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March  2013, 18(2): 437-452. doi: 10.3934/dcdsb.2013.18.437

Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay

Mikhail Kamenskii 1, and  Boris Mikhaylenko 1,

1. 

Voronezh State University, 1 Universitetskaya pl., 394006, Voronezh, Russian Federation, Russian Federation

Received  October 2011 Revised  April 2012 Published  November 2012

This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation.
Keywords: Equation of neutral type, adjoint Floquet solution, Malkin function., small delay, periodic solutions bifurcation, small perturbations.
Mathematics Subject Classification: Primary: 34K13, 34K18, 34K27, 34K40; Secondary: 46T9.
Citation: Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437
References:
[1]

R. R. Akhmerov, M. I. Kamenskii, V. S. Kozyakin and A. V. Sobolev, Periodic solutions to autonomous functional-differrential neutral-type equations with a small delay, Differential Equations, 10 (1974), 1923-1931 [in Russian].  Google Scholar

[2]

R. R.Akhmerov, M. I.Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, "Measures of Non-Compactness and Condensing Operators," Nauka, Novosibirsk, 1986 [in Russian].  Google Scholar

[3]

P. G. Ayzengendler, Exception theory application to a problem of bifurcation of solutions to non-linear equations, Scient. notes Mosc. reg. Krupskaya's ped. inst., 166 (1966), 253-273 [in Russian]. Google Scholar

[4]

P. G . Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to autonomous systems and differential equations in Banach spaces, USSR Academy of science reports, 176 (1967), 9-12 [in Russian]. Google Scholar

[5]

P. G. Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, I, IHE proceedings, 10 (1969), 3-10 [in Russian]. Google Scholar

[6]

P. G. Ayzengendler and M.M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, II, IHE proceedings, 11 (1969), 3-12 [in Russian]. Google Scholar

[7]

P. G. Ayzengendler and M. M. Vainberg, On periodic solutions to non-autonomous systems, USSR Academy of science reports, 165 (1965), 255-257 [in Russian]. Google Scholar

[8]

P. G. Ayzengendler and M. M. Vainberg, Theory of bifurcation of solutions to non-linear equations in multidimensional case, USSR Academy of science reports, 163 (1965), 543-546 [in Russian]. Google Scholar

[9]

A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Communication of Pure and Applied Analysis, 6 (2007), 103-111.  Google Scholar

[10]

C. N. Fang and Q. Y. Wang, Existence, uniqueness and stability of periodic solutions to a class of neutral functional differential equations, J. Fuzhou Univ. Nat. Sci. Ed., 37 (2009), 471-477 [in Chinese].  Google Scholar

[11]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Side," Nauka, Moscow, 1985 [in Russian].  Google Scholar

[12]

J. R. Graef and L. Kong, Periodic solutions for functional differential equations with sign-changing nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 597-616. doi: 10.1017/S0308210509000523.  Google Scholar

[13]

J. R. Graef and S. H. Saker, New oscillation criteria for generalized second-order nonlinear neutral functional differential equations, Dynam. Systems Appl., 19 (2010), 455-472.  Google Scholar

[14]

L. X. Guo, S. P. Lu, B. Du and F. Liang, Existence of periodic solutions to a second-order neutral functional differential equation with deviating arguments, J. Math. (Wuhan), 30 (2010), 839-847 [in Chinese].  Google Scholar

[15]

M. Kamenskii, O. Makarenkov and P. Nistri, Variables Scaling to Solve a Singular Bifurcation Problem with Application to Periodically perturbed Autonomous Systems, Journal of Dynamic and Differential Equations, 8 (2011), 135-153.  Google Scholar

[16]

M. Kamenskii, O. Makarenkov and P. Nistri, Periodic bifurcation for semilinear differential equations with Lipschitzian perturbations in Banach spaces, Advanced Nonlinear Studies, 8 (2008), 271-289.  Google Scholar

[17]

M. I. Kamenskii and B. A. Mikhaylenko, On a small perturbations of systems with multidimensional degeneracy, Aut. and Rem. Contr., 5 (2011), 148-160 [in Russian].  Google Scholar

[18]

M. A. Krasnoselskii, "Translation Operator Along the Trajectories of Differential Equations," Nauka, Moscow, 1966 [in Russian].  Google Scholar

