Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere

Previous Article
Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order
DCDS Home
This Issue
Next Article
Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium

doi: 10.3934/dcds.2020311

Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere

Elena Nozdrinova and Olga Pochinka

25/12 Bolshaya Pecherskaya Ulitsa, Nizhny Novgorod, 603155, Russian Federation

Received April 2020 Revised June 2020 Published August 2020

Fund Project: This work was supported by the Russian Science Foundation (project 17-11-01041), except for the section 3 which is partially supported by Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant (ag. 075-15-2019-1931) and section 5, the evidence of which was supported by Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (project 19-7-1-15-1)

In the present paper, a solution to the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere is obtained. It is precisely shown that with respect to the stable isotopic connectedness relation there exists countable many of equivalence classes of such systems. 43 words.

Keywords: Stable arc, flip, saddle-node, gradient-like diffeomorphism, two-dimensional sphere.

Mathematics Subject Classification: Primary: 37B25, 37B35, 37D05, 37D15.

Citation: Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020311

References:

[1]	V. S. Afraimovich and L. P. Shilnikov, On some global bifurcations associated with the disappearance of a saddle-node fixed point, Doc. USSR Acad. Sci., 219 (1974), 1281-1284. Google Scholar
[2]	V. S. Afraimovich and L. P. Shilnikov, On small periodic disturbances of autonomous systems, Doc. USSR Acad. Sci., 214 (1974), 739-742. Google Scholar
[3]	A. Andronov and L. Pontryagin, Rough systems, Doklady Akademii Nauk SSSR, 14 (1937), 247-250. Google Scholar
[4]	A. Banyaga, The structure of the group of equivariant diffeomorphism, Topology, 16 (1977), 279-283. doi: 10.1016/0040-9383(77)90009-X. Google Scholar
[5]	A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of two-dimensional manifolds, Differential Integral Equations, (1985), 33–37. Google Scholar
[6]	P. R. Blanchard, Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces, Duke Math. J., 47 (1980), 33-46. doi: 10.1215/S0012-7094-80-04704-3. Google Scholar
[7]	S. K. Boldsen, Different versions of mapping class groups of surfaces, preprint, arXiv: 0908.2221. Google Scholar
[8]	Kh. Bonatti, V. Z. Grines, V. S. Medvedev and O. V. Pochinka, Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices, Tr. Mat. Inst. Steklova, 256 (2007), 54-69. doi: 10.1134/S0081543807010038. Google Scholar
[9]	G. Fleitas, Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat., 8 (1977), 47-51. doi: 10.1007/BF02584749. Google Scholar
[10]	J. Franks, Necessary conditions for the stability of diffeomorphisms, Trans. A. M. S., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3. Google Scholar
[11]	N. Gourmelon, A Franks' lemma that preserves invariant manifolds, Ergodic Theory Dynam. Systems, 36 (2016), 1167-1203. doi: 10.1017/etds.2014.101. Google Scholar
[12]	V. Grines, E. Gurevich, O. Pochinka and D. Malyshev, On topological classification of Morse-Smale diffeomorphisms on the sphere $S^n$, preprint, arXiv: 1911.10234. Google Scholar
[13]	V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev and O. V. Pochinka, Global attractor and repeller of Morse–Smale diffeomorphisms, Proc. Steklov Inst. Math., 271 (2010), 103-124. doi: 10.1134/S0081543810040097. Google Scholar
[14]	V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer, Cham, 2016. doi: 10.1007/978-3-319-44847-3. Google Scholar
[15]	V. Z. Grines, O. V. Pochinka and S. Van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749. doi: 10.17323/1609-4514-2016-16-4-727-749. Google Scholar
[16]	F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. Google Scholar
[17]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042. Google Scholar
[18]	B. von Kerékjártó, Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919), 36-38. doi: 10.1007/BF01463232. Google Scholar
[19]	V. I. Lukyanov and L. P. Shilnikov, On some bifurcations of dynamical systems with homoclinic structures, Doc. USSR Academy of Sciences, 243 (1978), 26-29. Google Scholar
[20]	A. G. Mayer, Rough transformation of a circle into a circle, Scientific Notes - Gorky State Univ., 12 (1939), 215-229. Google Scholar
[21]	T. V. Medvedev, E. V. Nozdrinova, O. V. Pochinka and E. V. Shadrina, On a class of isotopic connectivity of gradient-like maps of the 2-sphere with saddles of negative orientation type, Russ. J. Nonlinear Dyn., 15 (2019), 199-211. doi: 10.20537/nd190209. Google Scholar
[22]	J. Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965. Google Scholar
[23]	S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5–71. Google Scholar
[24]	S. Newhouse, J. Palis and F. Takens, Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc., 82 (1976), 499-502. doi: 10.1090/S0002-9904-1976-14073-6. Google Scholar
[25]	S. Newhouse and M. M. Peixoto, There is a simple arc joining any two Morse-Smale flows, in Trois Études en Dynamique Qualitative, Astérisque, 31, Soc. Math. France, Paris, 1976, 15–41. Google Scholar
[26]	E. V. Nozdrinova, Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle, Russ. J. Nonlinear Dyn., 14 (2018), 543-551. doi: 10.20537/nd180408. Google Scholar
[27]	E. Nozdrinova and O. Pochinka, On the existence of a smooth arc without bifurcations joining source-sink diffeomorphisms on the 2-sphere, J. Phys. Conf. Ser., 990 (2018), 1-7. doi: 10.1088/1742-6596/990/1/012010. Google Scholar
[28]	J. Palis and C. Pugh, Fifty problems in dynamical systems, in Dynamical Systems, Lecture Notes in Math., 468, Springer, Berlin, 1975,345–353. doi: 10.1007/BFb0082633. Google Scholar
[29]	D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990. Google Scholar
[30]	S. Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626. doi: 10.2307/2033664. Google Scholar
[31]	S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

