January  2022, 42(1): 239-259. doi: 10.3934/dcds.2021114

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 Published  January 2022 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, 2022, 42 (1) : 239-259. 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. ChowC. 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. GoncharukY. 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. IlyashenkoY. 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. ChowC. 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. GoncharukY. 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. IlyashenkoY. 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

14]. 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]
14]. 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]
Figure 3.  Degenerate vector fields with structurally unstable unfoldings
Figure 4.  A vector field of class $ \mathbf{WG}_{1, 1}$
Figure 5.  Ensemble "lips"
Figure 6.  A vector field of class $ \mathbf{LEG}_{2}$
Figure 7.  An unfolding of a vector field $v_{0}\in \mathbf{LEG}_{2}$ satisfying assertions of Lemma 4.10 for $k = 2$
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