Branches of forced oscillations induced by a delayed periodic force. (English) Zbl 1412.70025
Summary: We study global continuation properties of the set of \(T\)-periodic solutions of parameterized second order delay differential equations with constant time lag on smooth manifolds. We apply our results to get multiplicity of \(T\)-periodic solutions. Our topological approach is mainly based on the notion of degree of a tangent vector field.
MSC:
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 34K13 | Periodic solutions to functional-differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 37N05 | Dynamical systems in classical and celestial mechanics |
Keywords:
periodic solutions; forced motion; delay differential equations; multiplicity of periodic solutions; forced motion on manifolds; degree of a tangent vector fieldReferences:
| [1] | P. Albers, U. Frauenfelder and F. Schlenk, An iterated graph construction and periodic orbits of Hamiltonian delay equations, preprint (2018), .; <element-citation publication-type=”other“> Albers, P.Frauenfelder, U.Schlenk, F.An iterated graph construction and periodic orbits of Hamiltonian delay equationsPreprint2018 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1802.07449v2 · Zbl 1491.37052 |
| [2] | P. Albers, U. Frauenfelder and F. Schlenk, What might a Hamiltonian delay equation be?, preprint (2018), .; <element-citation publication-type=”other“> Albers, P.Frauenfelder, U.Schlenk, F.What might a Hamiltonian delay equation be?Preprint2018 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1802.07453v2 · Zbl 1451.34088 |
| [3] | B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn, Delay Differential Equations. Recent Advances and new Directions, Springer, New York, 2009.; Balachandran, B.; Kalmár-Nagy, T.; Gilsinn, D. E., Delay Differential Equations. Recent Advances and new Directions (2009) · Zbl 1162.34003 |
| [4] | P. Benevieri, A. Calamai, M. Furi and M. P. Pera, Delay differential equations on manifolds and applications to motion problems for forced constrained systems, Z. Anal. Anwend. 28 (2009), no. 4, 451-474.; Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P., Delay differential equations on manifolds and applications to motion problems for forced constrained systems, Z. Anal. Anwend., 28, 4, 451-474 (2009) · Zbl 1195.34108 |
| [5] | P. Benevieri, A. Calamai, M. Furi and M. P. Pera, On general properties of retarded functional differential equations on manifolds, Discrete Contin. Dyn. Syst. 33 (2013), no. 1, 27-46.; Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P., On general properties of retarded functional differential equations on manifolds, Discrete Contin. Dyn. Syst., 33, 1, 27-46 (2013) · Zbl 1271.34062 |
| [6] | P. Benevieri, A. Calamai, M. Furi and M. P. Pera, Global continuation of forced oscillations of retarded motion equations on manifolds, J. Fixed Point Theory Appl. 16 (2014), no. 1-2, 273-300.; Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P., Global continuation of forced oscillations of retarded motion equations on manifolds, J. Fixed Point Theory Appl., 16, 1-2, 273-300 (2014) · Zbl 1321.34091 |
| [7] | P. Benevieri, A. Calamai, M. Furi and M. P. Pera, On general properties of n-th order retarded functional differential equations, Rend. Istit. Mat. Univ. Trieste 49 (2017), 73-93.; Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P., On general properties of n-th order retarded functional differential equations, Rend. Istit. Mat. Univ. Trieste, 49, 73-93 (2017) · Zbl 1448.34128 |
| [8] | L. Bisconti and M. Spadini, About the notion of non-T-resonance and applications to topological multiplicity results for ODEs on differentiable manifolds, Math. Methods Appl. Sci. 38 (2015), no. 18, 4760-4773.; Bisconti, L.; Spadini, M., About the notion of non-T-resonance and applications to topological multiplicity results for ODEs on differentiable manifolds, Math. Methods Appl. Sci., 38, 18, 4760-4773 (2015) · Zbl 1342.34058 |
| [9] | T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover Publications, Mineola, 2005.; Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations (2005) · Zbl 1209.34001 |
| [10] | A. Calamai, M. P. Pera and M. Spadini, Multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force, Nonlinear Anal. 151 (2017), 252-264.; Calamai, A.; Pera, M. P.; Spadini, M., Multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force, Nonlinear Anal., 151, 252-264 (2017) · Zbl 1381.34087 |