[19]

W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. Math., 70 (1959), 490-529.  Google Scholar

[20]

L. P. Luo, Oscillation theorems for nonlinear neutral hyperbolic partial functional differential equations, J. Math. (Wuhan), 30 (2010), 1023-1028 [in Chinese].  Google Scholar

[21]

O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations, J. Math. Anal. Appl., 338 (2008), 1401-1417. doi: 10.1016/j.jmaa.2007.05.086.  Google Scholar

[22]

I. G. Malkin, "Some Problems of Non-Linear Oscillations Theory," State publishers of technics and theory literature, Moscow, 1956 [in Russian].  Google Scholar

[23]

M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits, Methods Appl. Anal., 7 (2000), 85-104.  Google Scholar

[24]

A. E. Rodkina and B. N. Sadovskiy, On differentiability of translation operator along the trajectories of neutral-type equation, Math. fac. proc., 12 (1974), 31-37 [in Russian].  Google Scholar

[25]

G. Sansone, "Equazioni Differenziali Nel Campo Reale," p.1., Seconda edizione, Bologna, 1948.  Google Scholar

[26]

S. N. Shimanov, Oscillations of quasi-linear autonomous systems with delay, IHE proceedings. Radiophisics, 3 (1960), 456-466 [in Russian]. Google Scholar

[27]

S. N. Shimanov, To the oscillation theory of quasi-linear systems with delay, AMM., V.XXII (1959), 836-844 [in Russian]. Google Scholar

[28]

S. L. Wan, J. Yang, C. H. Feng and J. M. Huang, Existence of periodic solutions to higher-order nonlinear neutral functional differential equations with infinite delay, Pure Appl. Math. (Xi'an), 25 (2009), 556-562, 594 [in Chinese].  Google Scholar

[29]

C. Wang, Y. Li and Y. Fei, Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales, Math. Comput. Modelling, 52 (2010), 1451-1462.  Google Scholar

[30]

C. Wang and J. Wei, Hopf bifurcation for neutral functional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 1269-1277.  Google Scholar

[31]

F. Wei and K. Wang, The periodic solution of functional differential equations with infinite delay, Nonlinear Anal. Real World Appl., 11 (2010), 2669-2674. doi: 10.1016/j.nonrwa.2009.09.014.  Google Scholar

[32]

Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional differential equation, Int. J. Comput. Math. Sci., 5 (2011), 8-12.  Google Scholar

show all references

References:
[1]

R. R. Akhmerov, M. I. Kamenskii, V. S. Kozyakin and A. V. Sobolev, Periodic solutions to autonomous functional-differrential neutral-type equations with a small delay, Differential Equations, 10 (1974), 1923-1931 [in Russian].  Google Scholar

[2]

R. R.Akhmerov, M. I.Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, "Measures of Non-Compactness and Condensing Operators," Nauka, Novosibirsk, 1986 [in Russian].  Google Scholar

[3]

P. G. Ayzengendler, Exception theory application to a problem of bifurcation of solutions to non-linear equations, Scient. notes Mosc. reg. Krupskaya's ped. inst., 166 (1966), 253-273 [in Russian]. Google Scholar

[4]

P. G . Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to autonomous systems and differential equations in Banach spaces, USSR Academy of science reports, 176 (1967), 9-12 [in Russian]. Google Scholar

[5]

P. G. Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, I, IHE proceedings, 10 (1969), 3-10 [in Russian]. Google Scholar

[6]

P. G. Ayzengendler and M.M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, II, IHE proceedings, 11 (1969), 3-12 [in Russian]. Google Scholar

[7]

P. G. Ayzengendler and M. M. Vainberg, On periodic solutions to non-autonomous systems, USSR Academy of science reports, 165 (1965), 255-257 [in Russian]. Google Scholar

[8]

P. G. Ayzengendler and M. M. Vainberg, Theory of bifurcation of solutions to non-linear equations in multidimensional case, USSR Academy of science reports, 163 (1965), 543-546 [in Russian]. Google Scholar

[9]

A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Communication of Pure and Applied Analysis, 6 (2007), 103-111.  Google Scholar

[10]

