Abstract
We study small C 1-perturbations of systems of differential equations that have a weakly hyperbolic invariant set. We show that the weakly hyperbolic invariant set is stable even if the Lipschitz condition fails.
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Pliss, V.A. and Sell, G.R., Perturbations of Attractors of Differential Equations, J. Differential Equations, 1991, vol. 92, pp. 100–124.
Pliss, V.A. and Sell, G.R., Approximation Dynamics and the Stability of Invariant Sets, J. Differential Equations, 1997, vol. 149, pp. 1–51.
Pliss, V.A., Integral’nye mnozhestva periodicheskikh sistem differentsial’nykh uravnenii (Integral Sets of Periodic Systems of Differential Equations), Moscow: Nauka, 1977.
Monakov, V.N., The Arrangement of the Integral Surfaces of Weakly Nonlinear Systems of Differential Equations, Vestnik Leningrad. Univ., 1973, no. 1, pp. 68–74.
Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, New York, 1955. Translated under the title Teoriya obyknovennykh differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1958.
Kelley, Al., Stability of the Center-Stable Manifold, J. Math. Anal. Appl., 1967, vol. 18, pp. 336–344.
Kelley, Al., The Stable, Center-Stable, Center, Center-Unstable, Unstable Manifolds, J. Differential Equations, 1967, vol. 3, pp. 546–570.
Pliss, V.A., A Reduction Principle in the Theory of Stability of Motion, Izv. Akad. Nauk Ser. Mat., 1964, no. 28, pp. 1297–1324.
Fenichel, N., Persistence and Smoothness of Invariant Manifolds for Flows, Indiana Univ. Math. J., 1971, vol. 21, pp. 193–226.
Hirsch, M.W., Pugh, C.C., and Shub, M., Invariant Manifolds, Lecture Notes in Mathematics, New York, 1977, vol. 583.
Sacker, R.J., A Perturbation Theorem for Invariant Manifolds and Holder Continuity, J. Math. Mech., 1969, vol. 18, pp. 705–762.
Sell, G.R., The Structure of a Flow in the Vicinity of an Almost Periodic Motion, J. Differential Equations, 1978, vol. 27, pp. 359–393.
Smale, S., Differentiable Dynamical Systems, Bull. Amer. Math. Soc., 1967, vol. 73, pp. 747–817.
Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York, 1988.
Coppel, W.A., Dichotomies in Stability Theory, Lecture Notes in Math., New York, 1978, vol. 629.
Pliss, V.A. and Sell, G.R., Approximations of the Long-Time Dynamics of the Navier–Stokes Equations, Differential Equations and Geometric Dynamics: Control Science and Dynamical Systems: Lecture Notes, 1993, vol. 152, pp. 247–277.
Titi, E.S., On Approximate-Inertial Manifolds of the Navier–Stokes Equations, J. Math. Anal. Appl., 1990, vol. 149, pp. 540–557.
Begun, N.A., On the Stability of Leaf Invariant Sets of Two-Dimensional Periodic Systems, Vestnik St.-Peterburg. Univ. Ser. 1 Mat. Mekh. Astron., 2012, no. 4, pp. 3–12.
Begun, N.A., On the Closedness of a Leaf Invariant Set of Perturbed System, Differ. Uravn. Protsessy Upr., 2013, no. 1, pp. 80–88.
Begun, N.A., On the Stability of Invariant Sets of Leaves of Three-Dimensional Periodic Systems, Vestnik St.-Peterburg. Univ. Ser. 1 Mat. Mekh. Astron., 2014, no. 3, pp. 12–19.
Begun, N.A., Perturbations of Weakly Hyperbolic Invariant Sets of a Two-Dimension Periodic System, Vestnik St.-Peterburg. Univ. Ser. 1 Mat. Mekh. Astron., 2015, no. 1, pp. 23–33.
Kostrikin, A.I. and Manin, Yu.I., Lineinaya algebra i geometriya (Linear Algebra and Geometry), Moscow: Nauka, 1986.
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Original Russian Text © N.A. Begun, V.A. Pliss, J.R. Sell, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 2, pp. 139–148.
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Begun, N.A., Pliss, V.A. & Sell, J.R. On the stability of hyperbolic attractors of systems of differential equations. Diff Equat 52, 139–148 (2016). https://doi.org/10.1134/S0012266116020014
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DOI: https://doi.org/10.1134/S0012266116020014