American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros
  • DCDS Home
  • This Issue
  • Next Article
    Impulsive control of conservative periodic equations and systems: Variational approach
August  2018, 38(8): 3803-3829. doi: 10.3934/dcds.2018165

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

Hui Liu a, , Yiming Long b,d, and  Yuming Xiao c,

a. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
b. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
c. 

School of Mathematics, Sichuan University, Chengdu 610064, China
d. 

Beijing Advanced Innovation Center for Imaging Technology, Capital Normal University, Beijing 100048, China

1 Partially supported by NSFC (Nos. 11401555, 11771341), Anhui Provincial Natural Science Foundation (No. 1608085QA01)
2 Partially supported by NSFC (Nos. 11131004, 11671215 and 11790271), MCME and LPMC of MOE of China, Nankai University and BAICIT of Capital Normal University
3 Corresponding author: Yuming Xiao. Supported by the Sichuan Science and Technology Program (No. 2018JY0140)

Received  July 2017 Revised  February 2018 Published  May 2018

Let $M = S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n≥2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathbb{R}P^n$ in [25] to general compact space forms.

Keywords: Non-contractible closed geodesics, resonance identity, compact space forms, morse theory, index iteration theory, systems of irrational numbers.
Mathematics Subject Classification: 53C22, 58E05, 58E10.
Citation: Hui Liu, Yiming Long, Yuming Xiao. The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3803-3829. doi: 10.3934/dcds.2018165
References:
[1]

D. V. Anosov, Geodesics in Finsler geometry, Proc. I. C. M. (Vancouver, B. C. 1974), 2 (1975), 293-297; Montreal (Russian), Amer. Math. Soc. Transl., 109 (1977), 81-85.  Google Scholar

[2]

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7.  Google Scholar

[3]

V. Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993), 1-10.  doi: 10.1142/S0129167X93000029.  Google Scholar

[4]

V. Bangert and W. Klingenberg, Homology generated by iterated closed geodesics, Topology., 22 (1983), 379-388.  doi: 10.1016/0040-9383(83)90033-2.  Google Scholar

[5]

V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346 (2010), 335-366. doi: 10.1007/s00208-009-0401-1.  Google Scholar

[6]

R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 171-206.  doi: 10.1002/cpa.3160090204.  Google Scholar

[7]

K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014. Google Scholar

[8]

H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler n-spheres, J. Diff. Equa., 233 (2007), 221-240.  doi: 10.1016/j.jde.2006.10.002.  Google Scholar

[9]

H. Duan and Y. Long, The index growth and multiplicity of closed geodesics, J. Funct. Anal., 259 (2010), 1850-1913.  doi: 10.1016/j.jfa.2010.05.003.  Google Scholar

[10]

H. Duan, Y. Long and W. Wang, Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differ. Geom., 104 (2016), 275-289.  doi: 10.4310/jdg/1476367058.  Google Scholar

[11]

H. Duan, Y. Long and W. Wang, The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calc. Var. and PDEs., 55 (2016), Art. 145, 28 pp. doi: 10.1007/s00526-016-1075-7.  Google Scholar

[12]

H. Duan, Y. Long and Y. Xiao, Two closed geodesics on $\mathbb{R}P$ with a bumpy Finsler metric, Calc. Var. and PDEs, 54 (2015), 2883-2894.  doi: 10.1007/s00526-015-0887-1.  Google Scholar

[13]

J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418.  doi: 10.1007/BF02100612.  Google Scholar

[14]

D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology, 8 (1969), 361-369.  doi: 10.1016/0040-9383(69)90022-6.  Google Scholar

[15]

D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969), 493-510.  doi: 10.4310/jdg/1214429070.  Google Scholar

[16]

N. Hingston, Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984), 85-116.  doi: 10.4310/jdg/1214438424.  Google Scholar

[17]

N. Hingston, On the growth of the number of closed geodesics on the two-sphere, Inter. Math. Research Notices., 9 (1993), 253-262.  doi: 10.1155/S1073792893000285.  Google Scholar

[18]

N. Hingston and H.-B. Rademacher, Resonance for loop homology of spheres, J. Differ. Geom., 93 (2013), 133-174.  doi: 10.4310/jdg/1357141508.  Google Scholar

[19]

A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk. SSSR, 37 (1973), 539-576; [Russian]; Math. USSR-Izv., 7 (1973), 535-571.  Google Scholar

[20]

W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978.  Google Scholar

[21]

W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120.  Google Scholar

[22]

C. Liu, The relation of the morse index of closed geodesics with the maslov-type index of symplectic paths, Acta Math. Sinica, 21 (2005), 237-248.  doi: 10.1007/s10114-004-0406-3.  Google Scholar

[23]

C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002), 9-28.   Google Scholar

[24]

H. Liu, The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, J. Differential Equations, 262 (2017), 2540-2553.  doi: 10.1016/j.jde.2016.11.015.  Google Scholar

[25]