C. N. Fang and Q. Y. Wang, Existence, uniqueness and stability of periodic solutions to a class of neutral functional differential equations, J. Fuzhou Univ. Nat. Sci. Ed., 37 (2009), 471-477 [in Chinese].  Google Scholar

[11]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Side," Nauka, Moscow, 1985 [in Russian].  Google Scholar

[12]

J. R. Graef and L. Kong, Periodic solutions for functional differential equations with sign-changing nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 597-616. doi: 10.1017/S0308210509000523.  Google Scholar

[13]

J. R. Graef and S. H. Saker, New oscillation criteria for generalized second-order nonlinear neutral functional differential equations, Dynam. Systems Appl., 19 (2010), 455-472.  Google Scholar

[14]

L. X. Guo, S. P. Lu, B. Du and F. Liang, Existence of periodic solutions to a second-order neutral functional differential equation with deviating arguments, J. Math. (Wuhan), 30 (2010), 839-847 [in Chinese].  Google Scholar

[15]

M. Kamenskii, O. Makarenkov and P. Nistri, Variables Scaling to Solve a Singular Bifurcation Problem with Application to Periodically perturbed Autonomous Systems, Journal of Dynamic and Differential Equations, 8 (2011), 135-153.  Google Scholar

[16]

M. Kamenskii, O. Makarenkov and P. Nistri, Periodic bifurcation for semilinear differential equations with Lipschitzian perturbations in Banach spaces, Advanced Nonlinear Studies, 8 (2008), 271-289.  Google Scholar

[17]

M. I. Kamenskii and B. A. Mikhaylenko, On a small perturbations of systems with multidimensional degeneracy, Aut. and Rem. Contr., 5 (2011), 148-160 [in Russian].  Google Scholar

[18]

M. A. Krasnoselskii, "Translation Operator Along the Trajectories of Differential Equations," Nauka, Moscow, 1966 [in Russian].  Google Scholar

[19]

W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. Math., 70 (1959), 490-529.  Google Scholar

[20]

L. P. Luo, Oscillation theorems for nonlinear neutral hyperbolic partial functional differential equations, J. Math. (Wuhan), 30 (2010), 1023-1028 [in Chinese].  Google Scholar

[21]

O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations, J. Math. Anal. Appl., 338 (2008), 1401-1417. doi: 10.1016/j.jmaa.2007.05.086.  Google Scholar

[22]

I. G. Malkin, "Some Problems of Non-Linear Oscillations Theory," State publishers of technics and theory literature, Moscow, 1956 [in Russian].  Google Scholar

[23]

M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits, Methods Appl. Anal., 7 (2000), 85-104.  Google Scholar

[24]

A. E. Rodkina and B. N. Sadovskiy, On differentiability of translation operator along the trajectories of neutral-type equation, Math. fac. proc., 12 (1974), 31-37 [in Russian].  Google Scholar

[25]

G. Sansone, "Equazioni Differenziali Nel Campo Reale," p.1., Seconda edizione, Bologna, 1948.  Google Scholar

[26]

S. N. Shimanov, Oscillations of quasi-linear autonomous systems with delay, IHE proceedings. Radiophisics, 3 (1960), 456-466 [in Russian]. Google Scholar

[27]

S. N. Shimanov, To the oscillation theory of quasi-linear systems with delay, AMM., V.XXII (1959), 836-844 [in Russian]. Google Scholar

[28]

S. L. Wan, J. Yang, C. H. Feng and J. M. Huang, Existence of periodic solutions to higher-order nonlinear neutral functional differential equations with infinite delay, Pure Appl. Math. (Xi'an), 25 (2009), 556-562, 594 [in Chinese].  Google Scholar

[29]

C. Wang, Y. Li and Y. Fei, Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales, Math. Comput. Modelling, 52 (2010), 1451-1462.  Google Scholar

[30]

C. Wang and J. Wei, Hopf bifurcation for neutral functional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 1269-1277.  Google Scholar

[31]

F. Wei and K. Wang, The periodic solution of functional differential equations with infinite delay, Nonlinear Anal. Real World Appl., 11 (2010), 2669-2674. doi: 10.1016/j.nonrwa.2009.09.014.  Google Scholar

[32]

Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional differential equation, Int. J. Comput. Math. Sci., 5 (2011), 8-12.  Google Scholar

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