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 and 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.

[2]

A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251. 

[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.

[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.

[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.

[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.
[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.

[8]

A. V. Dukov, Functional invariants in generic semilocal families of vector fields on two-dimensional sphere, in preparation.

[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.

[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.

[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.

[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.

[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. 

[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.

[15]

N. B. Goncharuk and Y. S. Ilyashenko, Large bifurcation supports, arXiv: 1804.04596, 2018.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222.  doi: 10.2307/1970100.

[24]

M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2.

[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.

[26]

V. S. Roitenberg, Non-Local Two-Parametric Bifurcations of Planar Vector Fields, Ph.D Thesis, Yaroslavl State Technical University, 2000.

[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. 

[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.

[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.

[30]

F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.  doi: 10.1016/0040-9383(71)90035-8.

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.

[2]

A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251. 

[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.

[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.

[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.

[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.
[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.

[8]

A. V. Dukov, Functional invariants in generic semilocal families of vector fields on two-dimensional sphere, in preparation.

[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.

[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.

[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.

[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.

[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. 

[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.

[15]

N. B. Goncharuk and Y. S. Ilyashenko, Large bifurcation supports, arXiv: 1804.04596, 2018.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222.  doi: 10.2307/1970100.

[24]

M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2.

[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.

[26]

V. S. Roitenberg, Non-Local Two-Parametric Bifurcations of Planar Vector Fields, Ph.D Thesis, Yaroslavl State Technical University, 2000.

[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. 

[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.

[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.

[30]

F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.  doi: 10.1016/0040-9383(71)90035-8.

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]
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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