# American Institute of Mathematical Sciences

## Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization

 Institute of Information Technology, Mathematics and Mechanics, Lobachevsky National Research State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia

* Corresponding author: lermanl@mm.unn.ru

To Jürgen, with the gratitude and best wishes

Received  December 2017 Revised  June 2018 Published  April 2019

We study a class of scalar differential equations on the circle $S^1$. This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes $\mathbb R_+$ and $\mathbb R_+$. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.

Citation: Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020076
##### References:
 [1] A. A. Andronov, Systémes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251. [2] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of the negative curvature, Trudy Mat. Inst. Steklov, 90 (1967), 209 pp. [3] S. Bochner, Sur les fonctions presque périodiques de Bohr, Comptes Rendus, 180 (1925), 1156-1158. [4] I. U. Bronstein, Extensions of Minimal Transformation Groups, Springer Netherlands, 1979. [5] D. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific Publishing Co. Pte. Ltd, Singapoure, 2004. doi: 10.1142/9789812563088. [6] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, Mass., 1965. [7] C. Corduneanu, Almost Periodic Oscillations and Waves, Springer Science+Business Media, LLC, 2009. doi: 10.1007/978-0-387-09819-7. [8] B. P. Demidovich., Lectures on Mathematical Theory of Stability, Nauka, Moscow, 1967 (in Russian). [9] S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, On models with non-rough Poincare homoclinic curves, Physica D, 62 (1993), 1-14. doi: 10.1016/0167-2789(93)90268-6. [10] S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, Homoclinic tangencies of an arbitrary order in Newhouse domains, in Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., 67 (1999), 69-128. (Russian) [Homoclinic tangencies of an arbitrary order in Newhouse domains] J. Math. Sci., 105 (2001), 1738-1778. [11] S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case), Russian Acad. Sci.: Dokl. Math., 47 (1993), 268-283. [12] V. Z. Grines, E. Gurevich and O. Pochinka, Topological classification of Morse-Smale diffeomorphisms without heteroclinic intersections, J. Math. Sci., 208 (2015), 81-90. doi: 10.1007/s10958-015-2425-2. [13] V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, "Developments in Mathematics", v.46, Springer International Publishing, 2016. doi: 10.1007/978-3-319-44847-3. [14] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York-Toronto-Sydney, 1964. [15] J. L. Kelley, General Topology, D. Van Nostrand Co., Inc., Princeton, N.J., 1957. [16] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Math. surveys and monographs, v.176, AMS, Providence, R.I., 2011. doi: 10.1090/surv/176. [17] E. A. Leontovich-Andronova, On mathematical achievements of the Gorky school by A.A. Andronov, in Proc. of the Ⅲ All-Union Mathematical Congress, 3, USSR Acad. of Sci. P.H., Moscow, 1958. [18] E. A. Leontovich-Andronova and A. G. Maier, On a scheme determining