01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
27.02.1952
E-mail:
,
Keywords:
dynamical systems; differential and topological dynamics of discrete dynamical systems in low dimensions; one-dimensional dynamics; chaotic dynamics.
Subject:
The problem of the coexistence of periods of periodic points of continuous maps of the circle was solved. Interdependence of arithmetic correlations between periods of periodic points with the degree of a continuous map of the circle is established. Criteria of the existence of homoclinic points of continuous endomorphisms of the circle and criteria of the disguishing of continuous endomorphisms of the circle with complicated dynamics (in the sense of A. N. Sharkovsky) are proved. The new concept of the investigation of skew products of interval maps based on the use of new set-valued functions (the $\Omega$-function and the $Bi$-function) of a skew product of interval maps was proposed. In the frames of this concept the dual nature of skew products of interval maps was explaned (it was established why some skew products of interval maps inherit the properties of interval maps, and others have the new properties which are not observed in interval maps). The criterion of the $\Omega$-stability of a skew product of interval maps in the space of $C^1$-smooth skew products of interval maps was proved. Nongenericity of $\Omega$-stability in the space of $C^1$-smooth skew products of interval maps was proved. The problem of the description of dynamics of the "most simple" continuous maps of dendrites with a closed set of brunch points of a finite order was formulated. A number of papers (joint with E. N. Makhrova) were devoted to the investigation of dynamics of monotone and piecewise monotone maps of dendrites with a closed set of periodic points. The possibility of the existence of piecewise monotone maps with fixed points and zero topological entropy possessing of the wandering homoclinic points; nonwandering, but not $\omega$-limit homoclinic points; $\omega$-limit homoclinic points on dendrites with a closed set of brunch points was determined.
Biography
Graduated from the Faculty of Mathematics and Mechanics of Gorky State University in 1974 (department of differential equations and mathematical analysis). PhD thesis was defended in 1981. A list of my works contains more than 70 titles. I deliver the lectures on contemporary problems of the theory of discrete dynamical systems for students, masters and post-graduates.
Main publications:
L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
L. S. Efremova, E. N. Makhrova, “One-dimensional dynamical systems”, Russian Math. Surveys, 76:5 (2021), 821–881.
L. S. Efremova, “Ramified continua as global attractors of $C^1$-smooth self-maps of a cylinder close to skew products”, J. Difference Eq. & Appliq., 29:9-12 (2023), 1244-1274
Lozi R., Efremova L.S., Abdelouhab M.-S., El Assad S., Plunachek M., “Foreword to the special issue of Journal of Difference Equations and Applications on ‘Lozi, Hénon, and other chaotic attractors, theory and applications’”, J. Difference Eq. & Appliq., 29:9-12 (2023), 861 - 875
5.
L. S. Efremova, “Simplest skew products on $n$-dimensional $(n\ge 2)$ cells, cylinders and tori”, Lobachevskii Journal of Mathematics, 43:7 (2022), 1598–1618
L. S. Efremova, “Small C1-smooth perturbations of skew products and the partial integrability property”, Applied Math. & Nonlinear Sci., 5:2 (2020), 317 - 328
L. S. Efremova, “Periodic Behavior of Maps Obtained by Small Perturbations of Smooth Skew Products"”, Discontinuity, Nonlinearity & Complexity, 9:4 (2020), 519 - 523
L. S. Efremova, A. D. Grekhneva, V. Zh. Sakbaev, “Phase Flows Generated by Cauchy Problem for Nonlinear Schrödinger Equation and Dynamical Mappings of Quantum States", Lobachevskii Phase Flows Generated by Cauchy Problem for Nonlinear Schrödinger Equation and Dynamical Mappings of Quantum States”, Lobachevskii Journal of Mathematics, 40:10 (2019), 1455 - 1469
L. S. Efremova, “The trace map and integrability of the multifunctions”, Journal of Physics: Conference Series, 990:1 (2018), 012003 , 10 pp.
13.
L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
14.
L. S. Efremova, “Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map”, Russian Math. (Iz. VUZ), 60:2 (2016), 77–81
15.
L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, Theoret. and Math. Phys., 185:2 (2015), 1582–1598
16.
S. S. Bel’mesova, L. S. Efremova, “On the concept of integrability for discrete dynamical systems. Investigation of wandering points of some trace map”, Nonlinear maps and their applications (Saragoza, Spain, 2013), Springer Proceedings in Math. & Stat., 112, eds. R. López-Ruiz, D. Fournier-Prunaret et al, Springer, Cham, 2015, 127 - 158
L. S. Efremova, “Remarks on the nonwandering set of skew products with a closed set of periodic points of the quotient map”, Nonlinear maps and their applications (Evora, Portugal, 2011), Springer Proceedings in Math. & Stat., 57, eds. K. Gracio, D. Fournier-Prunaret et al, Springer, New - York, 2014, 39 - 58
L. S. Efremova, “Absence of $C^1$-$\Omega$-explosion in the space of smooth simplest skew products”, Journal of Mathematical Sciences, 202:6 (2014), 794–808
19.
