Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems

Takáč, Peter

doi:10.1090/S0002-9939-1992-1098406-9

Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems
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Proc. Amer. Math. Soc. 115 (1992), 691-698 Request permission

Abstract:

We construct two simple examples of strongly monotone time-periodic dynamical systems (of period $\tau > 0$) possessing linearly stable subharmonic orbits of period $n\tau$ for any integer $n \geq 2$. The first example is an irreducible cooperative system of four ODE’s that models positive feedback. The second example is a one-dimensional reaction-diffusion PDE with periodic boundary conditions. Our construction employs Chebyshev’s polynomials.

References

Nicholas D. Alikakos and Peter Hess, On stabilization of discrete monotone dynamical systems, Israel J. Math. 59 (1987), no. 2, 185–194. MR 920081, DOI 10.1007/BF02787260
Nicholas D. Alikakos, Peter Hess, and Hiroshi Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations 82 (1989), no. 2, 322–341. MR 1027972, DOI 10.1016/0022-0396(89)90136-8
Herbert Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 593–676. MR 808425
Herbert Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 47–83. MR 799657, DOI 10.1515/crll.1985.360.47
Xu-Yan Chen and Hiroshi Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), no. 1, 160–190. MR 986159, DOI 10.1016/0022-0396(89)90081-8
E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math. 419 (1991), 125–139. MR 1116922
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
J. K. Hale and A. S. Somolinos, Competition for fluctuating nutrient, J. Math. Biol. 18 (1983), no. 3, 255–280. MR 729974, DOI 10.1007/BF00276091
P. Hess and H. Weinberger, Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses, J. Math. Biol. 28 (1990), no. 1, 83–98. MR 1036413, DOI 10.1007/BF00171520
Morris W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 267–285. MR 706104
Morris W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 1–64. MR 741723, DOI 10.1090/S0273-0979-1984-15236-4
Morris W. Hirsch, Attractors for discrete-time monotone dynamical systems in strongly ordered spaces, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 141–153. MR 827267, DOI 10.1007/BFb0075221
Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1–53. MR 921986, DOI 10.1515/crll.1988.383.1
N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795
Hiroshi Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$, Nonlinear diffusion equations and their equilibrium states, II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 13, Springer, New York, 1988, pp. 139–162. MR 956085, DOI 10.1007/978-1-4613-9608-6_{8}
Roger D. Nussbaum, Positive operators and elliptic eigenvalue problems, Math. Z. 186 (1984), no. 2, 247–264. MR 741305, DOI 10.1007/BF01161807
Peter Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), no. 1, 89–110. MR 997611, DOI 10.1016/0022-0396(89)90115-0
James F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38 (1980), no. 1, 80–103. MR 592869, DOI 10.1016/0022-0396(80)90026-1
H. L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Anal. 10 (1986), no. 10, 1037–1052. MR 857738, DOI 10.1016/0362-546X(86)90087-8
Hal L. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev. 30 (1988), no. 1, 87–113. MR 931279, DOI 10.1137/1030003
Hal L. Smith and Horst R. Thieme, Quasi convergence and stability for strongly order-preserving semiflows, SIAM J. Math. Anal. 21 (1990), no. 3, 673–692. MR 1046795, DOI 10.1137/0521036
Hal L. Smith and Horst R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal. 22 (1991), no. 4, 1081–1101. MR 1112067, DOI 10.1137/0522070
Peter Takáč, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1990), no. 1, 223–244. MR 1052057, DOI 10.1016/0022-247X(90)90040-M
Peter Takáč, Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology, Nonlinear Anal. 14 (1990), no. 1, 35–42. MR 1028245, DOI 10.1016/0362-546X(90)90133-2
P. Takáč, Domains of attraction of generic $\omega$-limit sets for strongly monotone semiflows, Z. Anal. Anwendungen 10 (1991), no. 3, 275–317. MR 1155611, DOI 10.4171/ZAA/452
Peter Takáč, Domains of attraction of generic $\omega$-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math. 423 (1992), 101–173. MR 1142485, DOI 10.1515/crll.1992.423.101
Peter Takáč, Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry, J. Differential Equations 100 (1992), no. 2, 355–378. MR 1194815, DOI 10.1016/0022-0396(92)90119-8