Concluding remarks to problem of Moser and conjecture of Mawhin
Abstract
Uniqueness, exact multiplicity and stability of periodic solutions to the periodically forced pendulum equation are discussed. All of this can be considered as a further specification of contributions to the problem of Moser and especially Mawhin's conjecture.
References
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2. J. Andres, Further remarks on problem of Moser and conjecture of Mawhin, To appear in Topol. Meth. Nonlin. Anal., 6, 1 (1995).
3. F. Battelli, K.J. Palmer, Chaos in the Duffing equation, J. Diff. Eqns, 101, 2 (1993), 276-301.
4. J.-M. Belley, G. Fournier, K. Saadi Drissi, Almost periodic weak solutions to forced pendulum type equations without friction, Aequationes Math., 44, 1 (1992), 100-108.
5. V.N. Belykh, N.F. Pedersen, O.H. Soerenson, Shunted-Josephson-junction model, Phys. Rev. B, 16, 11 (1977), 4853-4871.
6. A. Benseny, C. Olivé, High precision angles between invariant manifolds for rapidly forced Hamiltonian systems, In: Internat. Conf. on Diff. Eqns, Barcelona 1991 (Ed. by C. Perelló, C. Simo, J. Solà-Morales), World Scientific, Singapore, 1993, 315-319.
7. L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer, Berlin, 1959.
8. A. Delshams, T.M. Seara, Splitting of separatrices in rapidly forced systems, In: Internat. Conf. on Diff. Eqns, Barcelona 1991 (Ed. by C. Perelló, C. Simo, J. Solà-Morales), World Scientific, Singapore, 1993, 103-113.
9. F. Donati, Sur l'existence de quatre solutions périodiques pour l'équation du pendule forcé, C. R. Acad. Sci. Paris, 317, 1 (1993), 667-672.
10. F. Donati, Some remarks on periodic solutions of the forced pendulum equation, Diff. and Integral Eqns., 8, 1 (1995), 141-149.
11. I. Goldrish, Y. Imry, G. Wassenman, E. Ben-Jacob, Studies of the intermittent-type chaos in Ac- and Dc-driven Josephson junction, Phys. Rev. B, 29, 3 (1984), 1218-1231.
12. D.R. He, W.J. Yeh, Y.H. Kao, Transition from quasiperiodicity to chaos in a Josephson-junction analog, Phys. Rev. B, 30 (1984), 197.
13. K. Hocket, P. Holmes, Nonlinear oscillations, iterated maps, symbolic dynamics, and knotted orbits, Proceed. of the IEEE, 75, 8 (1987), 1071-1080.
14. P. Holmes, Poincaré, celestial mechanics, dynamical-systems theory and "chaos", Phys. Rep. 193, 3 (1990), 137-163.
15. P. Holmes, J. Marsden, J. Scheurle, Exponentially small splitting of separatrices with applications, Contemp. Maths, 89 (1988), 213-244.
16. B.A. Huberman, J.D. Crutchfield, N.H. Packard, Noise phenomena in Josephson junctions, Appl. Phys. Lett., 37 (1980), 750-772.
17. T.J. Kaper, S. Wiggins, On the structure of separatrix-swept regions in singularly-perturbed Hamiltonian systems, Diff. and Integral Eqns, 5, 6 (1992), 1363-1381.
18. R.L. Kautz, Chaotic states of Rf-biased Josephson junction, J. Appl. Phys., 52 (1981), 6241.
19. M. Levi, Beating modes in the Josephson junction, In: Chaos in Nonlinear Dynamical Systems (Ed. J. Chandra), SIAM, Philadelphia, 1984, 56-73.
20. M. Levi, F.C. Hoppenstaedt, W.L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math., 36 (1978), 167-198.
21. C. Olech, Asymptotic behavior of solutions of second order differential equations, Bull. Acad. Polon. Sci., 7 (1959), 319-326.
22. R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. U. M. I., (7), 3-B (1989), 533-546.
23. R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 2, 42 (1990), 505-516.
24. R. Ortega, The first interval of stability of a periodic equation of Duffing type, Proceed. of the Amer. Math. Soc., 115, 4 (1992), 1061-1067.