S. S. Bel'mesova, L. S. Efremova, “A one-parameter family of quadratic maps of a plane including Morse–Smale endomorphisms”, Russian Math. (Iz. VUZ), 57:8 (2013), 70–74
20.
L. S. Efremova, “A decomposition theorem for the space of $C^1$-smooth skew products with complicated dynamics of the quotient map”, Sb. Math., 204:11 (2013), 1598–1623
21.
L. S. Efremova, “Differential properties and attracting sets of a simplest skew product of interval maps”, Sb. Math., 201:6 (2010), 873–907
22.
L. S. Efremova, “Space of $C^1$-smooth skew products of maps of an interval”, Theoret. and Math. Phys., 164:3 (2010), 1208–1214
23.
L. S. Efremova, “On the nonwandering set and center of some skew products of mappings of the interval”, Russian Math. (Iz. VUZ), 50:10 (2006), 17–25
24.
L. S. Efremova, “$\Omega$-Stable Skew Products of Interval Maps Are Not Dense in $T^1(I)$”, Proc. Steklov Inst. Math., 236 (2002), 157–163
25.
L. S. Efremova, E. N. Makhrova, “The dynamics of monotone maps of dendrites”, Sb. Math., 192:6 (2001), 807–821
26.
L. S. Efremova, “On the concept of the $\Omega$-function of the skew product of interval mappings”, J. Math. Sci. (New York), 105:1 (2001), 1779–1798
27.
M. I. Voinova, L. S. Efremova, “Dynamics of elementary maps of dendrites”, Math. Notes, 63:2 (1998), 161–171
28.
L. S. Efremova, E. N. Makhrova, “The dynamics of a monotone mapping of an $n$-odd”, Russian Math. (Iz. VUZ), 41:10 (1997), 29–34
29.
L. S. Efremova, “Letter to the editor”, Math. Notes, 56:5 (1994), 1193–1194
30.
L. S. Efremova, “A class of twisted products of maps of an interval”, Math. Notes, 54:3 (1993), 890–898
31.
L. S. Efremova, “A quotient of periods other than a power of two leads to chaos in a neighbourhood”, Russian Math. Surveys, 40:1 (1985), 217–218
32.
O. N. Ageev, Ya. B. Vorobets, B. Weiss, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, L. S. Efremova, A. Yu. Zhirov, E. V. Zhuzhoma, B. S. Kashin, V. N. Kolokoltsov, A. V. Kochergin, L. M. Lerman, I. V. Mykytyuk, V. I. Oseledets, A. Yu. Plakhov, O. V. Pochinka, V. V. Ryzhikov, V. Zh. Sakbaev, A. G. Sergeev, Ya. G. Sinai, A. T. Tagi-Zade, S. V. Tikhonov, J.-P. Thouvenot, A. Ya. Helemskii, A. I. Shafarevich, “Anatolii Mikhailovich Stepin (obituary)”, Russian Math. Surveys, 77:2 (2022), 361–367
33.
O. V. Anashkin, P. M. Akhmet'ev, D. V. Balandin, M. K. Barinova, I. V. Boykov, A. N. Bezdenezhnyh, V. N. Belykh, P. A. Vel'misov, I. Yu. Vlasenko, O. E. Galkin, S. Yu. Galkina, V. K. Gorbunov, S. D. Glyzin, S. V. Gonchenko, A. S. Gorodetski, E. V. Gubina, E. Ya. Gurevich, A. A. Davydov, L. S. Efremova, R. V. Zhalnin, A. Yu. Zhirov, E. V. Zhuzhoma, N. I. Zhukova, S. Kh. Zinina, Yu. S. Ilyashenko, N. V. Isaenkova, A. O. Kazakov, A. V. Klimenko, S. A. Komech, Yu. A. Kordyukov, V. E. Kruglov, E. V. Kruglov, E. B. Kuznetsov, S. K. Lando, Yu. A. Levchenko, L. M. Lerman, S. I. Maksimenko, M. I. Malkin, D. S. Malyshev, V. K. Mamaev, T. Ph. Mamedova, V. S. Medvedev, T. V. Medvedev, D. I. Mints, T. M. Mitryakova, A. D. Morozov, A. I. Morozov, E. V. Nozdrinova, E. N. Pelinovsky, Ya. B. Pesin, A. S. Pikovsky, S. Yu. Pilyugin, G. M. Polotovsky, O. V. Pochinka, I. D. Remizov, P. E. Ryabov, A. S. Skripchenko, A. V. Slunyaev, S. V. Sokolov, L. A. Sukharev, E. A. Talanova, V. A. Timorin, S. B. Tikhomir, “To the 75th anniversary of Vyacheslav Zigmundovich Grines”, Zhurnal SVMO, 23:4 (2021), 472–476
On ω-limit sets of simplest skew products defined on n-dimensional cells M. Shalagin, L. S. Efremova III International Conference “Mathematical Physics, Dynamical Systems, Infinite-Dimensional Analysis”, dedicated to the 100th anniversary of V.S. Vladimirov, the 100th anniversary of L.D. Kudryavtsev and the 85th anniversary of O.G. Smolyanov July 6, 2023 16:10
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