topological structure of the orbit foliation, Doklady AN SSSR, 103 (1955), 557-560. [19] L. Lerman, On Nonautonomous Dynamical Systems of the Morse-Smale Type, Ph.D thesis, Gorky State Univ., Gorky, 1975 (in Russian). [20] L. M. Lerman, On nonautonomous dynamical systems of the Morse-Smale type, Uspekhi Mat. Nauk, XXX (1975), p.195 (Sessions of the Moscow Math. Soc.) (in Russian). [21] L. M. Lerman and L. P. Shilnikov, On the classification of structurally stable nonautonomous systems of the second order with a finite number of cells, Soviet Math. Dokl., 14 (1973), 444–448 [transl. from Russian] Dokl. Akad. Nauk SSSR, 209 (1973), 544–547). [22] L. M. Lerman and L. P. Shilnikov, Homoclinic structures in nonautonomous systems: Nonautonomous chaos, Chaos: Intern. J. Nonlin. Sci., 2 (1992), 447-454. doi: 10.1063/1.165887. [23] L. M. Lerman, Remarks on skew products over irrational rotations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3675-3689. doi: 10.1142/S0218127405014118. [24] B. M. Levitan, Almost Periodic Functions, GITTL, Moscow, 1953 (in Russian). [25] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982. [26] L. Markus, Asymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations Ⅲ, (S. Lefschetz, ed.), Annals of Mathematical Studies, 36, Princeton. Univ. Press, (1956), 17–29. [27] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, 21, Boston, MA: Academic Press, 1966. [28] B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems, SIAM J. Appl. Dynam. Syst., 10 (2011), 35-65. doi: 10.1137/100794110. [29] V. V. Nemytsky and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princetom, N.J., 1960. [30] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2. [31] S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Scient., 50 (1979), 101-151. [32] Z. Opial, Sur les équation différentielle presque-périodique sans sólution presque-périodique, Bull. L'Acad. Polon. des Sci. Ser. math., astr. et phys.: Math. Astron. Phys., 9 (1961), 673-676. [33] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120. doi: 10.1016/0040-9383(65)90018-2. [34] Ch. Potzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes in Math., 2002, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14258-1. [35] M. Rasmussen, Nonautonomous bifurcation patterns for one-dimensional differential equations, J. Diff. Equat., 234 (2007), 267-288. doi: 10.1016/j.jde.2006.11.002. [36] J. Scheurle, Chaotic solutions of systems with almost periodic forcing, J. Appl. Math. Phys. (ZAMP), 37 (1986), 12-26. doi: 10.1007/BF00955515. [37] G. Sell, Nonautonomous differential equations and topological dynamics, Ⅰ, Ⅱ, Trans. AMS, 127 (1967), 263-283. doi: 10.2307/1994645. [38] S. Smale, Morse inequalities for a dynamical system, Bull. AMS, 66 (1960), 43-49. doi: 10.1090/S0002-9904-1960-10386-2. [39] D. Stoffer, Transversal homoclinic points and hyperbolic sets for non-autonomous maps Ⅰ, Ⅱ, J. Appl. Math. Phys. (ZAMP), 39 (1988), 518-549, Ibid, 39 (1988), 783-812. doi: 10.1007/BF00948961. [40] A. Strauss and J. A. Yorke, On asymptotically autonomous differential equations, Math. Syst. Theory, 1 (1967), 176-182. doi: 10.1007/BF01705527.