25. K.J. Palmer, Exponential dichotomy and transversal homoclinic points, J. Diff. Eqns, 55 (1984), 225-256.
26. R. Reissig, Continua of periodic solutions of the Liénard equation, In: Constructive methods for nonlinear boundary value problems and nonlinear oscillations (Ed. J. Albrecht, L. Collatz, K. Kirchgässner), INSM 48, Birkhäuser, Basel, 1979, 126-133.
27. J.A. Sanders, The (driven) Josephson equation: an exercise in asymptotics, In: Asymptotic analysis II (Ed. F. Verhulst), LNM 985, Springer, Berlin, 1983.
28. G. Sansone, L'équation θ" + f(θ,α)h(θ') = g(θ) + p(t), J. Math. Pures Appl., 9, 43 (1964), 149-175.
29. G. Sansone, Existence et stabilité asymptotique uniforme d'une solution périodique de l'équation θ" + f(θ,α)h(θ') = g(θ) + p(t), In: Les vibration forcées dans les systèmes non-lineaires. Colloq. Internat, du Centre National de la Rech. Sci. no 148 (Marseille, September 1964), 1965, 97-106.
30. B.V. Schmitt, N. Sari, Solutions périodiques paires et harmoniques-impaires de l'équation du pendule forcé, J. Mécan. Théor. Appl., 3 (1984), 979-993.
31. B.V. Schmitt, N. Sari, Sur la structure de l'équation du pendule forcé, J. Mécan. Théor. Appl., 4, 5 (1985), 615-628.
32. G. Seifert, On stability in the large for periodic solutions of differential systems, Ann. Math., 2, 67 (1958), 83-89.
33. G. Seifert, The asymptotic behavior of solutions of pendulum type equations, Ann. Math., 1, 69 (1959), 75-87.
34. G. Tarantello, On the number of solutions for the forced pendulum equation, J. DifF. Eqns, 80 (1989), 79-93.
2. J. Andres, Further remarks on problem of Moser and conjecture of Mawhin, To appear in Topol. Meth. Nonlin. Anal., 6, 1 (1995).
3. F. Battelli, K.J. Palmer, Chaos in the Duffing equation, J. Diff. Eqns, 101, 2 (1993), 276-301.
4. J.-M. Belley, G. Fournier, K. Saadi Drissi, Almost periodic weak solutions to forced pendulum type equations without friction, Aequationes Math., 44, 1 (1992), 100-108.
5. V.N. Belykh, N.F. Pedersen, O.H. Soerenson, Shunted-Josephson-junction model, Phys. Rev. B, 16, 11 (1977), 4853-4871.
6. A. Benseny, C. Olivé, High precision angles between invariant manifolds for rapidly forced Hamiltonian systems, In: Internat. Conf. on Diff. Eqns, Barcelona 1991 (Ed. by C. Perelló, C. Simo, J. Solà-Morales), World Scientific, Singapore, 1993, 315-319.
7. L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer, Berlin, 1959.
8. A. Delshams, T.M. Seara, Splitting of separatrices in rapidly forced systems, In: Internat. Conf. on Diff. Eqns, Barcelona 1991 (Ed. by C. Perelló, C. Simo, J. Solà-Morales), World Scientific, Singapore, 1993, 103-113.
9. F. Donati, Sur l'existence de quatre solutions périodiques pour l'équation du pendule forcé, C. R. Acad. Sci. Paris, 317, 1 (1993), 667-672.
10. F. Donati, Some remarks on periodic solutions of the forced pendulum equation, Diff. and Integral Eqns., 8, 1 (1995), 141-149.
11. I. Goldrish, Y. Imry, G. Wassenman, E. Ben-Jacob, Studies of the intermittent-type chaos in Ac- and Dc-driven Josephson junction, Phys. Rev. B, 29, 3 (1984), 1218-1231.
12. D.R. He, W.J. Yeh, Y.H. Kao, Transition from quasiperiodicity to chaos in a Josephson-junction analog, Phys. Rev. B, 30 (1984), 197.
