# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021114
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Families of vector fields with many numerical invariants

 University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, ON, L5L 1C6, Canada

Received  March 2021 Revised  May 2021 Early access August 2021

Fund Project: Both authors are partially supported by RFBR grant No. 20-01-00420

We study bifurcations in finite-parameter families of vector fields on $S^2$. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable $3$-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of $(2D+1)$-parameter families such that the topological classification of these families has at least $D$ numerical invariants and used those examples to construct families with functional invariants of topological classification.

In this paper, we construct locally generic $4$-parameter families with any prescribed number of numerical invariants and use them to construct $5$-parameter families with functional invariants. We also describe a locally generic class of $3$-parameter families with a tail of an infinite number sequence as an invariant of topological classification.

Citation: Nataliya Goncharuk, Yury Kudryashov. Families of vector fields with many numerical invariants. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021114
##### References:
 [1] V. Afraimovich and T. Young, Relative density of irrational rotation numbers in families of circle diffeomorphisms, Ergod. Th. and Dynam. Sys., 18 (1998), 1-16.  doi: 10.1017/S0143385798097648.  Google Scholar [2] A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251.   Google Scholar [3] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. ${\rm Ma}\breve{\rm{i}}{\rm er}$, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press [A division of John Wiley & Sons] and Israel Program for Scientific Translations, 1973.  Google Scholar [4] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Ma$\breve{\rm{i}}$er, Theory of Bifurcations of Dynamic Systems on A Plane, Halsted Press [A division of John Wiley & Sons] and Israel Program for Scientific Translations, 1973.  Google Scholar [5] V. I. Arnold, V. S. Afrajmovich, Yu. S. Ilyashenko and L. P. Shilnikov, Dynamical Systems V. Bifurcation Theory and Catastrophe Theory, Encyclopaedia of Mathematical Sciences, Springer-Verlag Berlin Heidelberg, 1994. Available from: http://www.springer.com/gp/book/9783540181736. Google Scholar [6] S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.  Google Scholar [7] H. F. DeBaggis, Dynamical systems with stable structures, Contributions to the Theory of Nonlinear Oscillations, 2 (1952), 37–60. Available from: http://www.jstor.org/stable/j.ctt1bgz9z7.6.  Google Scholar [8] A. V. Dukov, Functional invariants in generic semilocal families of vector fields on two-dimensional sphere, in preparation. Google Scholar [9] A. V. Dukov, Bifurcations of the "heart" polycycle in generic 2-parameter families, Trans. Moscow Math. Soc., 79 (2018), 209-229.  doi: 10.1090/mosc/284.  Google Scholar [10] A. V. Dukov and Y. S. Ilyashenko, Numeric invariants in semilocal bifurcations, J. Fixed Point Theory Appl., 23 (2021), 15 pp. doi: 10.1007/s11784-020-00837-x.  Google Scholar [11] N. Goncharuk and Y. S. Ilyashenko, Various equivalence relations in global bifurcation theory, Proc. Steklov Inst. Math., 310 (2020), 86-106.  doi: 10.1134/S0081543820050065.  Google Scholar [12] N. Goncharuk, Y. S. Ilyashenko and N. Solodovnikov, Global bifurcations in generic one-parameter families with a parabolic cycle on ${S^2}$, Moscow Math. J., 19 (2019), 709–737. Available from: http://www.mathjournals.org/mmj/2019-019-004/2019-019-004-004.html.  Google Scholar [13] N. Goncharuk and Y. G. Kudryashov, Bifurcations of the polycycle "tears of the heart": Multiple numerical invariants, Moscow Math. J., 20 (2020), 323-341.   Google Scholar [14] N. Goncharuk, Y. G. Kudryashov and N. Solodovnikov, New structurally unstable families of planar vector fields, Nonlinearity, 34 (2021), 438-454.  doi: 10.1088/1361-6544/abb86e.  Google Scholar [15] N. B. Goncharuk and Y. S. Ilyashenko, Large bifurcation supports, arXiv: 1804.04596, 2018. Google Scholar [16] Y. S. Ilyashenko, Towards the general theory of global planar bifurcations, Mathematical Sciences with Multidisciplinary Applications. In Honor of Professor Christiane Rousseau. And In Recognition of the Mathematics for Planet Earth Initiative, 157 (2016), 269-299.  doi: 10.1007/978-3-319-31323-8_13.  Google Scholar [17] Y. S. Ilyashenko, Y. G. Kudryashov and I. Schurov, Global bifurcations in the two-sphere: A new perspective, Invent. Math., 213 (2018), 461-506.  doi: 10.1007/s00222-018-0793-1.  Google Scholar [18] Y. S. Ilyashenko and N. Solodovnikov, Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$, Moscow Math. J., 18 (2018), 93-115.  doi: 10.17323/1609-4514-2018-18-1-93-115.  Google Scholar [19] Y. S. Ilyashenko and S. Yu. Yakovenko, Finitely-smooth normal forms of local families of diffeomorphisms and vector fields, Russian Mathematical Surveys, 46 (1991), 1-43.  doi: 10.1070/RM1991v046n01ABEH002733.  Google Scholar [20] A. Kotova and V. Stanzo, On few-parameter generic families of vector fields on the two-dimensional sphere, in Concerning the Hilbert 16th Problem, American Mathematical Society Translation Series 2,165, Amer. Math. Soc., 1995,155–201. doi: 10.1090/trans2/165.  Google Scholar [21] Y. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^nd$ edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar [22] I. P. Malta and J. Palis, Families of vector fields with finite modulus of stability, in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, 1981,212–229. doi: 10.1007/BFb0091915.  Google Scholar [23] M. M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222.  doi: 10.2307/1970100.  Google Scholar [24] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2.  Google Scholar [25] M. M. Peixoto, Structural stability on two-dimensional manifolds. A further remark, Topology, 2 (1963), 179-180.  doi: 10.1016/0040-9383(63)90032-6.  Google Scholar [26] V. S. Roitenberg, Non-Local Two-Parametric Bifurcations of Planar Vector Fields, Ph.D Thesis, Yaroslavl State Technical University, 2000. Google Scholar [27] V. S. Roitenberg, On bifurcation of vector fields with a separatrix winding onto a polycycle formed by separatrices of two saddles of different types, Almanac of Contemporary Science and Education, 7 (2012), 116-121.   Google Scholar [28] M. V. Shashkov, On bifurcation of separatrix contours with two saddles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 911-915.  doi: 10.1142/S0218127492000525.  Google Scholar [29] V. Starichkova, Global bifurcations in generic one-parameter families on $\mathbb{S}^2$, Regul. Chaotic Dyn., 23 (2018), 767-784.  doi: 10.1134/S1560354718060102.  Google Scholar [30] F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.  doi: 10.1016/0040-9383(71)90035-8.  Google Scholar

