Tricomi's Φ-equation
Main Article Content
Keywords
Periodic solutions, attractors, stability
Abstract
We study an autonomous nonlinear differential equation that models the movement of a damped Φ-pendulum with constant forcing. In the dissipative case, two results are presented, on the one hand, using the application of Poincaré and energy functions, a sufficient criterion is established to guarantee the existence, uniqueness and asymptotic stability of a periodic solution of the second kind and on the other hand, a criterion is presented with which the basin of attraction of an asymptotically stable equilibrium is estimated analytically with the help of the Lasalle’s invariance principle. While in the conservative case there are necessary conditions for range of the period function to be defined in an unbounded interval. The results
obtained in the dissipative case are a generalization of those established by Tricomi in the newtonian case.
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References
[1] S. H. Strogatz, Nonlinear Dynamics and Chaos with applications to Physics,Biology, Chemestry and Engineering. Perseus Books Publishing, 1994. 12
[2] C. Chicone, Ordinary Differential equations with Applications, 3rd ed., Sprin-ger, Ed. Springer, 2006. 12
[3] M. W. Hirsch, S. Smale, and R. L. Devaney,Differential Equations, Dyna-mical Systems An Introduction to Chaos, 3rd ed. Elsevier, 2004. 12, 20
[4] J. Mawhin,Handbook of Differential Equations: Ordinary Differen-tial equations. Elsevier, 2004, vol. 1, ch. Global Results forthe Forced Pendulum Equation, pp. 533–589. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S1874572500800085 12,14
[5] F. Tricomi, “Integrazione di un’equazione differenziale presentatasi in elettro-tecnica,”Annali della Scuola Normale Superiore di Pisa- Classe di Scienze,vol. 2, pp. 1–20, 1933. 12
[6] ——, “Sur une équation différentielle de l’électrotechnique,”Comptes RendusMathematique Academie des Sciences, Paris., vol. 193, pp. 635–636, 1931. 12
[7] R. Martins, “The effect of inversely unstable solutions on the at-tractor of the forced pendulum equation with friction,”Journal ofDifferential Equations, vol. 212, pp. 351–365, 2005. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022039604002657 13
[8] ——, “The attractor of an equation of tricomi’s type,”Journal ofMathematical Analysis and Applications, vol. 342, 2008. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X08000371 13
[9] J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields, Springer, Ed. Springer, 1997. 13
[10] M. Levi, F. C. Hoppensteadt, and W. L. Miranker, “Dynamics of thejosephson junction,”Quarterly of Applied Mathematics, vol. 36, pp. 167–198,1978. [Online]. Available: http://www.jstor.org/stable/43636917. 13
[11] G. Carapella, G. Costabile, N. Martucciello, M. Cirillo, R. Latempa,A. Polcari, and G. Filatrella, “Experimental realization of a relativisticfluxon ratchet,”Physica C, vol. 382, pp. 337–341, 2002. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0921453402012327 13
[12] A. Kuijper, “p-laplacian driven image processing,”2007 IEEE InternationalConference on Image Processing, vol. 5, pp. V–257–V–260, Septiembre 2007.13
[13] J. Tome ̆cek, “Periodic solution of differential equation withφ-laplacianand state-dependent impulses,”Journal of Mathematical Analysisand Applications, vol. 450, pp. 1029–1046, 2017. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X17300781 13
[14] J. Burkotová, I. Rachůnková, M. Rohleder, and J. Stryja, “Existence anduniqueness of damped solutions of singular ivps with phi-laplacian,”Elec-tronic Journal of Qualitative Theory of Differential Equations, vol. 121, pp.1–28, 2016. [Online]. Available: https://doi.org/10.14232/ejqtde.2016.1.12114
[15] Z. Do ̆slá, M. Cecchi, and M. Marini, “Asymptotic problems for differen-tial equations with boundedφ-laplacian,”Electronic Journal of QualitativeTheory of Differential Equations, no. 9, pp. 1–18, 2009. 14
[16] H. D. R and P. Radu, “Existence, localization and multiplicity ofpositive solutions toφ-laplace equations and systems,”Taiwanese JournalOf Mathematics, vol. 20, no. 1, pp. 77–89, 2016. [Online]. Available:https://doi.org/10.11650/tjm.20.2016.5553 14
[17] U. Kaufmann and L. Milne, “Positive solutions for nonlinear problemsinvolving the one-dimensionalφ-laplacian,”Journal of Mathematical Analysisand Applications, vol. 461, no. 1, pp. 24–37, mayo 2018. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X17311447 14
[18] S. Pérez-González, J. Torregrosa, and P. J. Torres, “Existen-ce and uniqueness of limit cycles for generalizedφ-laplacian lié-nard equations,”Journal of Mathematical Analysis and Applica-tions, vol. 439, no. 2, pp. 745–765, 2016. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X16002225 14
[19] P. J. Torres, “Nondegeneracy of the periodically forced liénard dif-ferential equation withφ-laplacian,”Communications in ContemporaryMathematics, vol. 13, no. 2, pp. 283–292, 2011. [Online]. Available:https://www.worldscientific.com/doi/abs/10.1142/S0219199711004208 14
[20] J. A. Wright, J. H. Deane, M. Bartuccelli, and G. Gentile,“Basins of attraction in forced systems with time-varying dissi-pation,”Communications in Nonlinear Science and Numerical Si-mulation, vol. 29, no. 1-3, pp. 72–87, 2015. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S1007570415001471 14
[21] A. Cabada and F. A. Fernández, “Periodic solutions for some phi-laplacianand reflection equations,”Boundary Value Problems, vol. 2016, no. 1, p. 56,Feb 2016. [Online]. Available: https://doi.org/10.1186/s13661-016-0565-z 14
[22] L. Perko,Differential equations and dynamical systems. Springer-Verlag,1991. 18, 19, 22, 23|
[2] C. Chicone, Ordinary Differential equations with Applications, 3rd ed., Sprin-ger, Ed. Springer, 2006. 12
[3] M. W. Hirsch, S. Smale, and R. L. Devaney,Differential Equations, Dyna-mical Systems An Introduction to Chaos, 3rd ed. Elsevier, 2004. 12, 20
[4] J. Mawhin,Handbook of Differential Equations: Ordinary Differen-tial equations. Elsevier, 2004, vol. 1, ch. Global Results forthe Forced Pendulum Equation, pp. 533–589. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S1874572500800085 12,14
[5] F. Tricomi, “Integrazione di un’equazione differenziale presentatasi in elettro-tecnica,”Annali della Scuola Normale Superiore di Pisa- Classe di Scienze,vol. 2, pp. 1–20, 1933. 12
[6] ——, “Sur une équation différentielle de l’électrotechnique,”Comptes RendusMathematique Academie des Sciences, Paris., vol. 193, pp. 635–636, 1931. 12
[7] R. Martins, “The effect of inversely unstable solutions on the at-tractor of the forced pendulum equation with friction,”Journal ofDifferential Equations, vol. 212, pp. 351–365, 2005. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022039604002657 13
[8] ——, “The attractor of an equation of tricomi’s type,”Journal ofMathematical Analysis and Applications, vol. 342, 2008. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X08000371 13
[9] J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields, Springer, Ed. Springer, 1997. 13
[10] M. Levi, F. C. Hoppensteadt, and W. L. Miranker, “Dynamics of thejosephson junction,”Quarterly of Applied Mathematics, vol. 36, pp. 167–198,1978. [Online]. Available: http://www.jstor.org/stable/43636917. 13
[11] G. Carapella, G. Costabile, N. Martucciello, M. Cirillo, R. Latempa,A. Polcari, and G. Filatrella, “Experimental realization of a relativisticfluxon ratchet,”Physica C, vol. 382, pp. 337–341, 2002. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0921453402012327 13
[12] A. Kuijper, “p-laplacian driven image processing,”2007 IEEE InternationalConference on Image Processing, vol. 5, pp. V–257–V–260, Septiembre 2007.13
[13] J. Tome ̆cek, “Periodic solution of differential equation withφ-laplacianand state-dependent impulses,”Journal of Mathematical Analysisand Applications, vol. 450, pp. 1029–1046, 2017. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X17300781 13
[14] J. Burkotová, I. Rachůnková, M. Rohleder, and J. Stryja, “Existence anduniqueness of damped solutions of singular ivps with phi-laplacian,”Elec-tronic Journal of Qualitative Theory of Differential Equations, vol. 121, pp.1–28, 2016. [Online]. Available: https://doi.org/10.14232/ejqtde.2016.1.12114
[15] Z. Do ̆slá, M. Cecchi, and M. Marini, “Asymptotic problems for differen-tial equations with boundedφ-laplacian,”Electronic Journal of QualitativeTheory of Differential Equations, no. 9, pp. 1–18, 2009. 14
[16] H. D. R and P. Radu, “Existence, localization and multiplicity ofpositive solutions toφ-laplace equations and systems,”Taiwanese JournalOf Mathematics, vol. 20, no. 1, pp. 77–89, 2016. [Online]. Available:https://doi.org/10.11650/tjm.20.2016.5553 14
[17] U. Kaufmann and L. Milne, “Positive solutions for nonlinear problemsinvolving the one-dimensionalφ-laplacian,”Journal of Mathematical Analysisand Applications, vol. 461, no. 1, pp. 24–37, mayo 2018. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X17311447 14
[18] S. Pérez-González, J. Torregrosa, and P. J. Torres, “Existen-ce and uniqueness of limit cycles for generalizedφ-laplacian lié-nard equations,”Journal of Mathematical Analysis and Applica-tions, vol. 439, no. 2, pp. 745–765, 2016. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022247X16002225 14
[19] P. J. Torres, “Nondegeneracy of the periodically forced liénard dif-ferential equation withφ-laplacian,”Communications in ContemporaryMathematics, vol. 13, no. 2, pp. 283–292, 2011. [Online]. Available:https://www.worldscientific.com/doi/abs/10.1142/S0219199711004208 14
[20] J. A. Wright, J. H. Deane, M. Bartuccelli, and G. Gentile,“Basins of attraction in forced systems with time-varying dissi-pation,”Communications in Nonlinear Science and Numerical Si-mulation, vol. 29, no. 1-3, pp. 72–87, 2015. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S1007570415001471 14
[21] A. Cabada and F. A. Fernández, “Periodic solutions for some phi-laplacianand reflection equations,”Boundary Value Problems, vol. 2016, no. 1, p. 56,Feb 2016. [Online]. Available: https://doi.org/10.1186/s13661-016-0565-z 14
[22] L. Perko,Differential equations and dynamical systems. Springer-Verlag,1991. 18, 19, 22, 23|
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Published
Jun 15, 2018
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How to Cite
Castro G., D. A., & Gutiérrez G., A. (2018). Tricomi’s Φ-equation. Ingeniería Y Ciencia, 14(27), 11–28. https://doi.org/10.17230/ingciencia.14.27.1
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