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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[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.  doi: 10.3934/dcds.2005.13.843.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.2991/jnmp.2008.15.s3.7.  Google Scholar

[6]

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

[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.  doi: 10.1006/jmaa.1994.1364.  Google Scholar

[8]

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

[9]

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

[10]

J. K. Hale, "Ordinary Differential Equations,'' Second edition,, R. E. Krieger Publ. Co., (1980).   Google Scholar

[11]

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

[12]

S. Hastings, Some mathematical problems arising inneurobiology,, in, 80 (2010), 179.   Google Scholar

[13]

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

[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.  doi: 10.1364/OL.31.002888.  Google Scholar

[15]

M. A. Krasnosel'skiĭ, "The Operator of Translation Along the Trajectories of Differential Equations,'', Translations of Mathematical Monographs, 19 (1968).   Google Scholar

[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.  doi: 10.1137/S0036139998349298.  Google Scholar

[17]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,'', Appl. Math. Sci., 74 (1989).   Google Scholar

[18]

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

[19]

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

[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.   Google Scholar

[21]

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

[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.  doi: 10.1137/S0036139901391409.  Google Scholar

[23]

C. Zanini and F. Zanolin, Positive periodic solutions for ordinary differential equations arising in the study of nerve fiber models,, in, 69 (2005), 564.   Google Scholar

[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.   Google Scholar

[25]

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

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.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[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.  doi: 10.3934/dcds.2005.13.843.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.2991/jnmp.2008.15.s3.7.  Google Scholar

[6]

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

[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.  doi: 10.1006/jmaa.1994.1364.  Google Scholar

[8]

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

[9]

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

[10]

J. K. Hale, "Ordinary Differential Equations,'' Second edition,, R. E. Krieger Publ. Co., (1980).   Google Scholar

[11]

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

[12]

S. Hastings, Some mathematical problems arising inneurobiology,, in, 80 (2010), 179.   Google Scholar

[13]

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

[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.  doi: 10.1364/OL.31.002888.  Google Scholar

[15]

M. A. Krasnosel'skiĭ, "The Operator of Translation Along the Trajectories of Differential Equations,'', Translations of Mathematical Monographs, 19 (1968).   Google Scholar

[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.  doi: 10.1137/S0036139998349298.  Google Scholar

[17]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,'', Appl. Math. Sci., 74 (1989).   Google Scholar

[18]

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

[19]

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

[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.   Google Scholar

[21]

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

[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.  doi: 10.1137/S0036139901391409.  Google Scholar

[23]

C. Zanini and F. Zanolin, Positive periodic solutions for ordinary differential equations arising in the study of nerve fiber models,, in, 69 (2005), 564.   Google Scholar

[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.   Google Scholar

[25]

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

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