November  2012, 32(11): 4045-4067. doi: 10.3934/dcds.2012.32.4045

Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity

1. 

Dipartimento di Scienze Matematiche G. L. Lagrange, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

2. 

Dipartimento di Matematica e Informaticà, Universita di Udine, via delle Scienze 206, 33100 Udine, Italy

Received  September 2011 Revised  November 2011 Published  June 2012

We prove the existence of multiple periodic solutions as well as the presence of complex profiles (for a certain range of the parameters) for the steady-state solutions of a class of reaction-diffusion equations with a FitzHugh-Nagumo cubic type nonlinearity. An application is given to a second order ODE related to a myelinated nerve axon model.
Citation: Chiara Zanini, Fabio Zanolin. Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4045-4067. doi: 10.3934/dcds.2012.32.4045
References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

D. G. Aronson, N. V. Mantzaris and H. G. Othmer, Wave propagarion and blocking in inhomogeneous media, Discrete and Continuous Dynamical Systems, 13 (2005), 843-876. doi: 10.3934/dcds.2005.13.843.

[3]

D. G. Aronson and V. Padron, Pattern formation in a model of an injured nerve fiber, SIAM J. Appl. Math., 70 (2009), 789-802. doi: 10.1137/080732341.

[4]

S. M. Baer and J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: A continuum theory, J. Neurophys., 65 (1991), 874-890.

[5]

J. Belmonte-Beitia and P. J. Torres, Existence of dark soliton of the cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, J. Nonlinear Math. Phys., 15 (2008), 65-72. doi: 10.2991/jnmp.2008.15.s3.7.

[6]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022.

[7]

P.-L. Chen and J. Bell, Spine-density dependence of the qualitative behavior of a model of a nerve fiber with excitable spines, J. Math. Anal. Appl., 187 (1994), 384-410. doi: 10.1006/jmaa.1994.1364.

[8]

E. N. Dancer and S. Yan, Interior peak solutions for an elliptic system of FitzHugh-Nagumo type, J. Differential Equations, 229 (2006), 654-679. doi: 10.1016/j.jde.2006.02.001.

[9]

P. Grindrod and B. D. Sleeman, A model of a myelinated nerve axon: Threshold behaviour and propagation, J. Math. Biol., 23 (1985), 119-135. doi: 10.1007/BF00276561.

[10]

J. K. Hale, "Ordinary Differential Equations,'' Second edition, R. E. Krieger Publ. Co., Inc., Huntington, New York, 1980.

[11]

S. Hastings, Some mathematical problems from neurobiology, Amer. Math. Monthly, 82 (1975), 881-895. doi: 10.2307/2318490.

[12]

S. Hastings, Some mathematical problems arising inneurobiology, in "Mathematics of Biology," C.I.M.E. Summer Sch., 80, Springer, Heidelberg, (2010), 179-264.

[13]

J. Kennedy and J. A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513-2530. doi: 10.1090/S0002-9947-01-02586-7.

[14]

Y. Kominis and K. Hizanidis, Lattice solitons in self-defocusing optical media:Analytical solutions of the nonlinear Kronig-Penney model, Optics Letters, 31 (2006), 2888-2890. doi: 10.1364/OL.31.002888.

[15]

M. A. Krasnosel'skiĭ, "The Operator of Translation Along the Trajectories of Differential Equations,'' Translations of Mathematical Monographs, 19, American Mathematical Society, Providence, R.I., 1968.

[16]

T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316. doi: 10.1137/S0036139998349298.

[17]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,'' Appl. Math. Sci., 74, Springer-Verlag, New York, 1989.

[18]

H. P. McKean, Jr., Nagumo's equation, Advances in Math., 4 (1970), 209-223.

[19]

A. Mellet, J.-M. Roquejoffre and Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 26 (2010), 303-312.

[20]

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91.

[21]

J. X. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys., 73 (1993), 893-926. doi: 10.1007/BF01052815.

[22]

J. Yang, S. Kalliadasis, J. H. Merkin and S. K. Scott, Wave propagation in spatially distributed excitable media, SIAM J. Appl. Math., 63 (2002), 485-509. doi: 10.1137/S0036139901391409.

[23]

C. Zanini and F. Zanolin, Positive periodic solutions for ordinary differential equations arising in the study of nerve fiber models, in "Applied and Industrial Mathematics in Italy," Ser. Adv. Math. Appl. Sci., 69, World Sci. Publ., Hackensack, NJ, (2005), 564-575.

