June  2022, 42(6): 2585-2601. doi: 10.3934/dcds.2021204

Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

1. 

E.T.S. Ingenieros Informáticos, Universidad Politécnica de Madrid, Madrid 28660, Spain

2. 

Facultad de Ciencias Matemáticas and Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, Madrid 28040, Spain

* Corresponding author: Héctor Barge

Affectionally dedicated to María Jesús Chasco on the ocassion of her 65th birthday

Received  July 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

Fund Project: The authors are supported by Spanish Ministerio de Ciencia e Innovación grant PGC2018-098321-B-I00

In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in $ n $-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.

Citation: Héctor Barge, José M. R. Sanjurjo. Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2585-2601. doi: 10.3934/dcds.2021204
References:
[1]

E. Akin, M. Hurley and J. A. Kennedy, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc., 164 (2003), viii+130 pp. doi: 10.1090/memo/0783.

[2]

H. BargeA. Giraldo and J. M. R. Sanjurjo, Bifurcations, robustness and shape of attractors of discrete dynamical systems, J. Fixed Point Theory Appl., 22 (2020), 1-13.  doi: 10.1007/s11784-020-0770-3.

[3]

H. Barge and J. M. R. Sanjurjo, Dissipative flows, global attractors and shape theory, Topology Appl., 258 (2019), 392-401.  doi: 10.1016/j.topol.2019.03.011.

[4]

K. Borsuk, Theory of Shape, Monografie Matematyczne 59. Polish Scientific Publishers, Warsaw, 1975.

[5]

R. J. Daverman and G. A. Venema, Embeddings in Manifolds, Graduate Studies in Mathematics, 106. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/106.

[6]

C. H. Edwards, Concentricity in 3-manifolds, Trans. Amer. Math. Soc., 113 (1964), 406-423.  doi: 10.1090/S0002-9947-1964-0178459-X.

[7]

J. Franks and D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc., 352 (2000), 3305-3322.  doi: 10.1090/S0002-9947-00-02488-0.

[8]

B. Günther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc., 119 (1993), 321-329.  doi: 10.1090/S0002-9939-1993-1170545-4.

[9] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. 
[10]

L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242.  doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W.

[11]

D. S. Li and Z. Q. Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J., 67 (2018), 583-621.  doi: 10.1512/iumj.2018.67.7292.

[12]

E. E. Moise, Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics, Vol. 47. Springer-Verlag, New York-Heidelberg, 1977.

[13]

J. Naimark, Motions close to doubly asymptotic motions, Dokl. Akad. Nauk SSSR, 172 (1967), 1021-1024. 

[14]

J. C. Robinson, Global attractors: Topology and finite-dimensional dynamics, J. Dynam. Differential Equations, 11 (1999), 557-581.  doi: 10.1023/A:1021918004832.

[15]

C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer-Verlag, New York, 1972.

[16]

D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.

[17]

F. R. Ruiz del Portal and J. J. Sánchez-Gabites, Čech cohomology of attractors of discrete dynamical systems, J. Differential Equations, 257 (2014), 2826-2845.  doi: 10.1016/j.jde.2014.05.055.

[18]

R. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Chapter Ⅱ. Bifurcation-mapping method, J. Difference Equ. Appl., 15 (2009), 759-774.  doi: 10.1080/10236190802357735.

[19]

D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3.

[20]

J. J. Sánchez-Gabites, Dynamical systems and shapes, Rev. R. Acad. Cien. Serie A. Mat., 102 (2008), 127-159.  doi: 10.1007/BF03191815.

[21]

J. J. Sánchez-Gabites, Aplicaciones de Topología Geométrica y Algebraica al Estudio de Flujos Continuos en Variedades, PhD thesis, Universidad Complutense de Madrid, 2009.

[22]

J. J. Sánchez-Gabites, How strange can an attractor for a dynamical system in a 3-manifold look?, Nonlinear Anal., 74 (2011), 6162-6185.  doi: 10.1016/j.na.2011.05.095.

[23]

J. J. Sánchez-Gabites, Arcs, balls and spheres that cannot be attractors in $\mathbb{R}^3$, Trans. Amer. Math. Soc., 368 (2016), 3591-3627.  doi: 10.1090/tran/6570.

[24]

J. M. R. Sanjurjo, Multihomotopy, Čech spaces of loops and shape groups, Proc. London Math. Soc., 69 (1994), 330-344.  doi: 10.1112/plms/s3-69.2.330.

[25]

J. M. R. Sanjurjo, Global topological properties of the Hopf bifurcation, J. Differential Equations, 243 (2007), 238-255.  doi: 10.1016/j.jde.2007.05.001.

[26]

P. Seibert and J. S. Florio, On the foundations of bifurcation theory, Nonlinear Anal., 22 (1994), 927-944.  doi: 10.1016/0362-546X(94)90058-2.

[27]

E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

[28]

J. WangD. Li and J. Duan, On the shape Conley index theory of semiflows on complete metric spaces, Disc. Cont. Dyn. Sys., 36 (2016), 1629-1647.  doi: 10.3934/dcds.2016.36.1629.

show all references

References:
[1]

E. Akin, M. Hurley and J. A. Kennedy, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc., 164 (2003), viii+130 pp. doi: 10.1090/memo/0783.

[2]

H. BargeA. Giraldo and J. M. R. Sanjurjo, Bifurcations, robustness and shape of attractors of discrete dynamical systems, J. Fixed Point Theory Appl., 22 (2020), 1-13.  doi: 10.1007/s11784-020-0770-3.

[3]

H. Barge and J. M. R. Sanjurjo, Dissipative flows, global attractors and shape theory, Topology Appl., 258 (2019), 392-401.  doi: 10.1016/j.topol.2019.03.011.

[4]

K. Borsuk, Theory of Shape, Monografie Matematyczne 59. Polish Scientific Publishers, Warsaw, 1975.

[5]

R. J. Daverman and G. A. Venema, Embeddings in Manifolds, Graduate Studies in Mathematics, 106. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/106.

[6]

C. H. Edwards, Concentricity in 3-manifolds, Trans. Amer. Math. Soc., 113 (1964), 406-423.  doi: 10.1090/S0002-9947-1964-0178459-X.

[7]

J. Franks and D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc., 352 (2000), 3305-3322.  doi: 10.1090/S0002-9947-00-02488-0.

[8]

B. Günther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc., 119 (1993), 321-329.  doi: 10.1090/S0002-9939-1993-1170545-4.

[9] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. 
[10]

L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242.  doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W.

[11]

D. S. Li and Z. Q. Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J., 67 (2018), 583-621.  doi: 10.1512/iumj.2018.67.7292.

[12]

E. E. Moise, Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics, Vol. 47. Springer-Verlag, New York-Heidelberg, 1977.

[13]

J. Naimark, Motions close to doubly asymptotic motions, Dokl. Akad. Nauk SSSR, 172 (1967), 1021-1024. 

[14]

J. C. Robinson, Global attractors: Topology and finite-dimensional dynamics, J. Dynam. Differential Equations, 11 (1999), 557-581.  doi: 10.1023/A:1021918004832.

[15]

C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer-Verlag, New York, 1972.

[16]

D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.

[17]

F. R. Ruiz del Portal and J. J. Sánchez-Gabites, Čech cohomology of attractors of discrete dynamical systems, J. Differential Equations, 257 (2014), 2826-2845.  doi: 10.1016/j.jde.2014.05.055.

[18]

R. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Chapter Ⅱ. Bifurcation-mapping method, J. Difference Equ. Appl., 15 (2009), 759-774.  doi: 10.1080/10236190802357735.

[19]

D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3.

[20]

J. J. Sánchez-Gabites, Dynamical systems and shapes, Rev. R. Acad. Cien. Serie A. Mat., 102 (2008), 127-159.  doi: 10.1007/BF03191815.

[21]

J. J. Sánchez-Gabites, Aplicaciones de Topología Geométrica y Algebraica al Estudio de Flujos Continuos en Variedades, PhD thesis, Universidad Complutense de Madrid, 2009.

[22]

J. J. Sánchez-Gabites, How strange can an attractor for a dynamical system in a 3-manifold look?, Nonlinear Anal., 74 (2011), 6162-6185.  doi: 10.1016/j.na.2011.05.095.

[23]

J. J. Sánchez-Gabites, Arcs, balls and spheres that cannot be attractors in $\mathbb{R}^3$, Trans. Amer. Math. Soc., 368 (2016), 3591-3627.  doi: 10.1090/tran/6570.

[24]

J. M. R. Sanjurjo, Multihomotopy, Čech spaces of loops and shape groups, Proc. London Math. Soc., 69 (1994), 330-344.  doi: 10.1112/plms/s3-69.2.330.

[25]

J. M. R. Sanjurjo, Global topological properties of the Hopf bifurcation, J. Differential Equations, 243 (2007), 238-255.  doi: 10.1016/j.jde.2007.05.001.

[26]

P. Seibert and J. S. Florio, On the foundations of bifurcation theory, Nonlinear Anal., 22 (1994), 927-944.  doi: 10.1016/0362-546X(94)90058-2.

[27]

E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

[28]

J. WangD. Li and J. Duan, On the shape Conley index theory of semiflows on complete metric spaces, Disc. Cont. Dyn. Sys., 36 (2016), 1629-1647.  doi: 10.3934/dcds.2016.36.1629.

Figure 1.  Construction of the dyadic solenoid as an attractor of a homeomorphism of $ \mathbb{R}^3 $
Figure 2.  Construction of the Whitehead continuum as an attractor of a homeomorphism of $ \mathbb{R}^3 $
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