

Geometric Structure of Mutually Coupled PhaseLocked Loops
HisaAki Tanaka, Shin'ichi Oishi, Kazuo Horiuchi
IEEE Transactions on Circuit and Systems  I: Fundamental Theory and Applications, vol. 43, No.6, pp.438443, June 1996.
Abstract
Dynamical properties such as lockin or outoflock condition of mutually coupled phaselocked loops (PLL’s) are problems of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLL’s incorporating lag filters and triangular phase detectors. The fourthorder ordinary differential equation (ODE) governing the mutually coupled PLL’s is reduced to the equivalent thirdorder ODE due to the symmetry, where the system is anafyzed in the context of nonlinear dynamical system theory. An understanding as to how and when lockin can be obtained or outoflock behavior persists, is provided by the geometric structure of the invariant manifolds generated in the vector field from the thirdorder ODE. In addition, a connection to the recently developed theory on chaos and bifurcations from degenerated homoclinic points is also found to exist. The twoparameter diagrams of the onehomoclinic orbit are obtained by graphical solution of a set of nonlinear (finite dimensional) equations. Their graphical results useful in determining whether the system undergoes lockin or continues outoflock behavior, are verified by numerical simulations.
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References

H. Inose, H. Fujisaki, and T. Saito, “Theory of mutually synchronized systems,” Electron. Commun. Japan., vol. 49, no. 4, pp. 755763, Apr. 1966.

K. Dessouky and W. C. Lindsey, “Phase and frequency transfer between mutually synchronized oscillators,” IEEE Trans. Commun., vol. 32, pp. 110115, Feb. 1984.

T. Endo and L. O. Chua, “Chaos from phaselocked loops,” IEEE Trans. Circuits Syst., vol. 35, no. 8, pp.9871003, Aug. 1988.

—“Piecewiselinear analysis of highdamping chaotic phaselocked loops using Melnikov's method,” IEEE Trans. Circuit Syst., vol. 40, pp. 801807, Nov. 1993.

E. Bradley, “Using chaos to broaden the capture range of a phaselocked loop,” IEEE Trans. Circuits Syst., vol. 40, pp. 808818, Nov. 1993.

T. Endo and L. 0. Chua, “Chaos from twocoupled phaselocked loops,” IEEE Trans. Circuits Syst., vol. 37, pp. 11831187, Sept. 1990.

M. Hirsch, C. Pugh, and IM. Shub, Invariant Manifolds. Berlin: SpringerVerlag, 1977, LNM583.

K. Iori, E. Yanagida, and T. Matsumoto, “Nhomoclinic bifurcations of piecewiselinear vector fields,” in Structure and Bifurcations ojDynumical Systems, S. Ushiki, Ed. Singapore: World Scientific, (Advanced Series in Dynamical Systems), 1993, vol. 11, pp. 8297.

M. Kisaka. H. Kokubu. and H. Oka. “Bifurcations to Nhomoclinic orbits and .Iperiodic orbits in vector fields,” J. Dynam. Diff Eq., vol.5, pp. 305357, 1993

T. Matsumoto, M. Komuro, H. Kokubu, and R. Tokunaga, Bifurcations. Tokyo: SpringerVerlag, 1993.

H. Tanaka, “Chaos from orbitflip homoclinic orbits generated in a practical circuit,” Phys. Rev. Lett., vol. 70, pp. 13391342, Feb. 1995.

H. Kokubu and H. Oka, (in preparation).

A. J. Homburg, H. Kokubu, and M. Krupa, “The cusp horseshoe and its bifurcations in the unfolding of an inclinationflip homoclinic orbit,” J. Ergod. Th., vol. 14, no. 4, pp. 667693, 1994.


