-
Previous Article
Non-isothermal cyclic fatigue in an oscillating elastoplastic beam
- CPAA Home
- This Issue
-
Next Article
Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition
Free vibrations in space of the single mode for the Kirchhoff string
1. | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy |
References:
[1] |
M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," Chapters 16, 17, New York: Dover, National bureau of standards, 1964. |
[2] |
L. P. Bonorino, E. H. M. Brietzke, J. P. Lukaszczyk and C. A. Taschetto, Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation, J. Differential Equations, 214 (2005), 156-175.
doi: 10.1016/j.jde.2004.08.007. |
[3] |
G. F. Carrier, On the non-linear vibration problem of an elastic string, Q. Appl. Math., 3 (1945), 157-165. |
[4] |
T. Cazenave and F. B. Weissler, Unstable simple modes of the nonlinear string, Q. Appl. Math., 54 (1996), 287-305. |
[5] |
C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.
doi: 10.1016/0022-0396(87)90122-7. |
[6] |
C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, New York, 2006. |
[7] |
A. Cima, A. Gasull and F. Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations, 168 (2000), 180-199.
doi: 10.1006/jdeq.2000.3912. |
[8] |
W. R. Dean, Note on the evaluation of an elliptic integral of the third kind, J. London Math. Soc., 18 (1943), 130-132.
doi: 10.1112/jlms/s1-18.3.130. |
[9] |
R. W. Dickey, Stability of periodic solutions of the non linear string,, Q. Appl. Math., 38 (): 253.
|
[10] |
G. Gallavotti, "The Elements of Mechanics," Springer-Verlag, New York, 1983. Also available from: Ipparco Editore, 2007. http://ipparco.roma1.infn.it/pagine/deposito/2007/elements.pdf. |
[11] |
M. Ghisi and M. Gobbino, Stability of simple modes of the Kirchhoff equation, Nonlinearity, 14 (2001), 1197-1220.
doi: 10.1088/0951-7715/14/5/314. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," Chapters 16, 17, New York: Dover, National bureau of standards, 1964. |
[2] |
L. P. Bonorino, E. H. M. Brietzke, J. P. Lukaszczyk and C. A. Taschetto, Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation, J. Differential Equations, 214 (2005), 156-175.
doi: 10.1016/j.jde.2004.08.007. |
[3] |
G. F. Carrier, On the non-linear vibration problem of an elastic string, Q. Appl. Math., 3 (1945), 157-165. |
[4] |
T. Cazenave and F. B. Weissler, Unstable simple modes of the nonlinear string, Q. Appl. Math., 54 (1996), 287-305. |
[5] |
C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.
doi: 10.1016/0022-0396(87)90122-7. |
[6] |
C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, New York, 2006. |
[7] |
A. Cima, A. Gasull and F. Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations, 168 (2000), 180-199.
doi: 10.1006/jdeq.2000.3912. |
[8] |
W. R. Dean, Note on the evaluation of an elliptic integral of the third kind, J. London Math. Soc., 18 (1943), 130-132.
doi: 10.1112/jlms/s1-18.3.130. |
[9] |
R. W. Dickey, Stability of periodic solutions of the non linear string,, Q. Appl. Math., 38 (): 253.
|
[10] |
G. Gallavotti, "The Elements of Mechanics," Springer-Verlag, New York, 1983. Also available from: Ipparco Editore, 2007. http://ipparco.roma1.infn.it/pagine/deposito/2007/elements.pdf. |
[11] |
M. Ghisi and M. Gobbino, Stability of simple modes of the Kirchhoff equation, Nonlinearity, 14 (2001), 1197-1220.
doi: 10.1088/0951-7715/14/5/314. |
[1] |
Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations and Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029 |
[2] |
Josselin Garnier. The role of evanescent modes in randomly perturbed single-mode waveguides. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 455-472. doi: 10.3934/dcdsb.2007.8.455 |
[3] |
Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297 |
[4] |
Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971 |
[5] |
Eva Sincich, Mourad Sini. Local stability for soft obstacles by a single measurement. Inverse Problems and Imaging, 2008, 2 (2) : 301-315. doi: 10.3934/ipi.2008.2.301 |
[6] |
Riccardo Adami, Diego Noja, Cecilia Ortoleva. Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5837-5879. doi: 10.3934/dcds.2016057 |
[7] |
Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287 |
[8] |
Dongyi Liu, Genqi Xu. Input-output $ L^2 $-well-posedness, regularity and Lyapunov stability of string equations on networks. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022007 |
[9] |
Cyrine Fitouri, Alain Haraux. Boundedness and stability for the damped and forced single well Duffing equation. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 211-223. doi: 10.3934/dcds.2013.33.211 |
[10] |
Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451 |
[11] |
Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571 |
[12] |
Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure and Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721 |
[13] |
Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315 |
[14] |
Sergei A. Avdonin, Boris P. Belinskiy. Controllability of a string under tension. Conference Publications, 2003, 2003 (Special) : 57-67. doi: 10.3934/proc.2003.2003.57 |
[15] |
Eduardo Liz. Local stability implies global stability in some one-dimensional discrete single-species models. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 191-199. doi: 10.3934/dcdsb.2007.7.191 |
[16] |
Giuseppe Gaeta, Sebastian Walcher. Higher order normal modes. Journal of Geometric Mechanics, 2020, 12 (3) : 421-434. doi: 10.3934/jgm.2020026 |
[17] |
Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301 |
[18] |
Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213 |
[19] |
Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048 |
[20] |
Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]