H. Liu and Y. Xiao, Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Advances in Math., 318 (2017), 158-190.  doi: 10.1016/j.aim.2017.07.024.  Google Scholar

[26]

Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.  Google Scholar

[27]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914.  Google Scholar

[28]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[29]

Y. Long, Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc., 8 (2006), 341-353.  doi: 10.4171/JEMS/56.  Google Scholar

[30]

Y. Long and H. Duan, Multiple closed geodesics on 3-spheres, Advances in Math., 221 (2009), 1757-1803.  doi: 10.1016/j.aim.2009.03.007.  Google Scholar

[31]

Y. Long and W. Wang, Multiple closed geodesics on Riemannian 3-spheres, Calc. Var. and PDEs, 30 (2007), 183-214.  doi: 10.1007/s00526-006-0083-4.  Google Scholar

[32]

A. Oancea, Morse theory, closed geodesics, and the homology of free loop spaces, With an appendix by Umberto Hryniewicz. IRMA Lect. Math. Theor. Phys., 24, Free loop spaces in geometry and topology, 67-109, Eur. Math. Soc., Zürich, 2015. arXiv: 1406.3107, 2014.  Google Scholar

[33]

H.-B. Rademacher, On the average indices of closed geodesics, J. Diff. Geom., 29 (1989), 65-83.  doi: 10.4310/jdg/1214442633.  Google Scholar

[34]

H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992.  Google Scholar

[35]

H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems., 27 (2007), 957-969.  doi: 10.1017/S0143385706001064.  Google Scholar

[36]

H.-B. Rademacher, The second closed geodesic on Finsler spheres of dimension n>2, Trans. Amer. Math. Soc., 362 (2010), 1413-1421.  doi: 10.1090/S0002-9947-09-04745-X.  Google Scholar

[37]

Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622.  Google Scholar

[38]

I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010), 84-100.  doi: 10.1134/S1560354710010053.  Google Scholar

[39]

I. A. Taimanov, The spaces of non-contractible closed curves in compact space forms, Mat. Sb., 207 (2016), 105-118.  doi: 10.4213/sm8708.  Google Scholar

[40]

M. Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J. Diff. Geom., 11 (1976), 663-644.  doi: 10.4310/jdg/1214433729.  Google Scholar

[41]

W. Wang, Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008), 1566-1603.  doi: 10.1016/j.aim.2008.03.018.  Google Scholar

[42]

W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006.  Google Scholar

[43]

C. Westerland, Dyer-Lashof operations in the string topology of spheres and projective spaces, Math. Z., 250 (2005), 711-727.  doi: 10.1007/s00209-005-0778-9.  Google Scholar

[44]

C. Westerland, String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007), 309-325.  doi: 10.2140/agt.2007.7.309.  Google Scholar

[45]

Y. Xiao and Y. Long, Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions, Advances in Math., 279 (2015), 159-200.  doi: 10.1016/j.aim.2015.03.013.  Google Scholar

[46]

W. Ziller, The free loop space of globally symmetric spaces, Invent. Math., 41 (1977), 1-22.  doi: 10.1007/BF01390161.  Google Scholar

[47]

W. Ziller, Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar

show all references

References:
[1]

D. V. Anosov, Geodesics in Finsler geometry, Proc. I. C. M. (Vancouver, B. C. 1974), 2 (1975), 293-297; Montreal (Russian), Amer. Math. Soc. Transl., 109 (1977), 81-85.  Google Scholar

[2]

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7.  Google Scholar

[3]

V. Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993), 1-10.  doi: 10.1142/S0129167X93000029.  Google Scholar

[4]

V. Bangert and W. Klingenberg, Homology generated by iterated closed geodesics, Topology., 22 (1983), 379-388.  doi: 10.1016/0040-9383(83)90033-2.  Google Scholar

[5]

V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346 (2010), 335-366. doi: 10.1007/s00208-009-0401-1.  Google Scholar

[6]

R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 171-206.  doi: 10.1002/cpa.3160090204.  Google Scholar

[7]

K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014. Google Scholar

[8]

H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler n-spheres, J. Diff. Equa., 233 (2007), 221-240.  doi: 10.1016/j.jde.2006.10.002.  Google Scholar

[9]

H. Duan and Y. Long, The index growth and multiplicity of closed geodesics, J. Funct. Anal., 259 (2010), 1850-1913.  doi: 10.1016/j.jfa.2010.05.003.  Google Scholar

[10]

H. Duan, Y. Long and W. Wang, Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differ. Geom., 104 (2016), 275-289.  doi: 10.4310/jdg/1476367058.  Google Scholar

[11]

H. Duan, Y. Long and W. Wang, The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calc. Var. and PDEs., 55 (2016), Art. 145, 28 pp. doi: 10.1007/s00526-016-1075-7.  Google Scholar

[12]

H. Duan, Y. Long and Y. Xiao, Two closed geodesics on $\mathbb{R}P$ with a bumpy Finsler metric, Calc. Var. and PDEs, 54 (2015), 2883-2894.  doi: 10.1007/s00526-015-0887-1.  Google Scholar