show all references

##### References:
 [1] V. Afraimovich and T. Young, Relative density of irrational rotation numbers in families of circle diffeomorphisms, Ergod. Th. and Dynam. Sys., 18 (1998), 1-16.  doi: 10.1017/S0143385798097648.  Google Scholar [2] A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251.   Google Scholar [3] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. ${\rm Ma}\breve{\rm{i}}{\rm er}$, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press [A division of John Wiley & Sons] and Israel Program for Scientific Translations, 1973.  Google Scholar [4] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Ma$\breve{\rm{i}}$er, Theory of Bifurcations of Dynamic Systems on A Plane, Halsted Press [A division of John Wiley & Sons] and Israel Program for Scientific Translations, 1973.  Google Scholar [5] V. I. Arnold, V. S. Afrajmovich, Yu. S. Ilyashenko and L. P. Shilnikov, Dynamical Systems V. Bifurcation Theory and Catastrophe Theory, Encyclopaedia of Mathematical Sciences, Springer-Verlag Berlin Heidelberg, 1994. Available from: http://www.springer.com/gp/book/9783540181736. Google Scholar [6] S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.  Google Scholar [7] H. F. DeBaggis, Dynamical systems with stable structures, Contributions to the Theory of Nonlinear Oscillations, 2 (1952), 37–60. Available from: http://www.jstor.org/stable/j.ctt1bgz9z7.6.  Google Scholar [8] A. V. Dukov, Functional invariants in generic semilocal families of vector fields on two-dimensional sphere, in preparation. Google Scholar [9] A. V. Dukov, Bifurcations of the "heart" polycycle in generic 2-parameter families, Trans. Moscow Math. Soc., 79 (2018), 209-229.  doi: 10.1090/mosc/284.  Google Scholar [10] A. V. Dukov and Y. S. Ilyashenko, Numeric invariants in semilocal bifurcations, J. Fixed Point Theory Appl., 23 (2021), 15 pp. doi: 10.1007/s11784-020-00837-x.  Google Scholar [11] N. Goncharuk and Y. S. Ilyashenko, Various equivalence relations in global bifurcation theory, Proc. Steklov Inst. Math., 310 (2020), 86-106.  doi: 10.1134/S0081543820050065.  Google Scholar [12] N. Goncharuk, Y. S. Ilyashenko and N. Solodovnikov, Global bifurcations in generic one-parameter families with a parabolic cycle on ${S^2}$, Moscow Math. J., 19 (2019), 709–737. Available from: http://www.mathjournals.org/mmj/2019-019-004/2019-019-004-004.html.  Google Scholar [13] N. Goncharuk and Y. G. Kudryashov, Bifurcations of the polycycle "tears of the heart": Multiple numerical invariants, Moscow Math. J., 20 (2020), 323-341.   Google Scholar [14] N. Goncharuk, Y. G. Kudryashov and N. Solodovnikov, New structurally unstable families of planar vector fields, Nonlinearity, 34 (2021), 438-454.  doi: 10.1088/1361-6544/abb86e.  Google Scholar [15] N. B. Goncharuk and Y. S. Ilyashenko, Large bifurcation supports, arXiv: 1804.04596, 2018. Google Scholar [16] Y. S. Ilyashenko, Towards the general theory of global planar bifurcations, Mathematical Sciences with Multidisciplinary Applications. In Honor of Professor Christiane Rousseau. And In Recognition of the Mathematics for Planet Earth Initiative, 157 (2016), 269-299.  doi: 10.1007/978-3-319-31323-8_13.  Google Scholar [17] Y. S. Ilyashenko, Y. G. Kudryashov and I. Schurov, Global bifurcations in the two-sphere: A new perspective, Invent. Math., 213 (2018), 461-506.  doi: 10.1007/s00222-018-0793-1.  Google Scholar [18] Y. S. Ilyashenko and N. Solodovnikov, Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$, Moscow Math. J., 18 (2018), 93-115.  doi: 10.17323/1609-4514-2018-18-1-93-115.  Google Scholar [19] Y. S. Ilyashenko and S. Yu. Yakovenko, Finitely-smooth normal forms of local families of diffeomorphisms and vector fields, Russian Mathematical Surveys, 46 (1991), 1-43.  doi: 10.1070/RM1991v046n01ABEH002733.  Google Scholar [20] A. Kotova and V. Stanzo, On few-parameter generic families of vector fields on the two-dimensional sphere, in Concerning the Hilbert 16th Problem, American Mathematical Society Translation Series 2,165, Amer. Math. Soc., 1995,155–201. doi: 10.1090/trans2/165.  Google Scholar [21] Y. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^nd$ edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar [22] I. P. Malta and J. Palis, Families of vector fields with finite modulus of stability, in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, 1981,212–229. doi: 10.1007/BFb0091915.  Google Scholar [23] M. M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222.  doi: 10.2307/1970100.  Google Scholar [24] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2.  Google Scholar [25] M. M. Peixoto, Structural stability on two-dimensional manifolds. A further remark, Topology, 2 (1963), 179-180.  doi: 10.1016/0040-9383(63)90032-6.  Google Scholar [26] V. S. Roitenberg, Non-Local Two-Parametric Bifurcations of Planar Vector Fields, Ph.D Thesis, Yaroslavl State Technical University, 2000. Google Scholar [27] V. S. Roitenberg, On bifurcation of vector fields with a separatrix winding onto a polycycle formed by separatrices of two saddles of different types, Almanac of Contemporary Science and Education, 7 (2012), 116-121.   Google Scholar [28] M. V. Shashkov, On bifurcation of separatrix contours with two saddles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 911-915.  doi: 10.1142/S0218127492000525.  Google Scholar [29] V. Starichkova, Global bifurcations in generic one-parameter families on $\mathbb{S}^2$, Regul. Chaotic Dyn., 23 (2018), 767-784.  doi: 10.1134/S1560354718060102.  Google Scholar [30] F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.  doi: 10.1016/0040-9383(71)90035-8.  Google Scholar
]. A similar figure was earlier published in Invent. Math. [17]">Figure 1.  A vector field with a polycycle "tears of the heart". This figure was first published in Nonlinearity [14]. A similar figure was earlier published in Invent. Math. [17]
]. This figure was first published in Nonlinearity [14]">Figure 2.  Vector fields with structurally unstable unfoldings from [14]. This figure was first published in Nonlinearity [14]
Degenerate vector fields with structurally unstable unfoldings
A vector field of class $\mathbf{WG}_{1, 1}$
Ensemble "lips"
A vector field of class $\mathbf{LEG}_{2}$
An unfolding of a vector field $v_{0}\in \mathbf{LEG}_{2}$ satisfying assertions of Lemma 4.10 for $k = 2$
 [1] BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85 [2] José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020 [3] C. Alonso-González, M. I. Camacho, F. Cano. Topological invariants for singularities of real vector fields in dimension three. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 823-847. doi: 10.3934/dcds.2008.20.823 [4] J. C. Artés, Jaume Llibre, J. C. Medrado. Nonexistence of limit cycles for a class of structurally stable quadratic vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 259-270. doi: 10.3934/dcds.2007.17.259 [5] Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621 [6] Ale Jan Homburg. Heteroclinic bifurcations of $\Omega$-stable vector fields on 3-manifolds. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 559-580. doi: 10.3934/dcds.1998.4.559 [7] Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83 [8] Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145 [9] Pablo Aguirre, Eusebius J. Doedel, Bernd Krauskopf, Hinke M. Osinga. Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1309-1344. doi: 10.3934/dcds.2011.29.1309 [10] Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481 [11] Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623 [12] Yunming Zhou, Desheng Shang, Tonghua Zhang. Seventeen limit cycles bifurcations of a fifth system. Conference Publications, 2007, 2007 (Special) : 1070-1081. doi: 10.3934/proc.2007.2007.1070 [13] Felicia Maria G. Magpantay, Xingfu Zou. Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections. Mathematical Biosciences & Engineering, 2010, 7 (2) : 421-442. doi: 10.3934/mbe.2010.7.421 [14] Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 [15] Dongchen Li, Dmitry V. Turaev. Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4399-4437. doi: 10.3934/dcds.2017189 [16] Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063 [17] Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511 [18] Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583 [19] Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031 [20] Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165

2020 Impact Factor: 1.392