111. doi: 10.3934/cpaa.2007.6.103 [20] Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001

2020 Impact Factor: 1.916