[13]

J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418.  doi: 10.1007/BF02100612.  Google Scholar

[14]

D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology, 8 (1969), 361-369.  doi: 10.1016/0040-9383(69)90022-6.  Google Scholar

[15]

D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969), 493-510.  doi: 10.4310/jdg/1214429070.  Google Scholar

[16]

N. Hingston, Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984), 85-116.  doi: 10.4310/jdg/1214438424.  Google Scholar

[17]

N. Hingston, On the growth of the number of closed geodesics on the two-sphere, Inter. Math. Research Notices., 9 (1993), 253-262.  doi: 10.1155/S1073792893000285.  Google Scholar

[18]

N. Hingston and H.-B. Rademacher, Resonance for loop homology of spheres, J. Differ. Geom., 93 (2013), 133-174.  doi: 10.4310/jdg/1357141508.  Google Scholar

[19]

A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk. SSSR, 37 (1973), 539-576; [Russian]; Math. USSR-Izv., 7 (1973), 535-571.  Google Scholar

[20]

W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978.  Google Scholar

[21]

W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120.  Google Scholar

[22]

C. Liu, The relation of the morse index of closed geodesics with the maslov-type index of symplectic paths, Acta Math. Sinica, 21 (2005), 237-248.  doi: 10.1007/s10114-004-0406-3.  Google Scholar

[23]

C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002), 9-28.   Google Scholar

[24]

H. Liu, The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, J. Differential Equations, 262 (2017), 2540-2553.  doi: 10.1016/j.jde.2016.11.015.  Google Scholar

[25]

H. Liu and Y. Xiao, Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Advances in Math., 318 (2017), 158-190.  doi: 10.1016/j.aim.2017.07.024.  Google Scholar

[26]

Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.  Google Scholar

[27]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914.  Google Scholar

[28]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[29]

Y. Long, Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc., 8 (2006), 341-353.  doi: 10.4171/JEMS/56.  Google Scholar

[30]

Y. Long and H. Duan, Multiple closed geodesics on 3-spheres, Advances in Math., 221 (2009), 1757-1803.  doi: 10.1016/j.aim.2009.03.007.  Google Scholar

[31]

Y. Long and W. Wang, Multiple closed geodesics on Riemannian 3-spheres, Calc. Var. and PDEs, 30 (2007), 183-214.  doi: 10.1007/s00526-006-0083-4.  Google Scholar

[32]

A. Oancea, Morse theory, closed geodesics, and the homology of free loop spaces, With an appendix by Umberto Hryniewicz. IRMA Lect. Math. Theor. Phys., 24, Free loop spaces in geometry and topology, 67-109, Eur. Math. Soc., Zürich, 2015. arXiv: 1406.3107, 2014.  Google Scholar

[33]

H.-B. Rademacher, On the average indices of closed geodesics, J. Diff. Geom., 29 (1989), 65-83.  doi: 10.4310/jdg/1214442633.  Google Scholar

[34]

H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992.  Google Scholar

[35]

H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems., 27 (2007), 957-969.  doi: 10.1017/S0143385706001064.  Google Scholar

[36]

H.-B. Rademacher, The second closed geodesic on Finsler spheres of dimension n>2, Trans. Amer. Math. Soc., 362 (2010), 1413-1421.  doi: 10.1090/S0002-9947-09-04745-X.  Google Scholar

[37]

Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622.  Google Scholar

[38]

I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010), 84-100.  doi: 10.1134/S1560354710010053.  Google Scholar

[39]

I. A. Taimanov, The spaces of non-contractible closed curves in compact space forms, Mat. Sb., 207 (2016), 105-118.  doi: 10.4213/sm8708.  Google Scholar

[40]

M. Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J. Diff. Geom., 11 (1976), 663-644.  doi: 10.4310/jdg/1214433729.  Google Scholar

[41]

W. Wang, Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008), 1566-1603.  doi: 10.1016/j.aim.2008.03.018.  Google Scholar

[42]

W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006.  Google Scholar

[43]

C. Westerland, Dyer-Lashof operations in the string topology of spheres and projective spaces, Math. Z., 250 (2005), 711-727.  doi: 10.1007/s00209-005-0778-9.  Google Scholar

[44]

C. Westerland, String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007), 309-325.  doi: 10.2140/agt.2007.7.309.  Google Scholar

[45]

Y. Xiao and Y. Long, Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions, Advances in Math., 279 (2015), 159-200.  doi: 10.1016/j.aim.2015.03.013.  Google Scholar

[46]

W. Ziller, The free loop space of globally symmetric spaces, Invent. Math., 41 (1977), 1-22.  doi: 10.1007/BF01390161.  Google Scholar

[47]

W. Ziller, Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar

[1]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[3]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449
[4]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[5]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[6]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[7]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[8]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[9]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[10]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[11]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363
[12]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381
[13]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018
[14]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[15]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434
[16]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[17]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[18]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[19]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383
[20]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (96)
  • HTML views (138)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences