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August  2020, 25(8): 3171-3181. doi: 10.3934/dcdsb.2020056

Global attraction in a system of delay differential equations via compact and convex sets

Departamento de Estatística, Análise Matemática e Optimización and Instituto de Matemáticas, Universidade de Santiago de Compostela, Facultade de Matemáticas, Campus Vida, 15782 Santiago de Compostela, Spain

Received  May 2019 Published  February 2020

We provide sufficient conditions for a concrete type of systems of delay differential equations (DDEs) to have a global attractor. The principal idea is based on a particular type of global attraction in difference equations in terms of nested, convex and compact sets. We prove that the solutions of the system of DDEs inherit the convergence to the equilibrium from an associated discrete dynamical system.

Citation: Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056
References:
[1]

H. A. El-Morshedy and V. Jiménez López, Global attractors for difference equations dominated by one-dimensional maps, J. Difference Equ. Appl., 14 (2008), 391-410.  doi: 10.1080/10236190701671632.  Google Scholar

[2]

H. A. El-Morshedy and A. Ruiz-Herrera, Geometric methods of global attraction in systems of delay differential equations, J. Differential Equations, 263 (2017), 5968-5986.  doi: 10.1016/j.jde.2017.07.001.  Google Scholar

[3]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

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A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in Dynam. Report. Expositions Dynam. Systems (N.S.) (eds. C. K. R. T. Jones, U. Kirchgraber and H.-O. Walther), Springer, Berlin, 1 (1992), 164–224.  Google Scholar

[5]

E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[6]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266.  doi: 10.1016/j.jde.2013.08.007.  Google Scholar

[7]

E. Liz and A. Ruiz-Herrera, Addendum to Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback" [J. Differential Equations, 255 (2013), 4244{4266], J. Differential Equations, 257 (2014), 1307{1309. doi: 10.1016/j.jde.2014.05.010.  Google Scholar

[8]

E. Liz and A. Ruiz-Herrera, Global dynamics of delay equations for populations with competition among immature individuals, J. Differential Equations, 260 (2016), 5926-5955.  doi: 10.1016/j.jde.2015.12.020.  Google Scholar

[9]

J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128.  doi: 10.1007/BF01790539.  Google Scholar

[10]

F. A. Valentine, Convex Sets, Robert E. Krieger Publishing Co., New York, 1976.  Google Scholar

[11]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.  Google Scholar

show all references

References:
[1]

H. A. El-Morshedy and V. Jiménez López, Global attractors for difference equations dominated by one-dimensional maps, J. Difference Equ. Appl., 14 (2008), 391-410.  doi: 10.1080/10236190701671632.  Google Scholar

[2]

H. A. El-Morshedy and A. Ruiz-Herrera, Geometric methods of global attraction in systems of delay differential equations, J. Differential Equations, 263 (2017), 5968-5986.  doi: 10.1016/j.jde.2017.07.001.  Google Scholar

[3]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[4]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in Dynam. Report. Expositions Dynam. Systems (N.S.) (eds. C. K. R. T. Jones, U. Kirchgraber and H.-O. Walther), Springer, Berlin, 1 (1992), 164–224.  Google Scholar

[5]

E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[6]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266.  doi: 10.1016/j.jde.2013.08.007.  Google Scholar

[7]

E. Liz and A. Ruiz-Herrera, Addendum to Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback" [J. Differential Equations, 255 (2013), 4244{4266], J. Differential Equations, 257 (2014), 1307{1309. doi: 10.1016/j.jde.2014.05.010.  Google Scholar

[8]

E. Liz and A. Ruiz-Herrera, Global dynamics of delay equations for populations with competition among immature individuals, J. Differential Equations, 260 (2016), 5926-5955.  doi: 10.1016/j.jde.2015.12.020.  Google Scholar

[9]

J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128.  doi: 10.1007/BF01790539.  Google Scholar

[10]

F. A. Valentine, Convex Sets, Robert E. Krieger Publishing Co., New York, 1976.  Google Scholar

[11]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.  Google Scholar

Figure 1.  The origin is not a strong attractor
Figure 2.  A possible set $ Q_{x, v, \varepsilon} $ is represented in gray. Distances are pointed out with dashed green lines. A particular $ v^* $ satisfying the hypotheses of the last assertion of Lemma 3.1 is also depicted (color figure online)
Figure 3.  Possible behaviour of $ x(t, \phi) $ (blue). The boundary of the set $ K_2 $ is represented in black. The boundaries of $ K_{2, \mu} $, for some values $ \mu>1 $ are represented in grey. The blue arrow represents $ x'(t, \phi) $, which "points to the interior" of a $ K_{2, \mu} $. $ f(K_2) $ is represented in red. The equilibrium $ z_* $ is depicted as a point inside $ f(K_2) $ (color figure online)
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