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##### References:
 [1] A. A. Andronov, Systémes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251. [2] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of the negative curvature, Trudy Mat. Inst. Steklov, 90 (1967), 209 pp. [3] S. Bochner, Sur les fonctions presque périodiques de Bohr, Comptes Rendus, 180 (1925), 1156-1158. [4] I. U. Bronstein, Extensions of Minimal Transformation Groups, Springer Netherlands, 1979. [5] D. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific Publishing Co. Pte. Ltd, Singapoure, 2004. doi: 10.1142/9789812563088. [6] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, Mass., 1965. [7] C. Corduneanu, Almost Periodic Oscillations and Waves, Springer Science+Business Media, LLC, 2009. doi: 10.1007/978-0-387-09819-7. [8] B. P. Demidovich., Lectures on Mathematical Theory of Stability, Nauka, Moscow, 1967 (in Russian). [9] S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, On models with non-rough Poincare homoclinic curves, Physica D, 62 (1993), 1-14. doi: 10.1016/0167-2789(93)90268-6. [10] S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, Homoclinic tangencies of an arbitrary order in Newhouse domains, in Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., 67 (1999), 69-128. (Russian) [Homoclinic tangencies of an arbitrary order in Newhouse domains] J. Math. Sci., 105 (2001), 1738-1778. [11] S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case), Russian Acad. Sci.: Dokl. Math., 47 (1993), 268-283. [12] V. Z. Grines, E. Gurevich and O. Pochinka, Topological classification of Morse-Smale diffeomorphisms without heteroclinic intersections, J. Math. Sci., 208 (2015), 81-90. doi: 10.1007/s10958-015-2425-2. [13] V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, "Developments in Mathematics", v.46, Springer International Publishing, 2016. doi: 10.1007/978-3-319-44847-3. [14] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York-Toronto-Sydney, 1964. [15] J. L. Kelley, General Topology, D. Van Nostrand Co., Inc., Princeton, N.J., 1957. [16] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Math. surveys and monographs, v.176, AMS, Providence, R.I., 2011. doi: 10.1090/surv/176. [17] E. A. Leontovich-Andronova, On mathematical achievements of the Gorky school by A.A. Andronov, in Proc. of the Ⅲ All-Union Mathematical Congress, 3, USSR Acad. of Sci. P.H., Moscow, 1958. [18] E. A. Leontovich-Andronova and A. G. Maier, On a scheme determining topological structure of the orbit foliation, Doklady AN SSSR, 103 (1955), 557-560. [19] L. Lerman, On Nonautonomous Dynamical Systems of the Morse-Smale Type, Ph.D thesis, Gorky State Univ., Gorky, 1975 (in Russian). [20] L. M. Lerman, On nonautonomous dynamical systems of the Morse-Smale type, Uspekhi Mat. Nauk, XXX (1975), p.195 (Sessions of the Moscow Math. Soc.) (in Russian). [21] L. M. Lerman and L. P. Shilnikov, On the classification of structurally stable nonautonomous systems of the second order with a finite number of cells, Soviet Math. Dokl., 14 (1973), 444–448 [transl. from Russian] Dokl. Akad. Nauk SSSR, 209 (1973), 544–547). [22] L. M. Lerman and L. P. Shilnikov, Homoclinic structures in nonautonomous systems: Nonautonomous chaos, Chaos: Intern. J. Nonlin. Sci., 2 (1992), 447-454. doi: 10.1063/1.165887. [23] L. M. Lerman, Remarks on skew products over irrational rotations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3675-3689. doi: 10.1142/S0218127405014118. [24] B. M. Levitan, Almost Periodic Functions, GITTL, Moscow, 1953 (in Russian). [25] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982. [26] L. Markus, Asymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations Ⅲ, (S. Lefschetz, ed.), Annals of Mathematical Studies, 36, Princeton. Univ. Press, (1956), 17–29. [27] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, 21, Boston, MA: Academic Press, 1966. [28] B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems, SIAM J. Appl. Dynam. Syst., 10 (2011), 35-65. doi: 10.1137/100794110. [29] V. V. Nemytsky and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princetom, N.J., 1960. [30] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2. [31] S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Scient., 50 (1979), 101-151. [32] Z. Opial, Sur les équation différentielle presque-périodique sans sólution presque-périodique, Bull. L'Acad. Polon. des Sci. Ser. math., astr. et phys.: Math. Astron. Phys., 9 (1961), 673-676. [33] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120. doi: 10.1016/0040-9383(65)90018-2. [34] Ch. Potzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes in Math., 2002, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14258-1. [35] M. Rasmussen, Nonautonomous bifurcation patterns for one-dimensional differential equations, J. Diff. Equat., 234 (2007), 267-288. doi: 10.1016/j.jde.2006.11.002. [36] J. Scheurle, Chaotic solutions of systems with almost periodic forcing, J. Appl. Math. Phys. (ZAMP), 37 (1986), 12-26. doi: 10.1007/BF00955515. [37] G. Sell, Nonautonomous differential equations and topological dynamics, Ⅰ, Ⅱ, Trans. AMS, 127 (1967), 263-283. doi: 10.2307/1994645. [38] S. Smale, Morse inequalities for a dynamical system, Bull. AMS, 66 (1960), 43-49. doi: 10.1090/S0002-9904-1960-10386-2. [39] D. Stoffer, Transversal homoclinic points and hyperbolic sets for non-autonomous maps Ⅰ, Ⅱ, J. Appl. Math. Phys. (ZAMP), 39 (1988), 518-549, Ibid, 39 (1988), 783-812. doi: 10.1007/BF00948961. [40] A. Strauss and J. A. Yorke, On asymptotically autonomous differential equations, Math. Syst. Theory, 1 (1967), 176-182. doi: 10.1007/BF01705527.
Bifurcation from infinities, $\mu = 0,$ $\mu > 0$
Bifurcation at the violation of Assumption 3
Foliation inside the rectangle
Neighborhoods: crosses correspond to u-solutions, bold points correspond to s-solutions
Construction of asymptotically autonomous ODE: crosses correspond to u-solutions, bold points correspond to s-solutions
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