ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Nonlinear Semigroups for Delay Equations in Hilbert Spaces, Inertial Manifolds and Dimension Estimates

Author(s):

Mikhail Mikhailovich Anikushin

PhD, junior researcher at the Department
of Applied Cybernetics, Faculty of Mathematics and Mechanics,
St. Petersburg State University

demolishka@yandex.ru

Abstract:

We study the well-posedness of nonautonomous nonlinear delay equations in $\mathbb{R}^{n}$ as evolutionary equations in a proper Hilbert space. We present a construction of solving operators (nonautonomous case) or nonlinear semigroups (autonomous case) for a large class of such equations. The main idea can be easily extended for certain PDEs with delay. Our approach has lesser limitations and much more elementary than some previously known constructions of such semigroups and solving operators based on the theory of accretive operators. In the autonomous case we also study differentiability properties of these semigroups in order to apply various dimension estimates using the Hilbert space geometry. However, obtaining effective dimension estimates for delay equations is a nontrivial problem and we explain it by means of a scalar delay equation. We also discuss our adjacent results concerned with inertial manifolds and their construction for delay equations.

Keywords

References:

  1. Anikushin M. M. Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces. arXiv preprint. 2022. arXiv:2012. 03821v3
  2. Anikushin M. M. Frequency theorem and inertial manifolds for neutral delay equations. arXiv preprint. 2022. arXiv:2003. 12499v5
  3. Anikushin M. M., Romanov A. O. Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO. arXiv preprint. 2022. arXiv:2102. 11711v3
  4. Anikushin M. M. Frequency theorem for parabolic equations and its relation to inertial manifolds theory. J. Math. Anal. Appl. 2021. Vol. 505, no. 1, 125454
  5. Anikushin M. M. Almost automorphic dynamics in almost periodic cocycles with one-dimensional inertial manifold. Differ. Uravn. Protsessy Upr. 2021. No. 2, P. 13-48 (In Russ. )
  6. Anikushin M. M. A non-local reduction principle for cocycles in Hilbert spaces. J. Differ. Equations. 2020. Vol. 269, no. 9, P. 6699-6731
  7. Anikushin M. M. On the Liouville phenomenon in estimates of fractal dimensions of forced quasi-periodic oscillations. Vestn. St. Petersbg. Univ., Math. 2019. Vol. 52, no. 3. P. 234-243
  8. Anikushin M. M., Reitmann V., Romanov A. O. Analytical and numerical estimates of the fractal dimension of forced quasiperiodic oscillations in control systems. Differ. Uravn. Protsessy Upr. 2019. no. 2, P. 162-183 (In Russ. )
  9. Anikushin M. M. On the Smith reduction theorem for almost periodic ODEs satisfying the squeezing property. Rus. J. Nonlin. Dyn. Vol. 15, no. 1, P. 97-108
  10. Bá tkai A., Piazzera S. Semigroups for Delay Equations. A K Peters, Wellesley, 2005
  11. Breda D., Van Vleck E. Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations. Numer. Math. 2014. Vol 126, no. 2, P. 225-257
  12. Breda D. Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents. Differ. Integral Equ. 2010. Vol. 23, no. 9/10, P. 935-956
  13. CarvalhoA. N., Langa J. A., Robinson J. C. Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. Springer Science & Business Media, 2012
  14. Chepyzhov V. V., Ilyin A. A. On the fractal dimension of invariant sets; applications to Navier-Stokes equations. Discrete Contin. Dyn. Syst. 2004. Vol. 10, no. 1& 2, P. 117-136
  15. Chicone C. Inertial and slow manifolds for delay equations with small delays. J. Differ. Equations. 2003. Vol 190, no. 2. P. 364-406
  16. Chow S. -N., Lu K., Sell G. R. Smoothness of inertial manifolds. J. Math. Anal. Appl. 1992. Vol. 169, P. 283-312
  17. Chueshov I. Dynamics of Quasi-stable Dissipative Systems. Berlin: Springer, 2015