[24]

C. Zanini and F. Zanolin, Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model, Nonlinear Anal. Real World Appl., 9 (2008), 141-153.

[25]

C. Zanini and F. Zanolin, Complex dynamics in a nerve fiber model with periodic coefficients, Nonlinear Anal. Real World Appl., 10 (2009), 1381-1400. doi: 10.1016/j.nonrwa.2008.01.024.

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

D. G. Aronson, N. V. Mantzaris and H. G. Othmer, Wave propagarion and blocking in inhomogeneous media, Discrete and Continuous Dynamical Systems, 13 (2005), 843-876. doi: 10.3934/dcds.2005.13.843.

[3]

D. G. Aronson and V. Padron, Pattern formation in a model of an injured nerve fiber, SIAM J. Appl. Math., 70 (2009), 789-802. doi: 10.1137/080732341.

[4]

S. M. Baer and J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: A continuum theory, J. Neurophys., 65 (1991), 874-890.

[5]

J. Belmonte-Beitia and P. J. Torres, Existence of dark soliton of the cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, J. Nonlinear Math. Phys., 15 (2008), 65-72. doi: 10.2991/jnmp.2008.15.s3.7.

[6]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022.

[7]

P.-L. Chen and J. Bell, Spine-density dependence of the qualitative behavior of a model of a nerve fiber with excitable spines, J. Math. Anal. Appl., 187 (1994), 384-410. doi: 10.1006/jmaa.1994.1364.

[8]

E. N. Dancer and S. Yan, Interior peak solutions for an elliptic system of FitzHugh-Nagumo type, J. Differential Equations, 229 (2006), 654-679. doi: 10.1016/j.jde.2006.02.001.

[9]

P. Grindrod and B. D. Sleeman, A model of a myelinated nerve axon: Threshold behaviour and propagation, J. Math. Biol., 23 (1985), 119-135. doi: 10.1007/BF00276561.

[10]

J. K. Hale, "Ordinary Differential Equations,'' Second edition, R. E. Krieger Publ. Co., Inc., Huntington, New York, 1980.

[11]

S. Hastings, Some mathematical problems from neurobiology, Amer. Math. Monthly, 82 (1975), 881-895. doi: 10.2307/2318490.

[12]

S. Hastings, Some mathematical problems arising inneurobiology, in "Mathematics of Biology," C.I.M.E. Summer Sch., 80, Springer, Heidelberg, (2010), 179-264.

[13]

J. Kennedy and J. A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513-2530. doi: 10.1090/S0002-9947-01-02586-7.

[14]

Y. Kominis and K. Hizanidis, Lattice solitons in self-defocusing optical media:Analytical solutions of the nonlinear Kronig-Penney model, Optics Letters, 31 (2006), 2888-2890. doi: 10.1364/OL.31.002888.

[15]

M. A. Krasnosel'skiĭ, "The Operator of Translation Along the Trajectories of Differential Equations,'' Translations of Mathematical Monographs, 19, American Mathematical Society, Providence, R.I., 1968.

[16]

T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316. doi: 10.1137/S0036139998349298.

[17]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,'' Appl. Math. Sci., 74, Springer-Verlag, New York, 1989.

[18]

H. P. McKean, Jr., Nagumo's equation, Advances in Math., 4 (1970), 209-223.

[19]

A. Mellet, J.-M. Roquejoffre and Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 26 (2010), 303-312.

[20]

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91.

[21]

J. X. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys., 73 (1993), 893-926. doi: 10.1007/BF01052815.

[22]

J. Yang, S. Kalliadasis, J. H. Merkin and S. K. Scott, Wave propagation in spatially distributed excitable media, SIAM J. Appl. Math., 63 (2002), 485-509. doi: 10.1137/S0036139901391409.

[23]

C. Zanini and F. Zanolin, Positive periodic solutions for ordinary differential equations arising in the study of nerve fiber models, in "Applied and Industrial Mathematics in Italy," Ser. Adv. Math. Appl. Sci., 69, World Sci. Publ., Hackensack, NJ, (2005), 564-575.

[24]

C. Zanini and F. Zanolin, Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model, Nonlinear Anal. Real World Appl., 9 (2008), 141-153.

[25]

C. Zanini and F. Zanolin, Complex dynamics in a nerve fiber model with periodic coefficients, Nonlinear Anal. Real World Appl., 10 (2009), 1381-1400. doi: 10.1016/j.nonrwa.2008.01.024.

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