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Abstract
we consider a lattice system of identical oscillators that are all coupled to one another with a diffusive coupling that has a time lag. We use the natural splitting of the system into synchronized manifold and transversal manifold to estimate the value of the time lag for which the stability of the system follows from that without a time lag. Each oscillator has a unique periodic solution that is attracting.
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References
- AFRAIMOVICH, V.S. BYKOV, V. and SHILNKOV, L. 1983. On Structurally Unstable Attracting Limit sets of Lorenz attractor Type, Tran. Of Moscow Math. Soc. 44: 153-216.
- AFRAIMOVICH, V.S, Verchev, N. N. and RIABONVICH, M.I. 1986. Stochastic Synchronization of oscillations in dissipative systems. Radio Phy. Quantum Electron. 29.
- BOSE, F.G. 1989. Stability Conditions for the General Limear Difference – Differential Equation with Constant Coefficients and One Constant Delay. J. Math. Anal. Appl. 140: 136-176.
- CHOW, S.N., LIN, W. 1997 Synchronization, stability and normal hyperbolicity. Resenhas IME-USP. 3: 139-158.
- HALANAY A. 1996. ''Differential equation, stability of oscillators Time-lags'' Academic press.
- HALE, J. 1997. Diffusive coupling, dissipation and synchronization J. Dyn. Differ. Equ. 9: 1-52.
- HALE, J. 1996. Attracting, Manifolds for evolutionary equations CDSNS 96-257.
- ROSSONI, E., CHEN, Y., DING, M., FENG, J. 2005. Stability of synchronous oscillations in systems of HH neurons with delayed diffusive and pulsed Coupling, Phys. Rev. E 71, 061904.
- A. GRAHAM, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Limited, 1981.
- WASIKE, A.A.M., OGANA, W. 2002. Periodic solutions of a system of delay differential equations for a small delay Science and Technology, 7: 295-302.
- WASIKE, A.A.M. 2002. Periodic solutions of systems of delay differential equations Indian Journal of Mathematics, 44 No. 1: 95-117.
- WASIKE, A.A.M. 2003. Synchronization and oscillator death in diffusively coupled lattice oscillators International Journal of Mathematics Science, 2(1): 67-82.
- WASIKE, A.A.M., ROTICH, P.T. 2007. Synchronization, Persistence in diffusively coupled lattice oscillators SQU Journal for Science, 12(1): 41-52.
References
AFRAIMOVICH, V.S. BYKOV, V. and SHILNKOV, L. 1983. On Structurally Unstable Attracting Limit sets of Lorenz attractor Type, Tran. Of Moscow Math. Soc. 44: 153-216.
AFRAIMOVICH, V.S, Verchev, N. N. and RIABONVICH, M.I. 1986. Stochastic Synchronization of oscillations in dissipative systems. Radio Phy. Quantum Electron. 29.
BOSE, F.G. 1989. Stability Conditions for the General Limear Difference – Differential Equation with Constant Coefficients and One Constant Delay. J. Math. Anal. Appl. 140: 136-176.
CHOW, S.N., LIN, W. 1997 Synchronization, stability and normal hyperbolicity. Resenhas IME-USP. 3: 139-158.
HALANAY A. 1996. ''Differential equation, stability of oscillators Time-lags'' Academic press.
HALE, J. 1997. Diffusive coupling, dissipation and synchronization J. Dyn. Differ. Equ. 9: 1-52.
HALE, J. 1996. Attracting, Manifolds for evolutionary equations CDSNS 96-257.
ROSSONI, E., CHEN, Y., DING, M., FENG, J. 2005. Stability of synchronous oscillations in systems of HH neurons with delayed diffusive and pulsed Coupling, Phys. Rev. E 71, 061904.
A. GRAHAM, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Limited, 1981.
WASIKE, A.A.M., OGANA, W. 2002. Periodic solutions of a system of delay differential equations for a small delay Science and Technology, 7: 295-302.
WASIKE, A.A.M. 2002. Periodic solutions of systems of delay differential equations Indian Journal of Mathematics, 44 No. 1: 95-117.
WASIKE, A.A.M. 2003. Synchronization and oscillator death in diffusively coupled lattice oscillators International Journal of Mathematics Science, 2(1): 67-82.
WASIKE, A.A.M., ROTICH, P.T. 2007. Synchronization, Persistence in diffusively coupled lattice oscillators SQU Journal for Science, 12(1): 41-52.