show all references

References:

[1]	V. S. Afraimovich and L. P. Shilnikov, On some global bifurcations associated with the disappearance of a saddle-node fixed point, Doc. USSR Acad. Sci., 219 (1974), 1281-1284. Google Scholar
[2]	V. S. Afraimovich and L. P. Shilnikov, On small periodic disturbances of autonomous systems, Doc. USSR Acad. Sci., 214 (1974), 739-742. Google Scholar
[3]	A. Andronov and L. Pontryagin, Rough systems, Doklady Akademii Nauk SSSR, 14 (1937), 247-250. Google Scholar
[4]	A. Banyaga, The structure of the group of equivariant diffeomorphism, Topology, 16 (1977), 279-283. doi: 10.1016/0040-9383(77)90009-X. Google Scholar
[5]	A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of two-dimensional manifolds, Differential Integral Equations, (1985), 33–37. Google Scholar
[6]	P. R. Blanchard, Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces, Duke Math. J., 47 (1980), 33-46. doi: 10.1215/S0012-7094-80-04704-3. Google Scholar
[7]	S. K. Boldsen, Different versions of mapping class groups of surfaces, preprint, arXiv: 0908.2221. Google Scholar
[8]	Kh. Bonatti, V. Z. Grines, V. S. Medvedev and O. V. Pochinka, Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices, Tr. Mat. Inst. Steklova, 256 (2007), 54-69. doi: 10.1134/S0081543807010038. Google Scholar
[9]	G. Fleitas, Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat., 8 (1977), 47-51. doi: 10.1007/BF02584749. Google Scholar
[10]	J. Franks, Necessary conditions for the stability of diffeomorphisms, Trans. A. M. S., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3. Google Scholar
[11]	N. Gourmelon, A Franks' lemma that preserves invariant manifolds, Ergodic Theory Dynam. Systems, 36 (2016), 1167-1203. doi: 10.1017/etds.2014.101. Google Scholar
[12]	V. Grines, E. Gurevich, O. Pochinka and D. Malyshev, On topological classification of Morse-Smale diffeomorphisms on the sphere $S^n$, preprint, arXiv: 1911.10234. Google Scholar
[13]	V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev and O. V. Pochinka, Global attractor and repeller of Morse–Smale diffeomorphisms, Proc. Steklov Inst. Math., 271 (2010), 103-124. doi: 10.1134/S0081543810040097. Google Scholar
[14]	V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer, Cham, 2016. doi: 10.1007/978-3-319-44847-3. Google Scholar
[15]	V. Z. Grines, O. V. Pochinka and S. Van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749. doi: 10.17323/1609-4514-2016-16-4-727-749. Google Scholar
[16]	F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. Google Scholar
[17]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042. Google Scholar
[18]	B. von Kerékjártó, Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919), 36-38. doi: 10.1007/BF01463232. Google Scholar
[19]	V. I. Lukyanov and L. P. Shilnikov, On some bifurcations of dynamical systems with homoclinic structures, Doc. USSR Academy of Sciences, 243 (1978), 26-29. Google Scholar
[20]	A. G. Mayer, Rough transformation of a circle into a circle, Scientific Notes - Gorky State Univ., 12 (1939), 215-229. Google Scholar
[21]	T. V. Medvedev, E. V. Nozdrinova, O. V. Pochinka and E. V. Shadrina, On a class of isotopic connectivity of gradient-like maps of the 2-sphere with saddles of negative orientation type, Russ. J. Nonlinear Dyn., 15 (2019), 199-211. doi: 10.20537/nd190209. Google Scholar
[22]	J. Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965. Google Scholar
[23]	S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5–71. Google Scholar
[24]	S. Newhouse, J. Palis and F. Takens, Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc., 82 (1976), 499-502. doi: 10.1090/S0002-9904-1976-14073-6. Google Scholar
[25]	S. Newhouse and M. M. Peixoto, There is a simple arc joining any two Morse-Smale flows, in Trois Études en Dynamique Qualitative, Astérisque, 31, Soc. Math. France, Paris, 1976, 15–41. Google Scholar
[26]	E. V. Nozdrinova, Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle, Russ. J. Nonlinear Dyn., 14 (2018), 543-551. doi: 10.20537/nd180408. Google Scholar
[27]	E. Nozdrinova and O. Pochinka, On the existence of a smooth arc without bifurcations joining source-sink diffeomorphisms on the 2-sphere, J. Phys. Conf. Ser., 990 (2018), 1-7. doi: 10.1088/1742-6596/990/1/012010. Google Scholar
[28]	J. Palis and C. Pugh, Fifty problems in dynamical systems, in Dynamical Systems, Lecture Notes in Math., 468, Springer, Berlin, 1975,345–353. doi: 10.1007/BFb0082633. Google Scholar
[29]	D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990. Google Scholar
[30]	S. Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626. doi: 10.2307/2033664. Google Scholar
[31]	S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

