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Abstract
We consider the synchronization and cessation of oscillation of a positive even number of planar oscillators that are coupled to their nearest neighbours on one, two, and three dimensional integer lattices via a linear and symmetric diffusion-like path. Each oscillator has a unique periodic solution that is attracting. We show that for certain coupling strength there are both symmetric and antisymmetric synchronization that corresponds to symmetric and antisymmetric non-constant periodic solutions respectively. Symmetric synchronization persists for all coupling strengths while the antisymmetric case exists for only weak coupling strength and disappears to the origin after a certain coupling strength.
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References
- ARONSON, D.G., DOEDEL, E.J., and OTHMER, H.G. 1987. An analytical amd numerical study of the bifurcations in a system of linearly coupled oscillators. Physica D 25: 20-104.
- BAR-ELL, K. 1985. On the stability of coupled chemical oscillators. Physica D. 14: 242-252.
- CHN, J.W. 1960. Theory of growth and interface motion in crystalline material, Acta Metallurgica 8: 554-562.
- CHUA, L.O, and ROSKA, T. 1993. The CNN Paradigm. IEEE Trans. Circuits Syst., 40: 147-156.
- M.F. CROWLEY, M.F., EPSTEIN, I. 1989. Experimental and theoretical studies of a coupled chemical oscillator : Phase Death, Multistability, and in-Phase and Out of phase and Entrainment. J. Phys. Chem. 93: 2496-2502.
- ERMENTOUT, G.B. KOPPEL, N. 1994 Inhibition-produced patterning in chains of coupled nonlinear oscillators. SIAM J. Appl. Maths. 54: 478-507.
- H. FUJISAKA, H. and YAMADA, H. 1983. Stability Theory of Synchronized Motion in coupled-Oscillator Systems Prog. Theor. Phys. 69: 32-47.
- GRAHAM, A. 1981. Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Limited.
- HOUSEHOLDER. A.S. 1964. The Theory of Matrices in Numerical Analysis. Blaisdell.
- MALLET-PARET, J. and CHOW, S.-N. 1995. Pattern formation and spatial chaos in lattice dynamical systems II. IEEE Trans. Circuits Syst. 42: 752-756.
- PERKO, L. 1998. Differential Equation and dynamical system 2nd. Ed. Springer-Verlag, Newyork, Berlin Heidelberg. 19.
- POINCACARE, H. 1891 Memoire sur les courbes dfnites par une equation differentielle,. J. Mathematiques, 7: 375-421.
References
ARONSON, D.G., DOEDEL, E.J., and OTHMER, H.G. 1987. An analytical amd numerical study of the bifurcations in a system of linearly coupled oscillators. Physica D 25: 20-104.
BAR-ELL, K. 1985. On the stability of coupled chemical oscillators. Physica D. 14: 242-252.
CHN, J.W. 1960. Theory of growth and interface motion in crystalline material, Acta Metallurgica 8: 554-562.
CHUA, L.O, and ROSKA, T. 1993. The CNN Paradigm. IEEE Trans. Circuits Syst., 40: 147-156.
M.F. CROWLEY, M.F., EPSTEIN, I. 1989. Experimental and theoretical studies of a coupled chemical oscillator : Phase Death, Multistability, and in-Phase and Out of phase and Entrainment. J. Phys. Chem. 93: 2496-2502.
ERMENTOUT, G.B. KOPPEL, N. 1994 Inhibition-produced patterning in chains of coupled nonlinear oscillators. SIAM J. Appl. Maths. 54: 478-507.
H. FUJISAKA, H. and YAMADA, H. 1983. Stability Theory of Synchronized Motion in coupled-Oscillator Systems Prog. Theor. Phys. 69: 32-47.
GRAHAM, A. 1981. Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Limited.
HOUSEHOLDER. A.S. 1964. The Theory of Matrices in Numerical Analysis. Blaisdell.
MALLET-PARET, J. and CHOW, S.-N. 1995. Pattern formation and spatial chaos in lattice dynamical systems II. IEEE Trans. Circuits Syst. 42: 752-756.
PERKO, L. 1998. Differential Equation and dynamical system 2nd. Ed. Springer-Verlag, Newyork, Berlin Heidelberg. 19.
POINCACARE, H. 1891 Memoire sur les courbes dfnites par une equation differentielle,. J. Mathematiques, 7: 375-421.