| [11] | A. Calamai, M. P. Pera and M. Spadini, Multiplicity of forced oscillations for scalar retarded functional differential equations, Math. Methods Appl. Sci. 41 (2018), no. 5, 1944-1953.; Calamai, A.; Pera, M. P.; Spadini, M., Multiplicity of forced oscillations for scalar retarded functional differential equations, Math. Methods Appl. Sci., 41, 5, 1944-1953 (2018) · Zbl 1395.34074 |
| [12] | J. Cronin, Differential Equations, 2nd ed., Monogr. Textb. Pure Appl. Math. 180, Marcel Dekker, New York, 1994.; Cronin, J., Differential Equations (1994) · Zbl 0798.34001 |
| [13] | T. Erneux, Applied Delay Differential Equations, Surv. Tutor. Appl. Math. Sci. 3, Springer, New York, 2009.; Erneux, T., Applied Delay Differential Equations (2009) · Zbl 1201.34002 |
| [14] | M. Furi, Second order differential equations on manifolds and forced oscillations, Topological Methods in Differential Equations and Inclusions (Montreal 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472, Kluwer Academic, Dordrecht (1995), 89-127.; Furi, M., Second order differential equations on manifolds and forced oscillations. Topological Methods in Differential Equations and Inclusions, Miscellaneous, 89-127 (1995) · Zbl 0843.58008 |
| [15] | M. Furi, M. P. Pera and M. Spadini, Forced oscillations on manifolds and multiplicity results for periodically perturbed autonomous systems, J. Comput. Appl. Math. 113 (2000), no. 1-2, 241-254.; Furi, M.; Pera, M. P.; Spadini, M., Forced oscillations on manifolds and multiplicity results for periodically perturbed autonomous systems, J. Comput. Appl. Math., 113, 1-2, 241-254 (2000) · Zbl 0936.37038 |
| [16] | M. Furi, M. P. Pera and M. Spadini, Periodic solutions of retarded functional perturbations of autonomous differential equations on manifolds, Commun. Appl. Anal. 15 (2011), no. 2-4, 381-394.; Furi, M.; Pera, M. P.; Spadini, M., Periodic solutions of retarded functional perturbations of autonomous differential equations on manifolds, Commun. Appl. Anal., 15, 2-4, 381-394 (2011) · Zbl 1235.34188 |
| [17] | M. Furi and M. Spadini, Multiplicity of forced oscillations for the spherical pendulum, Topol. Methods Nonlinear Anal. 11 (1998), no. 1, 147-157.; Furi, M.; Spadini, M., Multiplicity of forced oscillations for the spherical pendulum, Topol. Methods Nonlinear Anal., 11, 1, 147-157 (1998) · Zbl 0913.58049 |
| [18] | M. Furi and M. Spadini, Branches of forced oscillations for periodically perturbed second order autonomous ODEs on manifolds, J. Differential Equations 154 (1999), no. 1, 96-106.; Furi, M.; Spadini, M., Branches of forced oscillations for periodically perturbed second order autonomous ODEs on manifolds, J. Differential Equations, 154, 1, 96-106 (1999) · Zbl 0929.34032 |
| [19] | M. Furi and M. Spadini, Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud. 9 (2009), no. 2, 263-276.; Furi, M.; Spadini, M., Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud., 9, 2, 263-276 (2009) · Zbl 1193.34141 |
| [20] | V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, 1974.; Guillemin, V.; Pollack, A., Differential Topology (1974) · Zbl 0361.57001 |
| [21] | M. W. Hirsch, Differential Topology, Grad. Texts in Math. 33, Springer, New York, 1976.; Hirsch, M. W., Differential Topology (1976) · Zbl 0356.57001 |
| [22] | M. Lewicka and M. Spadini, On the genericity of the multiplicity results for forced oscillations on compact manifolds, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 4, 357-369.; Lewicka, M.; Spadini, M., On the genericity of the multiplicity results for forced oscillations on compact manifolds, NoDEA Nonlinear Differential Equations Appl., 6, 4, 357-369 (1999) · Zbl 0952.34032 |
| [23] | J. Mawhin, Seventy-five years of global analysis around the forced pendulum equation, Proceedings of Equadiff 9, Masaryk University, Brno (1998), 115-145.; Mawhin, J., Seventy-five years of global analysis around the forced pendulum equation, Proceedings of Equadiff 9, 115-145 (1998) · Zbl 1076.01500 |
| [24] | J. Mawhin, Global results for the forced pendulum equation, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam (2004), 533-589.; Mawhin, J., Global results for the forced pendulum equation, Handbook of Differential Equations, 533-589 (2004) · Zbl 1091.34019 |
| [25] | J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, 1965.; Milnor, J. W., Topology from the Differentiable Viewpoint (1965) · Zbl 0136.20402 |
| [26] | P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513.; Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