Figure 1. Conjugate arcs

Download full-size image

Download as PowerPoint slide

Figure 2. Saddle-node bifurcation

Download full-size image

Download as PowerPoint slide

Figure 3. Strongly stable and unstable foliations

Download full-size image

Download as PowerPoint slide

Figure 4. Flip bifurcation

Download full-size image

Download as PowerPoint slide

Figure 5. Illustration to the proof of lemma 3.1

Download full-size image

Download as PowerPoint slide

Figure 6. An example of an attractor $ A_f $ and repeller $ R_f $ for a gradient-like diffeomorphism $ f $

Download full-size image

Download as PowerPoint slide

Figure 7. For the shown diffeomorphism, there are two ways to choose the pair $ A_f,\,R_f $: 1) $ A_f = cl\,W^u_\sigma,\,R_f = \alpha $ and 2) $ A_f = \omega\cup f(\omega),\,R_f = cl\,W^s_\sigma $

Download full-size image

Download as PowerPoint slide

Figure 8. Illustration to the lemma 3.2

Download full-size image

Download as PowerPoint slide

Figure 9. Phase portrait of diffeomorphism $ \chi_m $

Download full-size image

Download as PowerPoint slide

Figure 10. Diffeomorphism $ \phi_{1, 3} $

Download full-size image

Download as PowerPoint slide

Figure 11. Illustration to the lemma 6.3, case 1)

Download full-size image

Download as PowerPoint slide

Figure 12. Illustration to the lemma 6.3, case 2)

Download full-size image

Download as PowerPoint slide

Figure 13. A diffeomorphism $ f\in G^+ $, for which $ m_f = 1 $, $ \mu_f = 3 $

Download full-size image

Download as PowerPoint slide

Figure 14. Transition from the diffeomorphism $ f\in G_m $ to the diffeomorphism $ g\in H_{m} $

Download full-size image

Download as PowerPoint slide

Figure 15. Curve $ C_\sigma $

Download full-size image

Download as PowerPoint slide

Figure 16. Curve $ C_{\tilde \sigma} $

Download full-size image

Download as PowerPoint slide

Figure 17.

$ f\in H_{k,m} $

Download full-size image

Download as PowerPoint slide

Figure 18.

$ f\in F_{1,3} $

Download full-size image

Download as PowerPoint slide

Figure 19.

$ f \in {F_{1,3}}$

Download full-size image

Download as PowerPoint slide

Figure 20. ϕ_{1, 3}

Download full-size image

Download as PowerPoint slide

Figure 21. Illustration to the lemma 9.1, the case 1)

Download full-size image

Download as PowerPoint slide

Figure 22. Illustration to the lemma 9.1, case 1)

Download full-size image

Download as PowerPoint slide

Figure 23. Illustration to the lemma 9.1, case 2)

Download full-size image

Download as PowerPoint slide

[1]	Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
[2]	Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[3]	Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[4]	Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[5]	Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[6]	W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[7]	Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069
[8]	Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947
[9]	Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073
[10]	Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[11]	Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755
[12]	Fanzhi Chen, Michael Herrmann. KdV-like solitary waves in two-dimensional FPU-lattices. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2305-2332. doi: 10.3934/dcds.2018095
[13]	Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801
[14]	Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1341-1367. doi: 10.3934/dcdss.2020076
[15]	Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[16]	Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011
[17]	Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257
[18]	Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2303-2326. doi: 10.3934/dcdss.2020092
[19]	Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609
[20]	Luca Bisconti, Davide Catania. Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1861-1881. doi: 10.3934/cpaa.2017090

2019 Impact Factor: 1.338

Tools

Metrics

Other articles
by authors

on AIMS
Elena Nozdrinova Olga Pochinka
on Google Scholar
Elena Nozdrinova Olga Pochinka

2019 Impact Factor: 1.338

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.338

Tools

Figures and Tables

2019 Impact Factor: 1.338

Tools

Metrics

Other articles
by authors

on AIMS
Elena Nozdrinova Olga Pochinka
on Google Scholar
Elena Nozdrinova Olga Pochinka

[Back to Top]