  18. Eden A., Zelik S. V., Kalantarov V. K. Counterexamples to regularity of Mané projections in the theory of attractors. Russian Math. Surveys. 2013. Vol. 68, no. 2, P. 199-226
  19. Faheem M., Rao M. R. M. Functional differential equations of delay type and nonlinear evolution in !!!! ERROR!!! IMAGE IS NOT ALLOWED! -spaces. J. Math. Anal. Appl. 1987. Vol. 123, no. 1, P. 73-103
  20. Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 1971. Vol. 21, no. 3, P. 193-226
  21. Hale J. K. Theory of Functional Differential Equations. Springer-Verlag, New York, 1977
  22. Katok A., Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1997
  23. Kostianko A., Li X., Sun C., Zelik S. Inertial manifolds via spatial averaging revisited. SIAM J. Math. Anal. 2022. Vol. 54, no. 1
  24. Kuznetsov N. V., Reitmann V. Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Switzerland: Springer International Publishing AG, 2021
  25. Kuznetsov N. V. The Lyapunov dimension and its estimation via the Leonov method. Phys. Lett. A. 2016. Vol. 380, no. 25-26, P. 2142-2149
  26. Leonov G. A., Kuznetsov N. V., Korzhemanova N. A., Kusakin D. V. Lyapunov dimension formula for the global attractor of the Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 2016. Vol. 41, P. 84-103
  27. Li M. Y., Muldowney J. S. Lower bounds for the Hausdorff dimension of attractors. J. Dyn. Differ. Equ. 1995. Vol. 7, no. 3, P. 457-469
  28. Likhtarnikov A. L., Yakubovich V. A. The frequency theorem for continuous one-parameter semigroups. Math. USSR-Izv. 1977. Vol. 41, no. 4, 895-911 (In Russ. )
  29. Mallet-Paret J., Nussbaum R. D. Tensor products, positive linear operators, and delay-differential equations. J. Dyn. Diff. Equat. 2013. Vol. 25, P. 843-905
  30. Mallet-Paret J., Sell G. R. The Poincaré -Bendixson theorem for monotone cyclic feedback systems with delay. J. Differ. Equations. 1996. Vol. 125, P. 441-489
  31. Mallet-Paret J. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Differ. Equations. 1976. Vol. 22, no. 2, P. 331-348
  32. Rosa R., Temam R. Inertial manifolds and normal hyperbolicity. Acta Appl. Math. 1996. Vol. 45, no. 1, 1-50
  33. Smith R. A. Some applications of Hausdorff dimension inequalities for ordinary differential equations. P. Roy. Soc. Edinb. A. 1986. Vol. 104, no. 3-4, P. 235-259
  34. Smith R. A. Orbital stability and inertial manifolds for certain reaction diffusion systems. Proc. Lond. Math. Soc. 1994. Vol. 3, no. 1, P. 91-120
  35. Smith R. A. Poincaré -Bendixson theory for certain retarded functional-differential equations. Differ. Integral Equ. 1992. Vol. 5, no. 1, P. 213-240
  36. Smith R. A. Certain differential equations have only isolated periodic orbits. Ann. Mat. Pura Appl. 1984. Vol. 137, no. 1, 217-244
  37. Smith R. A. Poincaré index theorem concerning periodic orbits of differential equations. Proc. Lond. Math. Soc. 1984. Vol. 3, no. 2, P. 341-362
  38. So J. W-H., Wu J. Topological dimensions of global attractors for semilinear PDE's with delays. Bull. Aust. Math. Soc. 1991. Vol. 43, no. 3, P. 407-422
  39. Suarez M. J., Schopf P. S. A delayed action oscillator for ENSO. J. Atmos. Sci. 1988. Vol. 45, no. 21, P. 3283-3287
  40. Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, 1997
  41. Thieullen P. Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 1992. Vol. 4, no. 1, 127-159
  42. Webb G. F., Badii M. Nonlinear nonautonomous functional differential equations in !!!! ERROR!!! IMAGE IS NOT ALLOWED! spaces. Nonlinear Anal-Theor. 1981. Vol. 5, no. 2, 203-223
  43. Webb G. F. Functional differential equations and nonlinear semigroups in !!!! ERROR!!! IMAGE IS NOT ALLOWED! -spaces. J. Differ. Equations. 1976. Vol. 20, no. 1, 71-89
  44. Zelik S. Inertial manifolds and finite-dimensional reduction for dissipative PDEs. P. Roy. Soc. Edinb. A. 2014. Vol. 144, no. 6, 1245-1327

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