13. K. Hocket, P. Holmes, Nonlinear oscillations, iterated maps, symbolic dynamics, and knotted orbits, Proceed. of the IEEE, 75, 8 (1987), 1071-1080.
14. P. Holmes, Poincaré, celestial mechanics, dynamical-systems theory and "chaos", Phys. Rep. 193, 3 (1990), 137-163.
15. P. Holmes, J. Marsden, J. Scheurle, Exponentially small splitting of separatrices with applications, Contemp. Maths, 89 (1988), 213-244.
16. B.A. Huberman, J.D. Crutchfield, N.H. Packard, Noise phenomena in Josephson junctions, Appl. Phys. Lett., 37 (1980), 750-772.
17. T.J. Kaper, S. Wiggins, On the structure of separatrix-swept regions in singularly-perturbed Hamiltonian systems, Diff. and Integral Eqns, 5, 6 (1992), 1363-1381.
18. R.L. Kautz, Chaotic states of Rf-biased Josephson junction, J. Appl. Phys., 52 (1981), 6241.
19. M. Levi, Beating modes in the Josephson junction, In: Chaos in Nonlinear Dynamical Systems (Ed. J. Chandra), SIAM, Philadelphia, 1984, 56-73.
20. M. Levi, F.C. Hoppenstaedt, W.L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math., 36 (1978), 167-198.
21. C. Olech, Asymptotic behavior of solutions of second order differential equations, Bull. Acad. Polon. Sci., 7 (1959), 319-326.
22. R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. U. M. I., (7), 3-B (1989), 533-546.
23. R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 2, 42 (1990), 505-516.
24. R. Ortega, The first interval of stability of a periodic equation of Duffing type, Proceed. of the Amer. Math. Soc., 115, 4 (1992), 1061-1067.
25. K.J. Palmer, Exponential dichotomy and transversal homoclinic points, J. Diff. Eqns, 55 (1984), 225-256.
26. R. Reissig, Continua of periodic solutions of the Liénard equation, In: Constructive methods for nonlinear boundary value problems and nonlinear oscillations (Ed. J. Albrecht, L. Collatz, K. Kirchgässner), INSM 48, Birkhäuser, Basel, 1979, 126-133.
27. J.A. Sanders, The (driven) Josephson equation: an exercise in asymptotics, In: Asymptotic analysis II (Ed. F. Verhulst), LNM 985, Springer, Berlin, 1983.
28. G. Sansone, L'équation θ" + f(θ,α)h(θ') = g(θ) + p(t), J. Math. Pures Appl., 9, 43 (1964), 149-175.
29. G. Sansone, Existence et stabilité asymptotique uniforme d'une solution périodique de l'équation θ" + f(θ,α)h(θ') = g(θ) + p(t), In: Les vibration forcées dans les systèmes non-lineaires. Colloq. Internat, du Centre National de la Rech. Sci. no 148 (Marseille, September 1964), 1965, 97-106.
30. B.V. Schmitt, N. Sari, Solutions périodiques paires et harmoniques-impaires de l'équation du pendule forcé, J. Mécan. Théor. Appl., 3 (1984), 979-993.
31. B.V. Schmitt, N. Sari, Sur la structure de l'équation du pendule forcé, J. Mécan. Théor. Appl., 4, 5 (1985), 615-628.
32. G. Seifert, On stability in the large for periodic solutions of differential systems, Ann. Math., 2, 67 (1958), 83-89.
33. G. Seifert, The asymptotic behavior of solutions of pendulum type equations, Ann. Math., 1, 69 (1959), 75-87.
34. G. Tarantello, On the number of solutions for the forced pendulum equation, J. DifF. Eqns, 80 (1989), 79-93.
AndresJ. (1996). Concluding remarks to problem of Moser and conjecture of Mawhin. Annales Mathematicae Silesianae, 10, 57-65. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14192
Jan Andres
Department of Mathematical Analysis, Faculty of Science, Palacký Univeristy, Czech Republic Czechia
Department of Mathematical Analysis, Faculty of Science, Palacký Univeristy, Czech Republic Czechia
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